Making Money with Bonds

In the article on yield curve strategies in general, I mentioned the two broad ways to make money with bonds:

1. Coupons (more generally, to incorporate synthetic strategies using, for example, swaps: interest payments)
2. Price changes

The curriculum breaks down the expected return on a bond in this manner:

\begin{align}E\left(R\right) &≈ Coupon\ income\\
\\
&\pm Rolldown\ return\\
\\
&\pm E\left(∆Price\ from\ investor’s\ view\ of\ benchmark\ yields\right)\\
\\
&\pm E\left(∆Price\ from\ investor’s\ view\ of\ yield\ spreads\right)\\
\\
&\pm E\left(∆Price\ from\ investor’s\ view\ of\ currency\ value\ changes\right)
\end{align}

In this article, we’ll largely ignore this breakdown; we’ll look at the price change on the bonds, not the yield income.  That’s the way of the curriculum.

What Is a Dynamic Yield Curve?

In general, a dynamic yield curve is one that we expect to change during our intended holding period.  Although the manner in which a yield curve change has myriad variations, we’ll break them down into a few simple building blocks.  Most changes will be (roughly) a combination of one of more of these building blocks, so the appropriate strategy will be (roughly) that same combination of the strategies for each of the separate blocks.  (At the end of the article, I’ll discuss what we do when we expect a wild change in the yield curve: one that cannot easily be constructed out of the blocks.)  The building blocks we will consider are a(n):

• Upward parallel shift (increase in level)
• Downward parallel shift (decrease in level)
• Steepening (increase in slope)
• Flattening (decrease in slope)
• Increase in curvature
• Decrease in curvature

Our Starting Yield Curve

Suppose that today’s yield curve looks like this:

Maturity, Years	YTM		Maturity, Years	YTM
1	2.220%		16	4.999%
2	2.521%		17	5.093%
3	2.799%		18	5.181%
4	3.057%		19	5.262%
5	3.296%		20	5.337%
6	3.518%		21	5.407%
7	3.722%		22	5.472%
8	3.912%		23	5.531%
9	4.087%		24	5.587%
10	4.249%		25	5.637%
11	4.399%		26	5.683%
12	4.539%		27	5.726%
13	4.667%		28	5.766%
14	4.787%		29	5.804%
15	4.897%		30	5.841%

Graphically:

Available Bonds

For simplicity, we’ll assume that we have available par bonds at all maturities from 1 year to 30 years.  As we’ll be talking quite a bit about duration and convexity, let’s take a look at the modified duration and convexity for each of these bonds:

Maturity, Years	Duration, Years	Convexity, Years2		Maturity, Years	Duration, Years	Convexity, Years2
1	0.98	1.91		16	10.84	154.28
2	1.93	5.61		17	11.19	166.63
3	2.84	10.95		18	11.52	178.78
4	3.71	17.74		19	11.83	190.72
5	4.54	25.81		20	12.11	202.39
6	5.33	34.98		21	12.37	213.80
7	6.06	45.06		22	12.61	224.91
8	6.76	55.91		23	12.84	235.72
9	7.41	67.36		24	13.04	246.23
10	8.01	79.27		25	13.23	256.41
11	8.57	91.51		26	13.41	266.28
12	9.10	103.98		27	13.58	275.83
13	9.58	116.57		28	13.73	285.07
14	10.03	129.19		29	13.87	294.00
15	10.45	141.78		30	14.00	302.61

Bullet, Ladder, and Barbell Portfolios

Throughout this article we’ll be talking about three portfolios in particular, each with an initial value of $10 million: a bullet portfolio (all of bonds in the portfolio having maturities very close to each other), a ladder portfolio (bonds at several maturities spread along the yield curve), and a barbell portfolio (a concentration of short-maturity bonds plus a concentration of long-maturity bonds, with no bonds with maturities in between those extremes).  The benchmark (modified) duration is 10.5 years, and each starting portfolio will have that same duration.  The specific portfolios are:

	Weights
Maturity, Years	Bullet	Ladder	Barbell
1		5.90%	26.88%
5		9.60%
10		13.21%
14	25.84%
15	33.52%	15.75%
16	40.65%
20		17.47%
25		18.64%
30		19.44%	73.12%
Total	100.00%	100.00%	100.00%
Duration, Years	10.50	10.50	10.50
Convexity, Years2	143.61	177.36	221.78

(For the bullet and ladder portfolios, I wanted to keep the weights as close to equal as possible; the criterion I used was to minimize the standard deviation of the weights, while keeping the duration equal to that of the benchmark.  With different criteria, other sets of weights are possible.)

Changes in Level

A change in the level of the yield curve is a nothing more than a parallel shift, upward or downward:

Although I have never seen nor heard the adjectives “bull” and “bear” applied to changes in level (they are used with changes in slope), there is no reason that they couldn’t be: a bull shift would be a downward parallel shift (so called because it will result in bond prices increasing), and a bear shift would be an upward parallel shift (accompanied by bond prices decreasing).  Thinking in this manner will help you when we get to the slope changes, and I think that there’s an advantage to using similar language for similar situations; it strengthens the memory.

Adjust Duration

When we anticipate an upward parallel shift (increase in level) in the yield curve, the time-honored strategy is to decrease the duration of our portfolio.  There are a number of ways that the duration of our portfolio can be decreased, including:

• Replace some or all of our bonds with bonds having shorter maturities
• Purchase put options on bonds
• Sell call options on bonds
• Enter into the short position in a bond futures or forward contract
• Enter into a plain vanilla interest rate swap as the fixed-rate payer / floating-rate receiver

When we anticipate a downward parallel shift (decrease in level), the indicated strategy is to increase the portfolio’s duration.  Of course, there are an equal number of ways that the duration of our portfolio can be increased, including:

• Replace some or all of our bonds with bonds having longer maturities
• Purchase call options on bonds
• Sell put options on bonds
• Enter into the long position in a bond futures or forward contract
• Enter into a plain vanilla interest rate swap as the fixed-rate receiver / floating-rate payer

All of these approaches are straightforward, and are described in detail in the article on yield curve strategies in general (see the link at the beginning of this article), so I won’t go into them again here.

Increase Convexity

Suppose, however, that the duration on the portfolio is constrained; e.g., the modified duration of the portfolio must be within 0.25 years of the modified duration of the benchmark.  Lengthening or shortening the duration by 3 months won’t make much of a difference when there is a parallel shift, so we need to look beyond duration.

Generally, the convexity of the portfolio is not constrained (the people who create investment policy statements (IPSs) generally aren’t particularly sophisticated, truth be told), so when we expect a parallel shift in the yield curve, we can benefit from an increase in the portfolio’s convexity: higher convexity gives a greater price increase when yields fall, and a smaller price decrease when yields rise.

One way to increase the convexity of our portfolio is to increase the dispersion of the cash flows.  Note that amongst the bullet, ladder, and barbell portfolios, the bullet has the lowest convexity, the ladder has middling convexity, and the barbell has the highest convexity.  Let’s take a look at the values of each of the bonds in our existing portfolios under the existing yield curve, a 50 bps upward shift, and a 75 bps downward shift:

	Bond Prices
Maturity, Years	Existing	+ 50 bps	− 75 bps
1	$1,000.00	$995.13	$1,007.39
5	$1,000.00	$977.61	$1,034.80
10	$1,000.00	$960.92	$1,062.38
14	$1,000.00	$951.40	$1,079.03
15	$1,000.00	$949.47	$1,082.53
16	$1,000.00	$947.69	$1,085.80
20	$1,000.00	$941.89	$1,096.81
25	$1,000.00	$936.92	$1,106.89
30	$1,000.00	$933.63	$1,114.09

(You can verify these prices with your calculator, or you can take my word for it.  It’s probably not a bad idea to verify one or two just to make sure that you’re happy with the table.)

The values of the bullet, ladder, and barbell portfolios under the existing yield curve, a 50 bps upward shift, and a 75 bps downward shift are:

Portfolio	Existing	+ 50 bps	− 75 bps
Bullet	$10,000,000	$9,492,464	$10,829,549
Ladder	$10,000,000	$9,496,342	$10,840,237
Barbell	$10,000,000	$9,501,611	$10,854,041

The duration approximation for the 50 bps upward shift is:

\[∆Price ≈ -\$10,000,000 × 10.5 × 0.5\% = -\$525,000\]

leaving a portfolio value of:

\[\$10,000,000  – \$525,000 = \$9,475,000\]

Because of the convexity, all of the portfolios under the 50 bps upward shift have a higher value than that, with the bullet (least convex)  having the lowest value and the barbell (most convex) the highest, as expected.

The duration approximation for the 75 bps downward shift is:

\[∆Price ≈ -\$10,000,000 × 10.5 × -0.75\% = \$787,500\]

giving a portfolio value of:

\[\$10,000,000  + \$787,500 = \$10,787,500\]

Because of the convexity, all of the portfolios under the 75 bps downward shift have a higher value than that, with the bullet having the lowest value and the barbell the highest; again, as expected.

What does this mean?  If we have a bullet portfolio and anticipate a parallel shift, one way we can increase our convexity and improve the performance is to switch to a ladder or barbell portfolio with the same duration, and if we have a ladder portfolio we can switch to a barbell portfolio.

One downside to this approach is that buying and selling bonds can incur significant transaction costs, as well as the possibility of having to recognize taxable gains, so it may not be the most efficient method to increase the convexity of our portfolio.  Another downside is that convexity is not free: it’s akin to buying insurance and, as with all other forms of insurance, it costs money.  In other words, the cost of the ladder portfolio should be higher than the cost of the bullet portfolio, and the cost of the barbell portfolio should be higher than the cost of the ladder portfolio.  A third downside is that the amount of convexity we can achieve is limited: the barbell portfolio will have the maximum possible convexity using only straight bonds.  If we want to increase convexity further, we have to look to other methods.

Putable Bonds

One possibility is to replace some or all of the straight bonds in our portfolio with putable bonds, which generally have greater convexity than otherwise comparable straight bonds.

Replacing straight bonds with putable bonds has the same disadvantages as changing from a bullet structure to a ladder or barbell structure, or from a ladder to a barbell: buying and selling bonds is costly, taxable gains might be incurred, buying convexity can be expensive, and there’s a limit to the additional convexity in putable bonds.  Another possible disadvantage is that putable bonds are generally less liquid than straight bonds, so you might not be able to buy them even if you really, really want to.

Options on Bonds

We saw in the article on static yield curve strategies that we can sell convexity whilst maintaining the duration of our portfolio by selling an appropriate mix of call options on bonds and put options on bonds.  Call options have positive duration and positive convexity, put options have negative duration, and out-of-the-money put options have positive convexity.  Therefore, if we want to buy convexity as protection against parallel shifts in the yield curve, we can buy an appropriate mix of call options on bonds and put options on bonds.  Buying options generally has lower transaction costs than buying and selling bonds, buying options does not trigger the recognition of taxable gains, and the amount of convexity we can purchase is virtually unlimited, constrained only by our budget for insurance.  An advantage of using options is that if we have some flexibility on the duration of our portfolio (say, within 0.25 years of the benchmark duration), we can choose the mix of calls and puts to add the convexity we want while adjusting the duration to the limit of that flexibility: increasing it when we expect a downward shift and decreasing it when we expect an upward shift.

