In computing modified (or effective) duration for a portfolio of securities, we change the **par** interest rate (the yield to maturity) at every maturity by some small amount up and down (±Δ*y*), and determine the percentage price change in the portfolio for 1% change in yield. In essence, we add ±Δ*y* to the entire par curve. (Look here for a refresher on the par curve, and for a refresher on bootstrapping the spot curve from the par curve, which we shall be doing a bit later.) Parallel shifts in the par curve look line this:

Modified (or effective) duration is useful for determining how the price of a security or a portfolio of securities will change when the (par) yield curve undergoes a parallel shift, but is less useful (i.e., *not useful at all*) when the yield curve changes in a manner other than a parallel shift (e.g., flattening or steepening). A more versatile measure of duration is needed in those cases; **key rate duration** is just such a measure.

The idea of a key rate duration (also known as *partial duration*) is that we don’t add ±Δ*y* to the entire par curve; we add ±Δ*y* *only to the YTM at a specific maturity* on the par curve, leaving the YTM at all other maturities unchanged; that specific maturity is called the **key rate**. Thus, we have the:

- ½-year key rate duration: ±Δ
*y* added to the ½-year YTM, while the YTMs at all other maturities remain unchanged
- 2-year key rate duration: ±Δ
*y* added to the 2-year YTM
- 5-year key rate duration: ±Δ
*y* added to the 5-year YTM
- 10-year key rate duration: ±Δ
*y* added to the 10-year YTM
- 20-year key rate duration: ±Δ
*y* added to the 20-year YTM
- 30-year key rate duration: ±Δ
*y* added to the 30-year YTM
- And so on

Shifts in the par curve to calculate key rate durations look like this:

Note that key rate durations can be *modified* (assuming no changes to the cash flows when the key rate changes) or *effective* (allowing that cash flows might change when the key rate changes). There’s probably no reason to concern yourself about this as far as the exam goes, but in real world applications you might as well assume that key rate durations are effective durations.

Once the effect of the key-rate shift is incorporated into the bond price, the duration is computed using the standard (modified or effective) duration formula:

\[Dur_{key\ rate}\ =\ \frac{P_–\ -\ P_+}{2P_0\Delta y}\]

In many respects, key rate durations behave exactly like all other durations (Macaulay duration, modified duration, effective duration, etc.), but in some ways their behavior is unusual. We’ll cover both.

Expected Behavior

*Key Rate Duration of a Portfolio*

As with Macaulay, modified, and effective duration (see here), the key rate duration of a portfolio is the weighted average of the key rate durations of its constituent bonds, where the weights are based on the market values of the constituent bonds:

\begin{align}Dur_{key\ rate_{port}}\ &=\ W_1\ ×\ Dur_{key\ rate_{bond\ 1}}\ +\ W_2\ ×\ Dur_{key\ rate_{bond\ 2}}\\

&+\ \cdots\ +\ W_n\ ×\ Dur_{key\ rate_{bond\ n}}\\

\\

&=\ \sum_{i=1}^n W_i\ ×\ Dur_{key\ rate_{bond\ i}}

\end{align}

where:

- \(W_i\): bond
*i*‘s **market** value ÷ **market** value of the portfolio

So, for example, the 5-year key rate duration of a portfolio of bonds is the weighted average of the 5-year key rate durations of the constituent bonds, the 20-year key rate duration of the portfolio is the weighted average of the 20-year key rate durations of the constituent bonds, and so on.

*Relationship of (Modified or Effective) Duration and Key Rate Durations*

Another characteristic of key rate durations that is expected (or, at least, should be, once you think of it) is that the (modified or effective) duration of a security or a portfolio is the sum of all of its (respectively, modified or effective) key rate durations:

\begin{align}Dur\ &=\ Dur_{key\ rate_1}\ +\ Dur_{key\ rate_2}\ +\ \cdots\ +\ Dur_{key\ rate_k}\ +\cdots\\

\\

&=\ \sum_{i=1}^{\infty}\ Dur_{key\ rate_i}

\end{align}

Note that this **is not** an infinite sum: eventually the index *i* will exceed the maturity of the security or portfolio; after that, all of the key rate durations will be zero (because, as we will see, all of the par rates and all of the spot rates at maturities shorter than the key rate maturity are unchanged).