Changes in Slope

For a normal (i.e., upward sloping) yield curve, the usual terminology for changes in slope make sense: what we call steepening makes the yield curve more steeply sloped upward (i.e., farther from horizontal, closer to vertical, long end up, short end down), and what we call flattening makes the yield curve less steeply sloped upward (i.e., closer to horizontal, farther from vertical, long end down, short end up, flatter).

The usual language falls apart, however, when we have an inverted (i.e., downward sloping) yield curve: a steepening (long end up, short end down) actually brings the curve closer to horizontal, while a flattening (long end down, short end up) makes the curve less horizontal (more downward).

Thus, we have the language of slope: what we call steepening is an increase in the slope, and what we call flattening is a decrease in the slope.

Personally, I prefer to think of steepening as a rotation counterclockwise (or anticlockwise, or widdershins), and flattening as a rotation clockwise (or deasil).

I encourage you to use whatever language works best for you.

When we talk about a change in slope (separate from a change in level), we generally agree to leave one point on the yield curve fixed, and move the other points around it.  (This is another reason I like the visual of a rotation.)  Although we could choose any point (i.e., the yield at any given maturity) to remain fixed, tradition has come to leaving one of two points fixed: either leaving the short end of the curve fixed or else leaving the long end of the curve fixed.  As we will see, the names we give to the specific types of steepening or flattening tell us which point is fixed.  Hang on.

In general, when there is a change in the slope of the yield curve, modified duration and convexity are of little use in telling us how the portfolio will behave (because those measures assume a parallel shift – a change in level only – not a change in slope); we need more sophistication in our analysis of our portfolio.  More on that in a moment.  First, let’s make sure that we understand changes in slope in general.

Steepening

When long-term yields rise, or short-term yields fall, or both, we say that the yield curve steepens: the slope of the yield curve increases.  Graphically, it looks something like this:

If long-term yields increase while short-term yields remain unchanged, we refer to the change as a bear steepening.  If long-term yields remain unchanged while short-term yields decrease, we refer to the change as a bull steepening.  Both are illustrated here:

Note that the general steepening illustrated above can be thought of as a bull steepening combined with a positive (bear) parallel shift:

or as a bear steepening combined with a negative (bull) parallel shift:

Without a duration constraint on our portfolio, we would want to shorten its duration when a steepening is expected: sell long-term (hence, long-duration) bonds, buy short-term (hence, short-duration) bonds.  If we have a duration constraint (so we’re pretty much restricted to the bullet, ladder, and barbell portfolios, above), we want primarily to limit our exposure to the 30-year bonds, so we would prefer the ladder to the barbell, and the bullet to the ladder.

For the graphs, above, the yield changes run:

• From 0 bps at 0 years to +100 bps at 30 years for the bear steepening
• From −50 bps at 0 years to +50 bps at 30 years for the general steepening
• From −100 bps at 0 years to 0 bps at 30 years for the bull steepening

The bond prices are:

	Bond Prices
Maturity, Years	Existing	Bull Steepening	General Steepening	Bear Steepening
1	$1,000.00	$1,009.55	$1,004.59	$999.67
5	$1,000.00	$1,038.76	$1,015.28	$992.47
10	$1,000.00	$1,055.21	$1,013.46	$973.73
14	$1,000.00	$1,055.40	$1,003.35	$954.55
15	$1,000.00	$1,054.08	$1,000.00	$949.47
16	$1,000.00	$1,052.30	$996.40	$944.33
20	$1,000.00	$1,041.52	$980.09	$923.57
25	$1,000.00	$1,022.42	$957.28	$898.09
30	$1,000.00	$1,000.00	$933.63	$873.92

The values of the bullet, ladder, and barbell portfolios under the existing yield curve, and the three steepenings are:

Portfolio	Existing	Bull Steepening	General Steepening	Bear Steepening
Bullet	$10,000,000	$10,536,962	$9,994,012	$9,486,940
Ladder	$10,000,000	$10,315,241	$9,791,734	$9,309,770
Barbell	$10,000,000	$10,025,664	$9,527,025	$9,077,271

These values verify what we expected: the bullet performs the best in each of the three scenarios, the ladder is in the middle, and the barbell performs the worst in each scenario.

Synthetic Adjustments

Another approach is to adjust your exposure to various maturities synthetically.  Here, you want to increase your exposure at the short end of the yield curve and decrease your exposure at the long end.  You can increase your exposure at the short end by:

• Taking the long position in futures or forwards on short-term bonds
• Buying call options on short-term bonds
• Selling put options on short-term bonds

You can decrease your exposure at the long end by:

• Taking the short position in futures or forwards on long-term bonds
• Buying put options on long-term bonds
• Selling call options on long-term bonds

If you have duration constraints, you’ll have to adjust the exposures so that the net result doesn’t violate those constraints.  And note that while futures, forwards, and options on long-term bonds are fairly common, futures, forwards, and options on short-term bonds may be much less common, so these strategies may be difficult or impossible to implement in practice.

Flattening

When short-term yields rise, or long-term yields fall, or both, we say that the yield curve flattens: the slope of the yield curve decreases.  Graphically, it looks something like this:

If short-term yields increase while long-term yields remain unchanged, we refer to the change as a bear flattening.  If short-term yields remain unchanged while long-term yields decrease, we refer to the change as a bull flattening.  Both are illustrated here:

Note that the general flattening illustrated above can be thought of as a bull flattening combined with a positive (bear) parallel shift:

or as a bear flattening combined with a negative (bull) parallel shift:

Without a duration constraint on our portfolio, we would want to lengthen its duration when a flattening is expected: sell short-term (hence, short-duration) bonds, buy long-term (hence, long-duration) bonds.  If we have a duration constraint (so we’re pretty much restricted to the bullet, ladder, and barbell portfolios, above), we want primarily to maintain as much exposure to the 30-year bonds as possible, so we would prefer the barbell to the ladder, and the ladder to the bullet.

For the graphs, above, the yield changes run:

• From +100 bps at 0 years to 0 bps at 30 years for the bear flattening
• From +50 bps at 0 years to −50 bps at 30 years for the general flattening
• From 0 bps at 0 years to −100 bps at 30 years for the bull flattening

The bond prices are:

	Bond Prices
Maturity, Years	Existing	Bull Flattening	General Flattening	Bear Flattening
1	$1,000.00	$1,000.33	$995.46	$990.63
5	$1,000.00	$1,007.60	$985.01	$963.04
10	$1,000.00	$1,027.15	$986.76	$948.31
14	$1,000.00	$1,048.27	$996.66	$948.27
15	$1,000.00	$1,054.08	$1,000.00	$949.47
16	$1,000.00	$1,060.06	$1,003.62	$951.06
20	$1,000.00	$1,085.44	$1,020.47	$960.73
25	$1,000.00	$1,119.77	$1,045.57	$978.29
30	$1,000.00	$1,156.52	$1,073.95	$1,000.00

The values of the bullet, ladder, and barbell portfolios under the existing yield curve, and the three flattenings are:

Portfolio	Existing	Bull Flattening	General Flattening	Bear Flattening
Bullet	$10,000,000	$10,550,067	$10,006,094	$9,498,087
Ladder	$10,000,000	$10,805,252	$10,229,878	$9,702,084
Barbell	$10,000,000	$11,145,325	$10,528,484	$9,974,817

These values verify what we expected: the bullet performs the worst in each of the three scenarios, the ladder is in the middle, and the barbell performs the best in each scenario.

Synthetic Adjustments

As with steepening, when you anticipate the yield curve flattening you can make adjustments using derivatives.  This time, you want to decrease your exposure at the short end of the yield curve and increase your exposure at the long end.  You can decrease your exposure at the short end by:

• Taking the short position in futures or forwards on short-term bonds
• Buying put options on short-term bonds
• Selling call options on short-term bonds

You can increase your exposure at the long end by:

• Taking the long position in futures or forwards on long-term bonds
• Buying call options on long-term bonds
• Selling put options on long-term bonds

Once again, mind any duration constraints, and make sure that the net result doesn’t violate those constraints.  And remember that derivatives on short-term bonds may be difficult to find.

Changes in Curvature

Changes in curvature of the yield curve look something like this:

Note that the change in curvature generally assumes that the ends of the yield curve are fixed, so that all of the action, if you will, takes place in the middle of the yield curve.  Using the bull/bear language, an increase in curvature would be a bear change, while a decrease in curvature wold be a bull change.  When a change in curvature is combined with a smaller magnitude parallel shift in the opposite direction (i.e., an increase in curvature plus a downward parallel shift, or a decrease in curvature plus an upward parallel shift), the combination is called a butterfly.  A positive butterfly looks like this:

while a negative butterfly looks like this:

(Note that the adjective – positive or negative – describes the parallel shift, or the direction in which the ends of the yield curve (the wings) move relative to the original yield curve.)

(Note, too, that , broadly, butterflies are neither bull nor bear necessarily.)

Increase in Curvature

When we expect that the curvature of the yield curve will increase, we want to reduce the exposure to the middle of the curve: mid-term bonds (the body of the butterfly).  Generally, this is accompanied by an increase in the exposure to the ends of the the yield curve (the wings), to keep the portfolio invested fully.  Note that there is no particular reason to change the (overall) duration of the portfolio; we can maintain the same duration while adjusting the exposures at different maturities.  Considering our three portfolios, the barbell should perform best (as it has no exposure to the middle of the yield curve), and the bullet should perform worst (as it has 100% exposure to the middle of the yield curve), while the ladder should perform somewhere in between.

For the graphs, above, the yield changes run:

• From 0 bps to +53.2 bps for the increased curvature
• From −26.6 bps to +26.6 bps for the negative butterfly

The bond prices are:

	Bond Prices
Maturity, Years	Existing	Increased Curvature	Negative Butterfly
1	$1,000.00	$999.05	$1,001.66
5	$1,000.00	$983.28	$995.20
10	$1,000.00	$959.47	$979.98
14	$1,000.00	$949.05	$974.44
15	$1,000.00	$948.00	$974.41
16	$1,000.00	$947.69	$975.06
20	$1,000.00	$952.44	$983.24
25	$1,000.00	$970.60	$1,005.18
30	$1,000.00	$1,000.00	$1,038.34

The values of the bullet, ladder, and barbell portfolios under the existing yield curve, the increased curvature, and the negative butterfly shift are:

Portfolio	Existing	Increased Curvature	Negative Butterfly
Bullet	$10,000,000	$9,481,453	$9,746,786
Ladder	$10,000,000	$9,710,104	$9,984,529
Barbell	$10,000,000	$9,997,452	$10,284,799

As expected: the bullet performs the worst in each of the three scenarios, the ladder is in the middle, and the barbell performs the best in each scenario.

Note, too, that because the bullet and barbell portfolios each have a modified duration of 10.50 years, you can create a custom portfolio that is short the bullet and long the barbell, which would enhance the performance even more.  For example, a portfolio that is short $2 million of the bullet and long $12 million of the barbell will still have a modified duration of 10.50 years, an initial value of $10 million, and a final value after the increase in curvature of:

\[-0.2 × \$9,481,453 + 1.2 × \$9,997,452 = \$10,100,651\]

and a final value after the negative butterfly of:

\[-0.2 × \$9,746,786 + 1.2 × \$10,284,799 = \$10,392,402\]

Decrease in Curvature

When we expect that the curvature of the yield curve will decrease, we want to increase the exposure to the middle of the curve: mid-term bonds (the body of the butterfly).  Generally, this is accompanied by a decrease in the exposure to the ends of the the yield curve (the wings), to keep the portfolio invested fully.  Once again, there is no particular reason to change the (overall) duration of the portfolio; we can maintain the same duration while adjusting the exposures at different maturities.  Considering our three portfolios, the barbell should perform worst (as it has no exposure to the middle of the yield curve), and the bullet should perform best (as it has 100% exposure to the middle of the yield curve), while the ladder should perform somewhere in between.

For the graphs, above, the yield changes run:

• From 0 bps to −53.2 bps for the decreased curvature
• From +26.6 bps to −26.6 bps for the positive butterfly

The bond prices are:

	Bond Prices
Maturity, Years	Existing	Decreased Curvature	Positive Butterfly
1	$1,000.00	$1,000.95	$998.35
5	$1,000.00	$1,017.07	$1,004.83
10	$1,000.00	$1,042.67	$1,020.52
14	$1,000.00	$1,054.51	$1,026.43
15	$1,000.00	$1,055.65	$1,026.36
16	$1,000.00	$1,056.06	$1,025.68
20	$1,000.00	$1,050.89	$1,017.16
25	$1,000.00	$1,030.72	$994.86
30	$1,000.00	$1,000.00	$963.80

The values of the bullet, ladder, and barbell portfolios under the existing yield curve, the increased curvature, and the negative butterfly shift are:

Portfolio	Existing	Decreased Curvature	Positive Butterfly
Bullet	$10,000,000	$10,555,203	$10,261,017
Ladder	$10,000,000	$10,307,107	$10,022,318
Barbell	$10,000,000	$10,002,553	$9,730,882

As expected: the bullet performs the best in each of the three scenarios, the ladder is in the middle, and the barbell performs the worst in each scenario.

We can create a custom portfolio here as well: a portfolio that is long $12 million of the bullet and short $2 million of the barbell will still have a modified duration of 10.50 years, an initial value of $10 million, and a final value after the decrease in curvature of:

\[1.2 × \$10,555,203 – 0.2 × \$10,002,553 = \$10,665,733\]

and a final value after the negative butterfly of:

\[1.2 × \$10,261,017 – 0.2 × \$9,730,882 = \$10,367,044\]

Summary

How do our standard portfolios – bullet, ladder, barbell – perform under various dynamic yield curve scenarios?

	Bullet	Ladder	Barbell
Upward Shift	Worst	Middle	Best
Downward Shift	Worst	Middle	Best
Steepening	Best	Middle	Worst
Flattening	Worst	Middle	Best
Increased Curvature	Worst	Middle	Best
Decreased Curvature	Best	Middle	Worst

What does this tell us?

If we know that there’s going to be a parallel shift in the yield curve, or a flattening of the yield curve, or an increase in the curvature, choose a barbell portfolio.  If we know that there’s going to be a steepening of the yield curve, or a decrease in the curvature, choose a bullet portfolio.  If we have no idea what’s going to happen to the yield curve, choose a ladder portfolio: never the best, but never the worst.

Making Money with Bonds

In the article on yield curve strategies in general, I mentioned the two broad ways to make money with bonds:

1. Coupons (more generally, to incorporate synthetic strategies using, for example, swaps: interest payments)
2. Price changes

The curriculum breaks down the expected return on a bond in this manner:

\begin{align}E\left(R\right) &≈ Coupon\ income\\
\\
&\pm Rolldown\ return\\
\\
&\pm E\left(∆Price\ from\ investor’s\ view\ of\ benchmark\ yields\right)\\
\\
&\pm E\left(∆Price\ from\ investor’s\ view\ of\ yield\ spreads\right)\\
\\
&\pm E\left(∆Price\ from\ investor’s\ view\ of\ currency\ value\ changes\right)
\end{align}

We’ll use this breakdown later in this article.

Earning a Return with a Static Yield Curve

First, a couple of calculations, then some theory.

Suppose that today’s yield curve looks like this:

Maturity, Years	Par Rate	Spot Rate	Forward Rate
1	2.220%	2.220%	2.220%
2	2.521%	2.525%	2.831%
3	2.799%	2.810%	3.382%
4	3.057%	3.078%	3.887%
5	3.296%	3.331%	4.348%
6	3.518%	3.570%	4.774%
7	3.722%	3.794%	5.150%
8	3.912%	4.008%	5.516%
9	4.087%	4.209%	5.836%
10	4.249%	4.401%	6.139%

(OK, technically it’s three yield curves.)

You buy a 10-year bond that pays a 4% coupon annually.  The yield on that bond is 4.249%, as the table above attests.  You hold that bond for one year, then sell it.  If the yield curve remains static for that year, then the yield on the bond when you sell it will be 4.087%: it will be a 9-year bond at that point.  The price at which you buy the bond is:

\[FV = 1,000\]

\[PMT = 40\]

\[i = 4.249\%\]

\[n = 10\]

\[Solve\ for\ PV = -980.05\]

The price at which you sell the bond is:

\[FV = 1,000\]

\[PMT = 40\]

\[i = 4.087\%\]

\[n = 9\]

\[Solve\ for\ PV = -993.56\]

Your holding period return is:

\[HPR = \frac{993.56 – 980.05 + 40}{980.05} = 5.459\%\]

Broken down according to our expected return formula, above:

\[Coupon\ return = \frac{40}{980.05} = 4.081\%\]

\[Rolldown\ return = \frac{993.56 – 980.05}{980.05} = 1.378\%\]

\[E\left(\%∆Price\ from\ investor’s\ view\ of\ benchmark\ yields\right) = 0\%\]

\[E\left(\%∆Price\ from\ investor’s\ view\ of\ yield\ spreads\right) = 0\%\]

\[E\left(\%∆Price\ from\ investor’s\ view\ of\ currency\ value\ changes\right) = 0\%\]

\[HPR = 4.081\% + 1.378\% + 0\% + 0\% + 0\% = 5.459\%\]

Suppose instead that, noting that your holding period is one year, you buy a 1-year bond paying an annual coupon of 3%.  The price when you buy the bond is 1,007.63 (you should verify this on your calculator), and 1,000 at maturity.  Your holding period return is:

\[HPR = \frac{1,000.00 – 1,007.63 + 30}{1,007.63} = 2.220\%\]

which is exactly the 1-year YTM.

This illustrates one strategy that you can employ when you think that the yield curve will be static for your expected holding period; it’s called riding the yield curve.

Static Yield Curve Strategies

Amongst the strategies that you can employ when you think that the yield curve will remain static are:

• Buy-and-hold
• Rolling down (or riding) the yield curve
• Carry trade
• Long position in bond futures
• Receive fixed, pay floating swap
• Selling convexity

Buy-and-Hold

While this is listed as a static yield curve strategy, as far as I can tell it’s more a properly a whatever-happens-to-the-yield-curve (or a who-cares-what-happens-to-the-yield-curve?) strategy.  You buy some bonds and hold them to maturity.  What happens to the yield curve doesn’t matter (unless you’re Silicon Valley Bank, but that’s a topic for an entirely different article), because the bond will be worth par at the end no matter what.  (OK: technically, what happens to the yield curve does matter a little if the bond pays coupons and you plan to reinvest those coupons.  But that’s not likely to have a huge impact on your overall return.  Furthermore, if that’s our concern, we’d prefer a buy-and-hold strategy when rates are rising, so that our reinvestment income increases.)

The only reason I can imagine that this is included under the rubric of static yield curve strategies is for comparison with other strategies in that category.  I believe that it’s unlikely that you’d see an exam question in which the correct answer to the question of which strategy to use with a static yield curve will be buy-and-hold.  Bear it in mind, of course, but don’t dwell on it.

Example

Suppose that you have a 2-year holding period.  You buy a 2-year par bond (with a coupon rate of 2.521%, paid annually) and hold it for 2 years.  If the yield curve remains static, then in one year when you get a coupon payment of 25.21, you’ll reinvest it for one year at 2.220% (the prevailing 1-year rate).  Your holding period return is:

\[HPR = \frac{1,000.00\ – 1,000.00 + 25.21\left(1.0222\right) + 25.21}{1,000.00} = 5.0980\%\]

The effective annual return is:

\[EAR = \left(1 + HPR\right)^{1/n}\ – 1 = 1.050980^{1/2}\ – 1 = 2.5173\%\]

Note that your effective annual return is less than your original YTM of 2.521%.  Why?  Because the reinvestment rate on the coupon you received after 1 year was less than your original YTM.

Rolling Down (Riding) the Yield Curve

When you expect the yield curve to remain static, then the duration of your portfolio matters very little (as the yield change will be only what exists in the current yield curve, which will likely be small, so the price change from the pull-to-par (shortening the maturity) will likely outweigh the price change from duration).  To take advantage of that fact, if the yield curve is normal (i.e., upward sloping), you can purchase bonds whose maturity is longer than your expected holding period, with the intention of selling those bonds at the end of the holding period.  The upward sloping yield curve ensures that you will be selling the bonds at a lower yield than the yield at which you buy them, and the longer maturity on the bonds will likely be accompanied by a higher coupon rate, both of which should enhance your return.

This strategy requires that you have some scope to increase the duration of your portfolio, possibly significantly, so it may not be practical in a portfolio that has tight restrictions on duration (e.g., that the portfolio duration may not differ from the benchmark duration by more than, say, 0.25 years).

Example

Suppose that you have a 2-year holding period.  You buy a 10-year par bond (with a coupon rate of 4.249%) and hold it for 2 years.  If the yield curve remains static, then in one year when you get a coupon payment of 42.49, which you’ll reinvest for one year at 2.220%.  At the end of two years, assuming a static (stable) yield curve, you’ll sell the (now 8-year) bond at a yield of 3.912%: the price will be calculated as:

\[FV = 1,000\]

\[PMT = 42.49\]

\[i = 3.912\%\]

\[n = 8\]

\[Solve\ for\ PV = -1,022.77\]

Your holding period return is:

\[HPR = \frac{1,022.77 – 1,000.00 + 42.49\left(1.0222\right) + 42.49}{1,000.00} = 10.8695\%\]

The effective annual return is:

\[EAR = \left(1 + HPR\right)^{1/n}\ – 1 = 1.108695^{1/2}\ – 1 = 5.2946\%\]

Notice the substantial increase in total return of this strategy compared to buying a 2-year bond: 5.2946% vs. 2.5173%.  The increase comes from a higher coupon as well as the price increase going from a yield of 4.249% to a yield of 3.912%.  This change in yield is the origin of the the terms rolling down the yield curve and riding the yield curve.  This approach would be notably less beneficial when the yield curve is flat or inverted.  Nobody wants to roll up the yield curve.