Unusual Behavior

Recalling the link between the par curve and the spot curve, we can determine the effect of key rate changes on bonds of various maturities and with various coupons. To illustrate these effects, I’ll use a simple example: a 4% flat yield curve with annual (effective) rates from 1 year to 10 years:

**Maturity (Years)** |
**Par Yield** |
**Spot Yield** |

1.0 |
4.0% |
4.0% |

2.0 |
4.0% |
4.0% |

3.0 |
4.0% |
4.0% |

4.0 |
4.0% |
4.0% |

5.0 |
4.0% |
4.0% |

6.0 |
4.0% |
4.0% |

7.0 |
4.0% |
4.0% |

8.0 |
4.0% |
4.0% |

9.0 |
4.0% |
4.0% |

10.0 |
4.0% |
4.0% |

Suppose that we want to compute the 5-year key rate duration of a bond. To do this, we add 50bp to the 5-year par rate (leaving all of the other par rates unchanged), and compute the effect on the spot rates, then subtract 50bp from the 5-year par rate (again, leaving all of the other par rates unchanged), and compute the effect on the spot rates. We’ll then use those modified spot rates to compute \(P_-\) and \(P_+\), and then use them in the duration formula.

Let’s start by adding 50bp to the 5-year par rate.

The 1-year spot yield is easy: it remains 4.0%. (Recall that the one-period par rate and the one-period spot rate are equal, and that the 1-year par rate hasn’t changed; only the 5-year par rate has changed.)

The 2-year spot yield is also easy: it, too, remains 4.0%, because the 2-year par yield is unchanged, and the 1-year spot yield is unchanged. If you want to run through the algebra, it’s:

\begin{align}$1,000\ &=\ \frac{$40.00}{1.04}\ +\ \frac{$1,040.00}{\left(1\ +\ s_2\right)^2}\\

\\

$1,000\ &=\ $38.46\ +\ \frac{$1,040.00}{\left(1\ +\ s_2\right)^2}\\

\\

$961.54\ &=\ \frac{$1,040.00}{\left(1\ +\ s_2\right)^2}\\

\\

\left(1\ +\ s_2\right)^2\ &=\ \frac{$1,040.00}{$961.54}\ =\ 1.0816\\

\\

1\ +\ s_2\ &=\ \sqrt{1.0816}\ =\ 1.0400\\

\\

s_2\ &=\ 4.0\%

\end{align}

The 3-year spot rate and the 4-year spot rate are also easy: they’re both still 4.0%, because their par yields are unchanged, as are all of the spot yields for shorter maturities. I’ll leave the algebra to you.

As an exciting (**!**) change of pace, the 5-year spot rate **isn’t 4.0%** (recall that we’ve added 50bp to the 5-year *par* rate). A 5-year bond paying a 4.5% (= 4.0% + 50bp) will therefore sell at par, so:

\begin{align}$1,000\ &=\ \frac{$45.00}{1\ +\ s_1}\ +\ \frac{$45.00}{\left(1\ +\ s_2\right)^2}\ +\ \frac{$45.00}{\left(1\ +\ s_3\right)^3}\ +\ \frac{$45.00}{\left(1\ +\ s_4\right)^4}\ +\ \frac{$1,045.00}{\left(1\ +\ s_5\right)^5}\\

\\

&=\ \frac{$45.00}{1.04}\ +\ \frac{$45.00}{1.04^2}\ +\ \frac{$45.00}{1.04^3}\ +\ \frac{$45.00}{1.04^4}\ +\ \frac{$1,045.00}{\left(1\ +\ s_5\right)^5}\\