Carry Trade

Carry trade involves much the same aspects as riding the yield curve: you purchase a bond with a longer maturity than your expected holding period, sell it at the end of your holding period, and earn the ensuing coupon payments plus the price change (which you hope is appreciation).  What distinguishes carry trade from simply riding the yield curve is the way the investment is financed.  When you ride the yield curve, it is understood that you purchase the bonds involved by using funds (i.e., cash) that are in the portfolio.  For a carry trade, the purchase is financed essentially by borrowing the funds for the holding period.  Often this borrowing is done using a repurchase agreement or repo: a contract in which you sell a security and and agree to repurchase it at a given (higher) price in the future, most commonly, the next day (an overnight repo).  Although the legal structure of a repo is that of a sale and repurchase, the economic substance of the agreement is that it is a collateralized loan: the difference between the repurchase price and the original sale price is the interest on the loan.  Carry trade is profitable if the coupon and price change on the long-term bond are greater than the cost of the financing.  Note that if the expected holding period is longer than one day, the overnight repo could simply be rolled over every day for the entire holding period.

Although an overnight repo is a common form of financing the purchase of a long-term bond for a carry trade, it is by no means the only way that a carry trade can be financed.  I’ll show you an example in which the financing will involve simply selling (issuing) a bond whose maturity is your expected holding period.  It will be a lot easier to analyze than rolling over an overnight repo every day for two years.

Example

A 2-year carry trade could be achieved by selling a 2-year bond and using the proceeds to buy a 10-year bond.  The holding period cost, as we saw in the buy-and-hold example, above, would be 5.0980%, while the holding period (gross) return, as we saw in the rolling-down-the-yield-curve example, above, would be 10.8695%, giving a net holding period return of:

\[HPR = 10.8695\% – 5.0980\% = 5.7716\%\]

The effective annual net return is:

\[EAR = \left(1 + HPR\right)^{1/n}\ – 1 = 1.057716^{1/2}\ – 1 = 2.8453\%\]

Long Position in Bond Futures

Taking a long position in futures (or forward) contracts is much the same as purchasing the underlying asset.  Indeed, for assets that don’t generate cash flows (and don’t have storage costs or convenience yields), a long futures position is equivalent to purchasing the asset.  For bond futures, it’s a bit different from purchasing actual bonds because the futures contracts don’t pay coupons, whereas the actual bonds do.  Therefore, whereas a rolling-down-the-yield-curve strategy (above) will earn both the coupon payments and any price appreciation on the bonds, a long position in a bond futures contract will gain only the price appreciation.  Against that, purchasing the bond requires an initial investment, whereas taking the long position in a futures contract does not.  (It does, however, require that you post a margin, which is often in the form of short-term risk-free (e.g., government) securities.  Therefore, the return you earn on the margin account will be a short-term risk-free rate, so there is an opportunity cost to consider.)

Taking the long position in bond futures will increase the duration of your fixed income portfolio.  However, if your view is that the yield curve will remain static, that increased duration will not be an issue as far as your return is concerned.  It may be an issue if your portfolio has a duration constraint (as was mentioned in the rolling-down-the-yield-curve strategy, above).

Example

Suppose that you take the long position in a 2-year government bond futures contract.  The (theoretical) underlying is a 10-year, option-free, 100,000-par bond that pays an annual coupon of 6%.  The future price is 108,703.10.  (I got this by discounting each of the payments on the bond at their appropriate spot rates, totting them up, subtracting the present values of the first two coupons (which we won’t receive as we don’t own the bond), then increasing that present value by the 2-year spot rate for two years.  You needn’t worry about the details; on the exam they’ll give you the price.)

Two years from today, the futures contract matures and the mark-to-market price of the underlying (now an 8-year bond) is 114,397.90.  The holding period return is:

\[HPR = \frac{114,397.90}{108,703.10}\ – 1 = 5.2389\%\]

The effective annual return is:

\[EAR = \left(1 + HPR\right)^{1/n}\ – 1 = 1.052389^{1/2}\ – 1 = 2.5860\%\]

Receive Fixed, Pay Floating Swap

You’ll recall from Level II that swaps are priced (i.e., the fixed rate is calculated) to prevent arbitrage: the underlying assumption, it turns out, is that interest rates will evolve according to the (1-period) forward curve.  In this situation, that assumption means that while today the 1-period rate is 2.220%, the expected 1-year rate starting one year from today is 2.831%, the expected 1-year rate starting two years from today is 3.382%, and so on.  (Look at the forward curve in the table at the top of this article.)  When the yield curve slopes upward, the fixed rate will be higher than the (initial) floating rate.  (Indeed, it should be the par rate for the tenor (maturity) of the swap).  Furthermore, if the yield curve remains static, then rates won’t evolve according to the forward curve, and the fixed-rate receiver will make out like a bandit.

Example

Suppose that you enter into a 2-year, annual pay, plain vanilla interest rate swap as the fixed rate receiver / floating rate payer.  Based on the yield curve at the top of this article, the fixed rate should be 2.521%, and the first year’s floating rate will be 2.220%.  At the end of the first year, your net payment will be a receipt of 0.301% (= 2.521% − 2.220%) times the notional value of the swap.  Should the yield curve remain static, then the net payment at the end of the second year will likewise be a receipt of 0.301% times the notional.

Selling Convexity

Convexity is useful when yields change: the more convex your portfolio, the greater the price increase when yields fall, and the smaller the price decrease when yields rise.  And, like most things that are useful, convexity costs money; a more convex portfolio is generally more expensive than a less convex, but otherwise comparable (e.g., identical duration), portfolio.

Convexity is not particularly useful, however, when yields don’t change.  When ∆y = 0, it doesn’t matter what the portfolio’s duration and convexity are: the price isn’t going to change (apart from the pull to par as time passes).  Therefore, a sound strategy when you expect the yield curve to be static is to sell convexity.  There are various ways to do this.  You can:

• Shorten the maturity of the bonds in your portfolio.  Shorter maturity bonds generally have lower convexity than longer maturity bonds.  However, with an upward sloping yield curve, shorter maturity bonds will also have lower yields than longer maturity bonds, so you may lose more than you gain by adopting this approach.
• Replace straight bonds with callable bonds, or prepayable bonds (such as MBSs) which have negative convexity at lower yields, and less convexity than straight bonds even at moderate yields.
• Sell call options and/or put options on bonds.  Call options and (out-of-the-money) put options have positive convexity, so selling them will reduce the convexity of the portfolio.  Call options increase the duration of the portfolio while put options decrease the duration.  By selling them in an appropriate mix, you can reduce the portfolio’s convexity while keeping its duration unchanged, or lengthening the duration, or shortening the duration.

Example

(Note: you won’t be doing any of these calculations on the exam; you’ll simply have to know that to sell convexity you sell some calls and sell some puts.  I’m including them simply to give you an understanding of how selling convexity works; i.e., how you figure out how many calls and puts to sell.  If you don’t care about the calculations, then 1) shame on you, and 2) skip this section.)

You own a portfolio with a market price of AUD 23,498,000, a modified duration of 8.32 years, and a (modified) convexity of 80.83 years².  You believe that the yield curve will remain unchanged for at least the next 6 months, so you decide to sell off half of your (money) convexity.  Your investment mandate requires you to match the benchmark duration of 8.20 years within ±0.25 years, so you decide to leave the duration of your portfolio unchanged.  You have available these options on 20-year government bonds:

Type	Strike	Price	Duration, Years	Convexity, Years2
Call	AUD 1,275.00	AUD 81.50	108.8	8,327.8
Put	AUD 1,275.00	AUD 70.38	−102.4	5,943.4

How many options should you buy/sell to achieve your goal?  (Assume options can be bought/sold only in lots of 100.)  How much will it cost?

Both calls and puts have positive convexity, so we’re going to be selling them.  First, we need to determine the ratio of puts to calls that we will sell.  We want to keep the duration of the portfolio unchanged, so the net money duration of the options must equal zero:

\[\#\ calls × call\ value × call\ duration + \#\ puts × put\ value × put\ duration = 0\]

\[\#\ calls × AUD\ 81.50 × 108.8\ years + \#\ puts × AUD\ 70.38 × -102.4\ years = 0\]

\[\#\ calls × 8,867.20 = \#\ puts × 7,206.91\]

\[\#\ puts = \frac{\#\ calls × 8,867.20}{7,206.91} = \#\ calls\left(\frac{8,867.20}{7,206.91}\right) = 1.2304\left(\#\ calls\right)\]

Therefore, for each call we sell, we need to sell 1.2304 puts to leave the duration unchanged.  We want to sell off half of the money convexity. so the calculation is:

\[\#\ calls × call\ value × call\ convexity + \#\ puts × put\ value × put\ convexity\\
= \frac12 × portfolio\ value × portfolio\ convexity\]

\[\#\ calls × AUD\ 81.50 × 8,327.8\ years^2 + \#\ puts × AUD\ 70.38 × 5,943.4\ years^2\\
= \frac{AUD\ 23,498,000 × 80.83\ years^2}{2}\]

\[\#\ calls\left(678,715.7\right) + \left(1.2304 × \#\ calls × 418,296.5\right) = 949,671,670\]

\[\#\ calls\left[678,715.7 + \left(1.2304 × 418,296.5\right)\right] = 949,671,670\]

\[\#\ calls\left(1,193,377\right) = 949,671,670\]

\[\#\ calls = \frac{949,671,670}{1,193,377} = 795.8\]

\[\#\ puts = 1.2304 × 795.8 = 979.1\]

Rounding to the nearest 100 options, we should sell 800 calls and 1,000 puts.  Because of the rounding, we’ll change the money duration slightly and be off a little on the money convexity:

\begin{align}∆money\ duration &= -800 × AUD\ 81.50 × 108.8\ years\\
\\
&\ \ \ \ \ -\  1,000 × AUD\ 70.38 × \left(-102.4\ years\right)\\
\\
&= -113,152\ AUD-years
\end{align}

\begin{align}∆money\ convexity &= -800 × AUD\ 81.50 × 8,327.8\ years^2\\
\\
&\ \ \ \ \ -\ 1,000 × AUD\ 70.38 × 5,943.4\ years^2\\
\\
&= -961,269,052\ AUD-years^2
\end{align}

This decreases the money duration by only 0.06% (0.0048 years, which is negligible), and reduces the money convexity by 50.61% (which is quite close to the 50% reduction we sought).

In 2019, CFA Institute revised all of their Level III Fixed Income readings, and, in particular, came up with a 10-page blue box example that was absurdly complex, which I simplified here.

That example’s gone.  Fortunately.  (In point of fact, I’m sorry that the idea is gone from the curriculum, because it was interesting and useful.  The problem was that the example was far too complicated for candidates, many of whom do not employ any fixed income in their day-to-day work, much less foreign fixed income.)

For 2022, CFA Institute replaced that Yield Curve Strategies reading with a new Yield Curve Strategies reading.  I’m going to address the new reading here, and in the companion articles linked here.  It’ll be interesting, but not as interesting as it could have been.

What are Yield Curve Strategies?

First, when we talk about yield curve strategies, we’re generally talking about investments in risk-free government bonds, either directly or synthetically.  This is distinguished from credit strategies in which we’re talking about investments in corporate bonds, for example, for which potential changes in credit spreads can play a significant rôle in our investment decision; the bonds we consider in yield curve strategies have essentially no credit spread component.  Thus, we may be talking about:

• Buying or selling government bonds (domestic or foreign)
• Taking the long or the short position in futures contracts on government bonds (domestic or foreign)
• Taking the long or the short position in forward contracts on government bonds (domestic or foreign) with reputable counterparties (i.e., those for whom the default risk is negligible)
• Buying or selling options on government bonds (domestic or foreign)
• Entering into plain vanilla interest rate swaps with reputable counterparties
• Entering into currency swaps with reputable counterparties

The objective in a yield curve strategy is, as you might expect for any investment strategy, to try to enhance the returns on our portfolio.  This may be accomplished by:

• Adjusting the overall duration of the portfolio
• Adjusting the key rate durations of the portfolio
• Adjusting the convexity of the portfolio
• Adjusting the cash flow of the portfolio

Which of these approaches we will use depends on our view of what will happen to the yield curve (or yield curves, if we’re considering investments denominated in more than one currency) over our investment horizon (and, of course, on the scope we’re allowed based on the investor’s investment policy statement (IPS)).  I’ll cover the possibilities, and help you to understand how to select an appropriate strategy given a particular view on the yield curve(s).

Making Money with Bonds

There are broadly two ways to make money with bonds:

1. Coupons (more generally, to incorporate synthetic strategies using, for example, swaps: interest payments)
2. Price changes

The curriculum breaks down the expected return on a bond in this manner:

\begin{align}E\left(R\right) &≈ Coupon\ income\\
\\
&\pm Rolldown\ return\\
\\
&\pm E\left(∆Price\ from\ investor’s\ view\ of\ benchmark\ yields\right)\\
\\
&\pm E\left(∆Price\ from\ investor’s\ view\ of\ yield\ spreads\right)\\
\\
&\pm E\left(∆Price\ from\ investor’s\ view\ of\ currency\ value\ changes\right)
\end{align}

For yield curve strategies we won’t be concerned with the fourth term (changes in yield spreads), and we’ll be concerned with the last term only when we discuss bonds denominated in foreign currencies, at the end of this article.  Our main focus, therefore, will be on coupon income, rolldown return (i.e., the price change in a bond that results from the passage of time, assuming that the yield curve does not change), and price changes that result from changes in the yield curve.

Your View of the Yield Curve

Broadly (i.e., really, really broadly), there are two possibilities for the yield curve in the future:

1. It can stay the same as it is today: a static yield curve
2. It can change: a dynamic yield curve

Your choice of strategy begins with your view of how the yield curve will look at the end of your holding period: the same as today, or different.  If you believe that the yield curve will remain static, you’ll choose from one set of strategies, but if you believe that the yield curve will change, you’ll choose from a different set of strategies.  I’ll cover both possibilities (and their corresponding strategy sets) in these articles:

• Yield Curve Strategies – Static Yield Curve
• Yield Curve Strategies – Dynamic Yield Curve

The Yield Curve

For the articles on yield curve strategies, I’ll start with this yield curve:

Maturity, Years	Par Rate	Spot Rate	Forward Rate
1	2.220%	2.220%	2.220%
2	2.521%	2.525%	2.831%
3	2.799%	2.810%	3.382%
4	3.057%	3.078%	3.887%
5	3.296%	3.331%	4.348%
6	3.518%	3.570%	4.774%
7	3.722%	3.794%	5.150%
8	3.912%	4.008%	5.516%
9	4.087%	4.209%	5.836%
10	4.249%	4.401%	6.139%

(OK, technically it’s three yield curves.)

(For some of the graphs I’ll extend the par curve to 30 years, but for calculations we’ll mainly stick to 10 years maximum.)

Here’s how they look for 10 years:

and here’s how they look for 30 years:

The Strategies in a Nutshell

Buying or Selling Government Bonds

This is about as straightforward as it gets: you have a bond portfolio, so you buy some bonds.  Then, perhaps, you sell some of those bonds and buy other bonds.

By selling one bond and buying another, you can:

• Adjust the overall duration of the portfolio (sell a bond with one duration, buy a bond with a different duration)
• Adjust the key rate durations of the portfolio (for example, by switching from a bullet portfolio to a barbell portfolio with the same duration)
• Adjust the convexity of the portfolio (for example, a laddered portfolio will have more convexity than a bullet portfolio with the same duration)
• Adjust the cash flow of the portfolio (sell a bond with one coupon rate, buy a bond with a different coupon rate)

You can also adjust your currency exposure by selling a bond denominated in one currency and buying a bond denominated in another currency.

Taking the Long or the Short Position in Futures Contracts on Government Bonds

Although this is extremely similar to buying or selling (respectively) government bonds, there are some significant differences:

• Futures contracts require no upfront payment (although you will likely have to post a margin); therefore, they are leveraged positions (which can increase or decrease your total return significantly)
• Futures contracts do not pay coupons, so their duration is generally greater than the duration of the underlying bond, and their convexity is generally less than the convexity of the underlying bond

Apart from these, taking a long position in a forward contract is much the same as buying the underlying bond, and taking the short position is much the same as selling the underlying bond.

Taking the Long or the Short Position in Forward Contracts on Government Bonds

This is nearly identical to taking the long or short position in futures contracts.  The main difference is that these are custom contracts, so you can adjust such characteristics as the underlying bonds, the time to maturity, and the collateral (if any).  Apart from that, they work much as the futures contracts (above) work.

Buying or Selling Options on Government Bonds

The effects of options on government bonds are:

• Long call options increase duration and increase convexity
• Short call options decrease duration and decrease convexity
• Long put options decrease duration
  • Long out-of-the-money puts increase convexity
  • Long in-the-money puts may decrease convexity negligibly
• Short put options increase duration
  • Short out-of-the-money puts decrease convexity
  • Short in-the-money puts may increase convexity negligibly

Entering into Plain Vanilla Interest Rate Swaps

The fixed leg on a plain vanilla interest rate swap has an effective duration that is roughly 75% of the maturity of the swap, while the floating leg has an effective duration that is roughly half the time between swap payments.  Therefore, if you enter into a pay-fixed, receive-floating swap you will reduce the duration (and the convexity) of your portfolio, and if you enter into a pay-floating, receive-fixed swap you will increase the duration (and the convexity) of your portfolio.

Note that this will also have an effect on your cash flow, but apart from the payment at the first settlement date, you cannot say for certain whether it will increase or decrease your cash flow.

Entering into Currency Swaps

A fixed-for-fixed currency swap isn’t likely to change the duration or convexity of the portfolio much, nor will a floating-for-floating currency swap; however, a fixed-for-floating currency swap can change the duration much as a plain vanilla interest rate swap does.  It can also change your cash flows, similar to a plain vanilla interest rate swap.

Of course, in this case the two legs of the swap will pay different currencies, so they’ll be subject to different yield curves.  Additionally, a currency swap will be subject to changes in currency exchange rates, which adds another dimension to the effect they will have on a fixed income portfolio.

One common use of currency swaps is to combine them with bonds denominated in foreign currencies, effectively changing the foreign currency cash flows into domestic currency cash flows  (i.e., they can be a tool to hedge currency exchange rate risk).  Depending on the scope allowed by the IPS and the manager’s view on exchange rates, the manager may choose to overhedge or underhedge the foreign currency payments; i.e., use the swap as tool to engage in active currency management.

As a gift to the candidates studying for the Level III exam in 2019, CFA Institute replaced the existing Level III Fixed Income readings with new Fixed Income readings.  Amongst the new readings was one entitled Yield Curve Strategies, and included in that reading was a section on Inter-Market Curve Strategies, including, in particular, inter-market curve positioning.

Their example on inter-market curve positioning is 10 pages long.

In this article I’ll cover the important points on inter-market curve positioning without all of the gruesome detail of that example.  Nevertheless, I believe that you will get enough to be able to handle any question on the topic that the exam might throw at you.

Note that I’ll be rounding all of the values to zero decimal places, which means that occasionally the addition / subtraction might appear to be off just slightly.  Pay it no attention.

To begin, we need a couple of yield curves, one for each of two currencies.  I’ll call them Australian Dollar (AUD) and Swiss Franc (CHF) for no particularly good reason; I stress that these are artifacts of my warped imagination, and that if they represent actual yield curves for these (or any other) currencies at any time in history or in the future, it’s nothing more than an amazing coincidence.  AUD will be our home currency.  An excerpt from these curves is:

Maturity, Years	YTM
	AUD	CHF
2	2.50%	1.50%
5	3.66%	2.79%
10	4.83%	4.37%
15	5.43%	5.44%
20	5.75%	6.17%
30	6.00%	7.00%

The full yield curves look like this:

Note that these are par curves, not spot curves.  (If you need a refresher on the difference between a par curve and a spot curve, look here.)  The modified durations of par bonds at these maturities – which will come into play presently – are:

Maturity, Years	Durmod, Years
	AUD	CHF
2	1.9390	1.9631
5	4.5312	4.6372
10	7.8590	8.0325
15	10.1694	10.1641
20	11.7954	11.4007
30	13.8378	12.4724

The strategy is a three-act play:

1. Intra-market carry trade in AUD: borrowing AUD at a low rate and lending AUD at a high rate.  The complication is that we want to be currency neutral (i.e., lending exactly the same amount we borrow) and duration neutral (i.e., have a net modified duration (properly, currency duration) of zero).
2. Intra-market carry trade in CHF: borrowing CHF at a low rate and lending CHF at a high rate.  As with the AUD carry trade, we want to be currency neutral and duration neutral.
3. Inter-market carry trade: either borrowing AUD at a low rate, converting it to CHF, and lending the CHF at a high rate, or borrowing CHF at a low rate, converting it to AUD, and lending AUD at a high rate.  And, once again, we want to be currency neutral and duration neutral (in both currencies, on both yield curves).