\\

$1,000\ &=\ $43.27\ +\ $41.61\ +\ $40.00\ +\ $38.47\ +\ \frac{$1,045.00}{\left(1\ +\ s_5\right)^5}\\

\\

$836.65\ &=\ \frac{$1,045.00}{\left(1\ +\ s_5\right)^5}\\

\\

\left(1\ +\ s_5\right)^5\ &=\ \frac{$1,045.00}{$836.65}\ =\ 1.249022\\

\\

1\ +\ s_5\ &=\ 1.249022^{1/5}\ =\ 1.045476\\

\\

s_5\ &=\ 4.5476\%

\end{align}

This result seems reasonable: the YTM is 4.5%, so we’ll get par ($1,000) if we discount all the cash flows at 4.5%. Because we’ve discounted the first four payments at 4% (less than 4.5%), we have to discount the final payment at more than 4.5% to average a 4.5% discount for everything. And because the first four payments are much smaller than the final payment, the difference on the final discount rate should be much smaller than the (50bp) difference on the first four: it’s 4.76bp.

Now comes the **really** interesting part (**!!**): the 6-year spot rate. Let’s do the algebra (recalling that **the 6-year par rate hasn’t changed**: it’s still 4.0%):

\begin{align}$1,000\ &=\ \frac{$40.00}{1.04}\ +\ \frac{$40.00}{1.04^2}\ +\ \frac{$40.00}{1.04^3}\ +\ \frac{$40.00}{1.04^4}\\

\\

&+\ \frac{$40.00}{1.045476^5}\ +\ \frac{$1,040.00}{\left(1\ +\ s_6\right)^6}\\

\\

$1,000\ &=\ $38.46\ +\ $36.98\ +\ $35.56\ +\ $34.19\ +\ $32.03\ +\ \frac{$1,040.00}{\left(1\ +\ s_6\right)^6}\\

\\

$822.78\ &=\ \frac{$1,040.00}{\left(1\ +\ s_6\right)^6}\\

\\

\left(1\ +\ s_6\right)^6\ &=\ \frac{$1,040.00}{$822.78}\ =\ 1.264009\\

\\

1\ +\ s_6\ &=\ 1.264009^{1/6}\ =\ 1.039820\\

\\

s_6\ &=\ 3.9820\%

\end{align}

Again, this result makes sense: the 5-year spot rate has increased, so the only way that the 6-year par rate can remain unchanged is for the 6-year spot rate to decrease. And, once again, because the final payment is much larger than the first 5 (coupon only) payments, the decrease should be small compared to the increase in the 5-year spot rate: the 5-year spot rate is 54.76bp higher than the 4% YTM, while the 6-year spot rate is 1.80bp lower than the YTM.

Note how the change in the 5-year key rate leads to some somewhat surprising results. For example, a 6-year zero-coupon bond will have a slightly **higher** price (because the 6-year spot rate is *lower*) when the 5-year key rate is *increased*; in other words, **a 6-year zero-coupon bond will have a negative 5-year key rate duration**. (Remember that when interest rates increase, bond prices normally decrease, and that duration is normally positive. When the price increases with a rate increase, duration must be negative.)

Continuing in this same tedious manner, we get:

**Maturity (Years)** |
**Par Yield** |
**Spot Yield** |

1.0 |
4.0000% |
4.0000% |

2.0 |
4.0000% |
4.0000% |

3.0 |
4.0000% |
4.0000% |

4.0 |
4.0000% |
4.0000% |

5.0 |
4.5000% |
4.5476% |

6.0 |
4.0000% |
3.9820% |

7.0 |
4.0000% |
3.9846% |

8.0 |
4.0000% |
3.9865% |

9.0 |
4.0000% |
3.9880% |

10.0 |
4.0000% |
3.9892% |

Graphically:

When we subtract 50bp from the 5-year par rate, we get:

**Maturity (Years)** |
**Par Yield** |
**Spot Yield** |

1.0 |
4.0000% |
4.0000% |

2.0 |
4.0000% |
4.0000% |

3.0 |
4.0000% |
4.0000% |

4.0 |
4.0000% |
4.0000% |

5.0 |
3.5000% |
3.4641% |

6.0 |
4.0000% |
4.0182% |

7.0 |
4.0000% |
4.0156% |

8.0 |
4.0000% |
4.0136% |

9.0 |
4.0000% |
4.0121% |

10.0 |
4.0000% |
4.0109% |

Graphically:

*Key Rate Durations of Various Bonds*

Not only does the key rate duration of a bond depend on the bond’s maturity (compared to the maturity of the key rate), it also depends on the bond’s coupon rate compared to its YTM; i.e., it depends on whether the bond is priced at par, at a premium, or at a discount. Using our 4% flat yield curve, here are the key rate durations for five 5-year, option-free bonds with varying coupon rates, along with the sum of their key rate durations, and their effective durations:

**Key Rate Duration (Years), 5-Year Bond** |

**Key Rate Maturity** |
**Coupon Rate** |

**0.0%** |
**2.0%** |
**4.0%** |
**6.0%** |
**8.0%** |

**1 Year** |
(0.0385) |
(0.0174) |
0.0000 |
0.0145 |
0.0268 |

**2 Years** |
(0.0785) |
(0.0354) |
0.0000 |
0.0296 |
0.0547 |

**3 Years** |
(0.1201) |
(0.0542) |
0.0000 |
0.0453 |
0.0838 |

**4 Years** |
(0.1633) |
(0.0737) |
0.0000 |
0.0616 |
0.1140 |

**5 Years** |
5.2081 |
4.7931 |
4.4519 |
4.1666 |
3.9243 |

**6+ Years** |
0.0000 |
0.0000 |
0.0000 |
0.0000 |
0.0000 |

**Sum** |
**4.8078** |
**4.6125** |
**4.4519** |
**4.3176** |
**4.2036** |

**Dur**_{Eff} |
4.8078 |
4.6125 |
4.4519 |
4.3176 |
4.2036 |

So, for example, a 5-year, zero-coupon bond has a 4-year key rate duration of −0.1633 years; if the 4-year par rate *increases* by 1%, the price of the bond will **increase** by approximately 0.1633%. A 5-year, 8% coupon bond has a 3-year key rate duration of 0.0838 years; if the 3-year par rate *decreases* by 1%, the price of the bond will **increase** by approximately 0.0838%.

The key ideas to glean from this table are:

*Par* bonds have key rate durations of *zero years* for any key rate maturity **shorter** than the bond’s maturity
*Discount* bonds have **negative** key rate durations for key rate maturities **shorter** than the bond’s maturity; in particular, *zero-coupon bonds* have *negative* key rate durations for key rate maturities **shorter** than the bond’s maturity
*Premium* bonds have **positive** key rate durations for key rate maturities **shorter** than the bond’s maturity
*All* bonds have key rate durations of *zero years* for any key rate maturity **longer** than the bond’s maturity
- The sum of the key rate durations for all key rate maturities equals the bond’s effective duration

Note that these last two points was already covered under Expected Behavior, above.

Misconceptions about Key Rate Durations

The primary misconception about key rate durations is that they correspond to a change in a single *spot rate*, instead of a change in a single **par rate**. To be clear:

- The key rate duration of a given bond for a given maturity is the ratio of the percentage change in that bond’s price to the change in
*the ***par** rate at that maturity, when *the ***par** rates at all other maturities remain unchanged.
- When the par rate
**at a given maturity** changes, and the par rates **at all other maturities** remain unchanged:
- The spot rates at maturities
**less** than the given maturity *will not change*
- The spot rate
**at** the given maturity will change *in the same direction* as change in the par rate, and *by an amount greater* than the change in the par rate
- The spot rates at maturities
**greater** than the given maturity will change *in the opposite direction* of the change in the par rate, and *by an amount (much) less* than the change in the par rate (and the change will be smaller at longer maturities)