I’ll take these three separately, in the order named.  And to keep matters simple, I’ll assume that all of the bonds mentioned are par bonds; i.e., their coupon rates equal their respective yields to maturity (YTMs).  I’ll also assume that the holding period is one year, and that the bonds pay coupons annually (so that there’s no reinvestment risk to complicate things).  Furthermore, I’ll assume that the yield on each bond will remain unchanged (so that they remain par bonds); i.e., the yield will result exclusively from the coupon payments.  Thus, I’m assuming that the yield curves one year from today will be (note the shorter maturities from the original table):

Maturity, Years	YTM
	AUD	CHF
1	2.50%	1.50%
4	3.66%	2.79%
9	4.83%	4.37%
14	5.43%	5.44%
19	5.75%	6.17%
29	6.00%	7.00%

While we’re discussing assumptions, let’s assume that the spot AUD/CHF exchange rate today is AUD 1.5169 = CHF 1.0, that the 1-year forward rate is AUD 1.5319 = CHF 1.0, and that one year from today the spot exchange rate is also AUD 1.5319 = CHF 1.0.  (Later, I’ll mention what happens when the bonds aren’t priced at par, and when the yield curves change, and when the expected future exchange rate is not the forward rate, but we’ll keep it simple for now.)

So that we don’t go completely bonkers (because we’re new at this, so we don’t want to risk losing a ton of money if we foul it up completely), I’ll impose limits of ±AUD 1,000,000 and ±CHF 1,000,000 on any single security.

A Word About Hedging

There is no (well, maybe some, actually, but not much) hedging.  None.  (Well, almost none.)  Specifically:

• No bond price hedging
• No interest rate hedging
• Maybe some exchange rate hedging on bonds with maturities longer than the expected holding period.

Zero!  (Well, practically zero.)  No hedging at all!  (Well, not much.)  If you hedge, you fail.  (OK: that’s a bit dramatic, but generally you don’t want to hedge.  Why?  Because the basis of carry trade is that interest rate parity won’t hold.  If it does, you simply earn the risk-free rate in your home currency for your holding period.  Hedging generally ensures that interest rate parity holds.)

You might hedge exchange rate risk on bonds with maturities longer than your expected holding period because that hedge will be based on interest rate parity for the holding period (and will lock in a forward exchange rate based on the short-term rate differential.)  If the longer-term rate differential is different, it may not exactly wipe out any expected yield gain.

In general, if you want to hedge, don’t waste your time with this stuff; buy Treasuries.  Be boring.  Wear beige.

Act 1: Intra-Market Carry Trade in AUD

The idea behind intra-market (i.e., one market) carry trade is pretty simple: to borrow at a low rate and lend at a high rate in the same currency.  Conventional wisdom is that you make more money when the yield curve is steeper, and less when its flatter, so we’ll start at the short end of the AUD yield curve (where it’s steeper) by:

• Borrowing the limit, AUD 1,000,000, at the 2-year rate of 2.50%
• Lending AUD 1,000,000 at the 10-year rate of 4.83%

The yield pickup is 2.33% (\(=\ AUD\ 23,300\)) per year.  To borrow the money we issue (or sell) 2-year bonds and to lend it we purchase 10-year bonds:

Because we have borrowed and lent the same amount, we are currency (i.e., AUD) neutral, but we are not duration neutral: our net AUD-duration is:

\[AUD\ 1,000,000\ ×\ 7.8590\ years\ -\ AUD\ 1,000,000\ ×\ 1.9390\ years\]

\[=\ 5,920,000\ AUD-years\]

To be duration neutral will, unfortunately, cost some money (i.e., yield): we construct another AUD trade that negates the AUD-duration of the first trade.  Think of it as buying insurance: we’re insuring ourselves against parallel shifts in the yield curve.  (Well, small ones at least.)  Because we need negative AUD-duration, we have to buy shorter-term bonds and sell longer-term bonds, which means that we’ll have a net negative yield on this portion of the strategy.  To implement it, we’ll

• Issue (sell) 30-year bonds
• Buy 15-year bonds

in equal AUD amounts:

We have to calculate the appropriate amount to make sure that we are net duration neutral, and we do so by solving this equation, where \(A\) is the unknown amount of AUD:

\begin{align}A\ ×\ 10.1694\ years\ -\ A\ ×\ 13.8378\ years\ &=\ -5,920,000\ AUD-years\\
\\
A\ ×\ \left(10.1694\ years\ -\ 13.8378\ years\right)\ &=\ -5,920,000\ AUD-years\\
\\
A\ ×\ \left(-3.6684\ years\right)\ &=\ -5,920,000\ AUD-years\\
\\
A\ &=\ \frac{-5,920,000\ AUD-years}{-3.6684\ years}\\
\\
A\ &=\ AUD\ 1,613,783
\end{align}

(This amount exceeds our position limit of AUD, which means that we need to adjust everything downward.  I’ll get to that in a moment; I don’t want to lose momentum here.)

Now, we have to see how much this trade is going to cost us, to see if we’re really making any money.  We’re borrowing at 6.00% and lending at 5.43%, for a net loss of 0.57% on AUD 1,613,783, or \(AUD\ 9,199\) per year.  So, we’re currency neutral, duration neutral, and netting:

\[AUD\ 23,300\ -\ AUD\ 9,199\ =\ AUD\ 14,101\]

per year.

Now, let’s deal with that pesky matter of exceeding our position limits.  This turns out to be easy: really easy.  Our positions in the 15-year and 30-year bonds are too big by a factor of 1.613783.  So we simply divide every position by that factor.  That also divides our profit by the same factor.  Thus:

• The short position in 2-year AUD bonds will be:

\(\dfrac{AUD\ 1,000,000}{1.613783}\ =\ AUD\ 619,662\)

• The long position in 10-year AUD bonds will be:

\(\dfrac{AUD\ 1,000,000}{1.613783}\ =\ AUD\ 619,662\)

• The long position in 15-year AUD bonds will be:

\(\dfrac{AUD\ 1,613,783}{1.613783}\ =\ AUD\ 1,000,000\)

• The short position in 30-year AUD bonds will be:

\(\dfrac{AUD\ 1,613,783}{1.613783}\ =\ AUD\ 1,000,000\)

• The profit will be:

\(\dfrac{AUD\ 14,101}{1.613783}\ =\ AUD\ 8,738\)

So far, so good.  We’re making money, and if we have a parallel yield curve shift, we’re covered.

Act 2: Intra-Market Carry Trade in CHF

We’ll do much the same thing with CHF as we did with AUD, making this the most boring part of the process.  For reasons that I will explain later, we’ll start off by:

• Borrowing CHF 1,000,000 at the 10-year rate of 4.37% (issuing (selling) 10-year bonds)
• Lending CHF 1,000,000 at the 30-year rate of 7.00% (buying 30-year bonds):

The yield pickup is 2.63% (\(=\ CHF\ 26,300\)) per year.  We are currency (i.e., CHF) neutral, but we are not duration neutral: our net CHF-duration is:

\[CHF\ 1,000,000\ ×\ 12.4724\ years\ -\ CHF\ 1,000,000\ ×\ 8.0325\ years\]

\[=\ 4,439,900\ CHF-years\]

To negate the CHF-duration of the first trade, we’ll:

• Issue (sell) 15-year bonds
• Buy 2-year bonds, in equal CHF amounts:

To get the amount of CHF for this trade, we have to solve this equation, where \(C\) is the unknown amount of CHF:

\begin{align}C\ ×\ 1.9631\ years\ -\ C\ ×\ 10.1641\ years\ &=\ -4,439,900\ CHF-years\\
\\
C\ ×\ \left(1.9631\ years\ -\ 10.1641\ years\right)\ &=\ -4,439,900\ CHF-years\\
\\
C\ ×\ \left(-8.2010\ years\right)\ &=\ -4,439,900\ CHF-years\\
\\
C\ &=\ \frac{-4,439,900\ CHF-years}{-8.2010\ years}\\
\\
C\ &=\ CHF\ 541,385
\end{align}

As this is within our position limits, we do not need to make any changes in the amounts here.  Now, we have to see how much this trade is going to cost us, to see if we’re really making any money. We’re borrowing at 5.44% and lending at 1.50%, for a net loss of 3.94% on CHF 541,385, or \(CHF\ 21,330\) per year. So, we’re currency neutral, duration neutral, and netting:

\[CHF\ 26,300\ -\ CHF\ 21,330\ =\ CHF\ 4,969\]

per year.

We’re on a roll!

Note that this is the one area in inter-market carry trade where we might hedge the AUD/CHF exchange rate.  The idea here is to make a profit exclusively in CHF and transfer that profit over to AUD; we’re not trying to profit on the transfer itself.  Later, when we do inter-market trades, the exchange rate movement will generally be a part of the profit we’re seeking, so we generally won’t hedge it there; here, we can hedge it or not, and it’s more likely that we will hedge it.

By the way, the reason that I chose to start with a 10-year/30-year (10/30) trade here rather than with a 2/10 trade as I did with AUD is that I set this scenario up in Excel and used an optimizer (Solver) to determine the set of trades that would be the most profitable.  In practice, if you’re doing inter-market (or just intra-market) carry trades, you’ll use some computer-based optimization system to determine the best set of trades.

Intermission: Total Intra-Market Profit

If our assumption holds that the yields for each of the bonds remain unchanged, as well as our assumption about the future exchange rate, then one year from today we’ll have made:

\begin{align}AUD\ 8,738\ +\ CHF\ 4,969\left(\frac{AUD\ 1.5319}{CHF\ 1.0}\right)\ &=\ AUD\ 8,738\ +\ AUD\ 7,613\\
\\
&=\ AUD\ 16,351
\end{align}

We’re making money, but we want more!

Act 3: Inter-Market Carry Trade

Now comes the interesting part: the inter-market trades.  Borrowing AUD, converting them to CHF, and investing the CHF, or else borrowing CHF, converting them to AUD, and investing the AUD.  This is the type of carry trade that you covered at Level II: borrow a low-rate currency, convert it to a high-rate currency, invest at the high rate, then later convert back to the low-rate currency and hope that the exchange rate change has been less than what is suggested by interest rate parity.

Here, we complicate those simple trades by imposing currency neutrality and duration neutrality on the inter-market carry trades.  Thus, if we borrow CHF 100,000 (at some maturity), convert it to AUD 1,516,900 (at the current spot exchange rate), and invest the AUD, we’ll also have to borrow AUD 1,516,900 (at some other maturity), convert it to CHF 100,000, and invest the CHF.  And then do another pair of borrowings/lendings to maintain duration neutrality.

You should have seen my note, above, about why I started the CHF intra-market carry trade with a 10/30 trade: I used an optimizer to determine the trades that would maximize the profit.  For the inter-market trades, I did the same.

Without constraints, it should be obvious that you would want to:

• Borrow 2-year CHF, convert it to AUD, and invest 2-year AUD, because the AUD 2-year rate is higher than the CHF 2-year rate
• Borrow 5-year CHF and invest 5-year AUD
• Borrow 10-year CHF and invest 10-year AUD
• Pretty much leave the 15-year maturity alone (as there’s no strong yield advantage), unless you need to mess with it for duration neutrality
• Borrow 20-year AUD and invest 20-year CHF
• Borrow 30-year AUD and invest 30-year CHF

Of course, these may not all hold true when the currency-neutral and duration-neutral constraints come into play (recall that in the intra-market trades we deliberately lost money on two trades to ensure duration neutrality), but they’re a starting point, and a reasonable reality check.

Using an optimizer, and bearing in mind that we have to keep our net positions within the imposed limits, I came up with these inter-market trades that are currency and duration neutral, and maximize the profit:

• Borrow AUD 380,338 at the AUD 2-year rate, convert them to CHF 250,734, and invest that at the CHF 2-year rate
• Borrow CHF 362,431 at the CHF 5-year rate, convert them to AUD 549,635, and invest that at the AUD 5-year rate
• Borrow AUD 288,712 at the AUD 20-year rate, convert them to CHF 190,330, and invest that at the CHF 20-year rate
• Borrow CHF 78,723 at the CHF 30-year rate, convert them to AUD 119,415, and invest that at the AUD 30-year rate

All of this results in a profit (remember: assuming no yield changes and a given future AUD/CHF exchange rate) of \(AUD\ 995\).

Final Result

The total profit would be:

\[AUD\ 17,346\ \left(=\ AUD\ 16,351\ +\ AUD\ 995\right)\]

Here’s how the positions add up:

Maturity, Years	AUD
	Intra-Market	Inter-Market	Total
2	(AUD 619,662)	(AUD 380,338)	(AUD 1,000,000)
5	AUD 0	AUD 549,635	AUD 549,635
10	AUD 619,662	AUD 0	AUD 619,662
15	AUD 1,000,000	AUD 0	AUD 1,000,000
20	AUD 0	(AUD 288,712)	(AUD 288,712)
30	(AUD 1,000,000)	AUD 119,415	(AUD 880,585)

and:

Maturity, Years	CHF
	Intra-Market	Inter-Market	Total
2	CHF 541,385	CHF 250,734	CHF 792,119
5	CHF 0	(CHF 362,341)	(CHF 362,341)
10	(CHF 1,000,000)	CHF 0	(CHF 1,000,000)
15	(CHF 541,385)	CHF 0	(CHF 541,385)
20	CHF 0	CHF 190,330	CHF 190,330
30	CHF 1,000,000	(CHF 78,723)	CHF 921,277

Graphically, here’s what’s happening, in toto:

Notice, in particular, that for the 30-year bonds the inter-market trade negates some of the corresponding intra-market trades; i.e., the intra-market AUD trade is short AUD 1,000,000 in the 30-year bond, while the inter-market trade is long AUD 119,415 in that same bond, and the intra-market CHF trade is long CHF 1,000,000 in the 30-year bond, while the inter-market trade is short CHF 78,723 in that same bond.  The curriculum uses the term “reverse” or “switch” to denote a situation in which the inter-market trade on a given bond is in the opposite direction of the intra-market trade on that same bond.

Complications

In the real world, things are a lot more complicated than the (already at least moderately complex) situation I described above.  Real-world complications include the possibility that the:

• Bonds used in the trade might not all be par bonds (indeed, it’s most likely that none of them will be par bonds)
• Yield curves might (indeed, probably will) change: move up, move down, flatten, steepen, whatever
• Expected future exchange rate is not the forward rate
• Strategy will incorporate more than two yield curves
• Bonds used in the trade are not fixed-rate bonds
• Coupons are received before the holding period ends

Premium / Discount Bonds

If the bonds used in carry trade are not par bonds (i.e., they’re premium bonds or discount bonds), then there will be a riding-the-yield-curve effect on the price of the bonds (even if the bond’s yield to maturity is unchanged) which will affect the carry trade return.  To be fair, even par bonds will see price changes when the yield curve is not flat.  When calculating the return on a carry trade transaction, be sure to remember to include the capital gain/loss on the bonds themselves.  And, of course, remember that premium bonds will have a higher coupon return, and discount bonds will have a lower coupon return.

Yield Curve Changes

If the yield curves are expected to change, then the prices of some (or all) of the bonds in the carry trade transaction will change accordingly.  Ideally, you can predict the yield curve changes accurately and incorporate them into your algorithm for determining the best carry trade transactions.  Nevertheless, this is a source of risk in carry trade.  Note that it will affect both aspects of the carry trade: intra-market and inter-market.

Exchange Rate Changes

As with yield curve changes, you can try to anticipate exchange rate changes and incorporate those changes into your carry trade algorithm.  As with everything else, the better you are at predicting the actual (spot) exchange rate at the end of your holding period, the better you will be at making an inter-market carry trade profit.

Three or More Yield Curves

The example in the curriculum has three currencies: USD, GBP, and EUR.  In the real world, you may be looking at dozens of currencies, including both stable, developed market currencies (e.g., USD, GBP, EUR, JPY, CHF, AUD, CAD, and so on) and less stable currencies, or currencies from less developed markets (Turkish lira (TRY), Brazilian real (BRL), Mexican peso (MXN), Indian rupee (INR), Argentinian peso (ARS), Thai bhat (THB), and so on).  This means that the model has to include yield curves for each of the currencies involved (current yield curves as well as projected yield curves) and exchange rates between each of the currencies involved and the investor’s home currency (again, current spot rates and projected spot rates).  The optimization becomes more complicated with each additional currency; in general, the complexity will increase as roughly the square of the number of currencies.

Floating-Rate Bonds

I confess that this would never have occurred to me, but there was a question on a CFA forum about what happens if the cost of borrowing changes.  This would occur if the bonds that are sold are floating-rate bonds.  Of course, it’s also possible for the cost of lending to change, if the bonds that are bought are floating-rate bonds.

In the real world this could happen (though I doubt it ever would: it is an added complication that, frankly, nobody wants), but it will never happen on the exam.  Please put it out of your mind.

Interim Coupons

If any of the bonds used in the carry trade pay coupons before your holding period is up, those coupons will have to be reinvested for the remainder of the holding period.  This probably isn’t a significant problem, as the coupons are generally small relative to the price of the bond itself, and the reinvestment return will be small relative to the coupons that are reinvested (so, really, really small relative to the price of the bond).  Nevertheless, it’s a complicating factor that a fixed income portfolio manager should consider as part of the carry trade strategy.  I’m quite sure, however, that it will not arise on the exam.

How Can This Be Tested?

This is a really good question.  Because this material was new in the 2019 Level III curriculum, and because 2019 is the year that CFA Institute decided that it would no longer publish the morning session actual exams and guideline answers, I have no definitive answer on how they tested this in 2019 or how they might test it in the future.  The best I can do is to speculate.  Therefore, do not take anything that follows as gospel; it’s nothing more nor less than my opinion (along with a soupçon of reasoning).

First, you will not have to go through all of the calculations included in the 10-page example in the curriculum.  That’s far too long for an exam question.  Nor do I expect that you will have to determine, on the whole, what transactions you will have to make, though you might have to do so with some clues given to you.  Here are some examples of what I think you could expect:

Example 1

Suppose that you are given four bonds denominated in a single currency, say, 2-year, 5-year, 7-year, and 10-year maturities, and you’re given the YTMs today (a normal (upward sloping) yield curve), the modified durations today, and the expected YTMs in one year (also a normal yield curve).  You’re further told that all carry trades will be pairs trades (i.e., sell one bond, buy the same amount of one other bond), and that the pairs trade to pick up yield will be a 2/7 trade for a given amount.  You might be asked to calculate the:

• Expected yield pickup on the 2/7 trade (in bps or in currency)
• Amount of the 5/10 trade to ensure that you’re duration neutral
• Expected yield loss on the 5/10 trade (in bps or in currency)
• Expected yield gain for the entire transaction (in bps or in currency)

Example 2

Suppose that you’re given yield curves for GBP and BRL, and all of the transactions for an intra-market carry trade in BRL.  For the inter-market carry trade, you might be asked to:

• Determine the position (buy or sell) in, say, the 10-year GBP bond based on the relative 10-year yield of BRL vs. GBP
• Calculate the expected yield gain (in bps or GBP or BRL) on the 10-year inter-market carry trade

Example 3

You might also have qualitative questions about intra-market or inter-market carry trade, such as explaining why you would deliberately enter into a specific trade knowing that you will lose yield on the trade (Answer: to maintain duration neutrality), or why you might have a position that is less than the allowable position limit (Answer: to maintain duration neutrality; I know, this is starting to get boring).

Final Thought

I encourage you to go through the example in the curriculum.  It’s long, and it’s tedious, and it’s boring, but at least now you have a good grasp on what they’re doing and why, so you’ll be able to step through it fairly quickly.  It has a number of questions which you should read and try to answer before reading their answers.  I think that it will serve you well.

The idea of immunization is fairly straightforward: you have a known set of liabilities due at known future dates, and you want to create a portfolio that will fund those liabilities with a minimum of risk, at a reasonable price.  Two types of immunization are discussed in the CFA curriculum: classical immunization and contingent immunization.  We’ll cover the former in the companion article and the latter in this article.

Contingent Immunization

The idea behind contingent immunization is that you have a known liability coming due at a known time, and you have more assets than you need to fund the liability; therefore, you can engage in active management of your asset portfolio, rather than the trite, passive, humdrum, boring life of classical immunization.

“Ah!” you say, “Some excitement has come into fixed income portfolio management!  Tell me more!”

You got it!

Let’s approach this by way of an example, which will illustrate concretely the sometimes abstract ideas of contingent immunization.

Example

Suppose that you have a liability of $2,000,000 coming due in 5 years, and that you have $1,600,000 in assets dedicated to funding this liability.  The annual return that you must earn on these assets is:

\[\left(\frac{\$2,000,000}{\$1,600,000}\right)^{1/5}\ -\ 1\ =\ 4.564\%\]

As luck would have it, you believe (with reasonable justification) that you can earn 5.5% annually on your assets: the current yield curve is flat at 5.5%.  Therefore, you really need only:

\[\frac{\$2,000,000}{1.055^5}\ =\ \$1,530,269\]

in assets to fund the liability.  Therefore, you have a cushion of $69,731 (= $1,600,000 − $1,530,269).  (You could also say that you have a cushion spread of 0.936% (= 5.5% − 4.564%).)

Because of this cushion, you can manage your assets actively, trying to earn a higher return while taking on some additional risk.  However, you have to mind that cushion carefully: if it should ever reach $0 (or, worse yet, become negative), you must switch your strategy immediately to classical immunization.  You can see, therefore, why this approach is called contingent immunization: employing it is contingent on having a positive cushion.

A common problem you can face in a contingent immunization situation is determining what happens to the cushion (and, consequently, to your strategy) if interest rates change abruptly.  Let’s take a look at that.

We’ll need some specific assets, so let’s say that we’ve invested in 8-year, annual-pay, $1,000 par, 5.5% coupon bonds having a YTM of 5.5%; therefore, each bond is selling at par; you own 1,600 bonds.  We’ll determine what happens to the cushion when rates decrease immediately by 50bp, when rates increase immediately by 50bp, and when rates increase immediately by 350bp.

50bp Decrease

The discount rate on the liability will decrease from 5.5% to 5.0%, so the present value of the liability will be:

\[\frac{\$2,000,000}{1.050^5}\ =\ \$1,567,052\]

The YTM on the bonds also will decrease from 5.5% to 5.0%, so the present value of each bond is $1,032.32 (FV = $1,000; PMT = $55; i = 5.0%; n = 8; solve for PV = −$1,032.32).  The value of the portfolio, therefore, is:

\[1,600\left(\$1,032.32\right)\ =\ \$1,651,706\]

In this case, the cushion is $84,654 (= $1,651,706 − $1,567,052); as it is still positive, we can continue with contingent immunization and can manage the portfolio actively.

50bp Increase

The discount rate on the liability will increase from 5.5% to 6.0%, so the present value of the liability will be:

\[\frac{\$2,000,000}{1.060^5}\ =\ \$1,494,516\]

The YTM on the bonds also will increase from 5.5% to 6.0%, so the present value of each bond is $968.95 (FV = $1,000; PMT = $55; i = 6.0%; n = 8; solve for PV = −$968.95). The value of the portfolio, therefore, is:

\[1,600\left(\$968.95\right)\ =\ \$1,550,322\]

In this case, the cushion is $55,806 (= $1,550,322 − $1,494,516); as it is still positive, we can continue with contingent immunization and can manage the portfolio actively.

350bp Increase

The discount rate on the liability will increase from 5.5% to 9.0%, so the present value of the liability will be:

\[\frac{\$2,000,000}{1.090^5}\ =\ \$1,299,863\]

The YTM on the bonds also will increase from 5.5% to 9.0%, so the present value of each bond is $806.28 (FV = $1,000; PMT = $55; i = 9.0%; n = 8; solve for PV = −$806.28). The value of the portfolio, therefore, is:

\[1,600\left(\$968.95\right)\ =\ \$1,290,050\]

Here, the cushion has become negative ($1,290,050 − $1,299,863 = −$9,813), so we must abandon contingent immunization and switch to classical immunization.  Oh, well: we tried.

Here’s how the value of the assets and liabilities compare given particular instantaneous parallel shifts in the yield curve:

Final Thoughts

In the example above we assumed that the yield curve was flat, so that the same rate was used to discount the liability (at one maturity) and the cash flows on the asset(s) (at a different maturity).  We also assumed that the yield curve would undergo a parallel shift.

In the real world you won’t be so lucky.  The discount rate for the liability will be different from the YTM for the bond (or bonds) in your portfolio, and you’ll have to mind all sorts of changes in the yield curve: steepening, flattening, humping, butterflying, whatever else-ing.  In short, it’s a lot more complex in the real world than it will be for the exam.

The idea of immunization is fairly straightforward: you have a known set of liabilities due at known future dates, and you want to create a portfolio that will fund those liabilities with a minimum of risk, at a reasonable price.  Two types of immunization are discussed in the CFA curriculum: classical immunization and contingent immunization.  We’ll cover the former in this article and the latter in the companion article.

Classical (Single-Period) Immunization

Classical immunization involves a single holding period (i.e., a single, known liability at a known future date); it has two goals:

1. Provide sufficient money on the date that the liability is due to be able to pay the liability.
2. Do not lose any value in the company if interest rates change.

To achieve the first goal, the present value of the funding bond portfolio must equal the present value of the liability, discounted at the expected rate of return on the portfolio.

To achieve the second goal, at a minimum the money duration of the bond portfolio must equal the money duration of the liability; equivalently, their basis point values (BPVs) must equal each other.  This condition will ensure that the change in the value of the portfolio will equal the change in the (present) value of the liability for a single, instantaneous, relatively small, parallel shift in the yield curve.  If the present value of the bond portfolio equals the present value of the liability, this condition is equivalent to saying that the effective duration of the bond portfolio must equal the effective duration of the liability.

Here’s where the curriculum and I disagree on a point: determining the duration of the liability.  The curriculum refers only to the “duration” of the liability (without characterizing what sort of duration they mean), and it says that the duration of the liability is the time to maturity.  So, without admitting to it, the authors are talking about the Macaulay duration of the liability.  Unfortunately, they then equate the (Macaulay) duration of the liability to the (effective) duration of the funding portfolio, as the basis for immunization against interest rate risk.

Unfortunately, it doesn’t work.  Macaulay duration is not a measure of interest rate risk.  (If you’re a bit foggy on the differences amongst Macaulay, modified, and effective duration, look here.)  Properly, we should compute the modified (or effective) duration of the liability (which will be slightly shorter than the Macaulay duration) and equate that to the effective duration of the funding portfolio.  The differences will be slight, I grant you, but as long as we’re learning how to do this, we might as well learn to do it correctly.

The effective duration of normal (i.e., fixed coupon) bonds is generally shorter than their maturity – perhaps much shorter, depending on the coupon rate – so matching the duration of the portfolio to the duration of the liability generally requires purchasing a bond whose maturity is longer than that of the liability.

Example

You have a liability of $2,000,000 coming due in 5 years; going with CFA Institute’s convention, we’ll assume that the (effective) duration of the liability is 5 years (though it’s really a bit shorter than 5 years).  You have two bonds that you can use to fund the liability:  a 4-year, zero-coupon, $2,000,000 par bond (really, 2,000, $1,000-par bonds) and an 8-year, zero-coupon, $2,000,000 par bond (again, really 2,000, $1,000-par bonds).  The current spot curve is:

Maturity	Spot Rate
1	2.000%
2	3.000%
3	3.800%
4	4.440%
5	4.952%
6	5.362%
7	5.689%
8	5.951%

The 4-year ($2,000,000 par) bond sells at:

\[\frac{\$2,000,000}{1.04440^4}\ =\ \$1,680,980\]

and the 8-year ($2,000,000 par) bond sells at:

\[\frac{\$2,000,000}{1.05951^8}\ =\ \$1,259,475\]

The modified (and effective) duration of the 4-year bond is:

\[\frac{4\ years}{1.04440}\ =\ 3.83\ years\]

and the modified (and effective) duration of the 8-year bond is:

\[\frac{8\ years}{1.05951}\ =\ 7.55\ years\]

The present value of the liability is:

\[\frac{\$2,000,000}{1.04952^5}\ =\ \$1,570,639\]

The weighted average of the bonds’ present values has to equal the liability’s present value:

\[w_4\left(\$1,680,980\right)\ +\ w_8\left(\$1,259,475\right)\ =\ \$1,570,639\]

and the weighted average of the bonds’ durations has to equal the liability’s duration:

\[w_4\left(3.83\right)\ +\ w_8\left(7.55\right)\ =\ 5.00\]

I’ll spare you the linear algebra; the solution is:

\begin{align}w_4\ &=\ 0.7068\\
\\
w_8\ &=\ 0.3037
\end{align}

Therefore, the market value of the 4-year bond you need to purchase is:

\[0.7068\left(\$1,680,980\right)\ =\ \$1,188,188\]

which has a par value of:

\[\$1,188,188\ ×\ 1.04440^4\ =\ \$1,413,684\]

The market value of the 8-year bond you need to purchase is:

\[0.3037\left(\$1,259,475\right)\ =\ \$382,451\]

which has a par value of:

\[\$382,451\ ×\ 1.05951^8\ =\ \$607,319\]

Suppose that there is an instantaneous 50bp increase in interest rates, then the present value of the liability will be:

\[\frac{\$2,000,000}{1.05452^5}\ =\ \$1,533,755\]

the present value of the 4-year bond will be:

\[\frac{\$1,413,684}{1.04940^4}\ =\ \$1,165,704\]

the present value of the 8-year bond will be:

\[\frac{\$607,319}{1.06451^8}\ =\ \$368,315\]

and the total value of the portfolio will be:

\[\$1,165,704\ +\ \$368,315 =\ \$1,534,019\]

Note that the difference of $264 (= $1,534,019 − $1,533,755) is the result of the slightly different convexities of the portfolio and the liability.

By the way, if we had used 4.76 years (= 5 years ÷ 1.04952) as the duration of the liability (which is not what CFA Institute uses, but is the correct way to analyze the problem), the weights would be w₄ = 0.7446 and w₈ = 0.2533.

Rebalancing

Because the durations and values of the liability and the portfolio will change over time, the portfolio will occasionally need to be rebalanced.  One common way to rebalance the portfolio is based on the rebalancing ratio: the dollar duration of the liability divided by the dollar duration of the portfolio.

Example (cont.)

Suppose that one year later the spot curve has shifted and flattened somewhat:

Maturity	Spot Rate
1	2.500%
2	3.400%
3	4.100%
4	4.640%
5	5.052%
6	5.362%
7	5.589%
8	5.751%

The present value of the liability is now:

\[\frac{\$2,000,000}{1.04640^4}\ =\ \$1,668,165\]

and, assuming that its duration is 4 years,  its dollar duration is:

\[\$1,668,165\ ×\ 4.00\ years\ ×\ 1\%\ =\ 66,727\ dollar-years\]

The present value of the (original) 4-year bond is now:

\[\frac{\$1,413,684}{1.04100^3}\ =\ \$1,253,142\]

its effective duration is:

\[\frac{3\ years}{1.04100}\ =\ 2.88\ years\]

and its dollar duration is:

\[\$1,253,142\ ×\ 2.88\ years\ ×\ 1\%\ =\ 36,090\ dollar-years\]

The present value of the (original) 8-year bond is now:

\[\frac{\$607,319}{1.05589^7}\ =\ \$415,036\]

its effective duration is:

\[\frac{7\ years}{1.05589}\ =\ 6.63\ years\]

and its dollar duration is:

\[\$415,036\ ×\ 6.63\ years\ ×\ 1\%\ =\ 27,517\ dollar-years\]

The total dollar duration of the portfolio is, therefore:

\[36,090\ dollar-years\ +\ 27,517\ dollar-years\ =\ 63,607\ dollar-years\]

The rebalancing ratio is:

\[\frac{66,727}{63,607}\ =\ 1.0491\]

Thus, you need to purchase an additional 4.91% of the 3-year bond and the 7-year bond:

\begin{align}\$1,253,142\ ×\ 4.91\%\ &=\ \$61,468\\
\\
\$415,036\ ×\ 4.91\%\ &=\ \$20,378
\end{align}

Note that by using the rebalancing ratio, we’re not explicitly trying to match the present value and effective duration of the portfolio to those of the liability.  This is a flaw in the rebalancing ratio approach that never seems to be mentioned anywhere.  Well, except here, of course.

(To be fair, the differences are insignificant in this example.  Nevertheless, it seems a little silly to use the rebalancing ratio when it’s just as easy to calculate the values that match the liability’s value and effective duration.  Oh, well.)