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# Cash Flow Matching

As an approach to asset-liability management (ALM), cash flow matching has advantages and disadvantages. The primary advantages are that:

- It works perfectly.
- It is relatively simple to understand.

The primary disadvantages are that:

- It can be costly (i.e., the interest that you earn on your assets may be low).
- It may be difficult to implement in practice.

The easiest way to explain cash flow matching is through an example, so here goes:

Suppose that you manage a pension fund that has liabilities of €2 million, €4 million, €7 million, and €11 million coming due in 1, 2, 3, and 4 years, respectively. You want to invest in bonds whose cash flows will exactly match the liabilities. The following bonds are available:

- 1-year, 2-year, 3-year, and 4-year zero-coupon bonds; the yields (YTMs) are 1%, 2%, 3%, and 4%, respectively
- 1-year annual-pay bonds with a coupon rate of 1%, selling at par
- 2-year annual-pay bonds with a coupon rate of 2%, selling at par
- 3-year annual-pay bonds with a coupon rate of 3%, selling at par
- 4-year annual-pay bonds with a coupon rate of 4%, selling at par

The easiest way to implement cash-flow matching is with the zero-coupon bonds: buy €2 million par of 1-year bonds, €4 million par of 2-year bonds, €7 million par of 3-year bonds, and €11 million par of 4-year bonds. This will match your cash flows perfectly – the 1-year bonds will mature in 1 year and pay €2 million, the 2-year bonds will mature in 2 years and pay €4 million, and so on, but it will be a costly way to match the cash flows; the outlay today to buy the bonds will be:

- €1,980,198 (= €2 million ÷ 1.01) for the 1-year bonds
- €3,844,675 (= €4 million ÷ 1.02
^{2}) for the 2-year bonds - €6,405,992 (= €7 million ÷ 1.03
^{3}) for the 3-year bonds - €9,402,846 (= €11 million ÷ 1.04
^{4}) for the 4-year bonds

The total cost for these bonds is €21,633,711.

The alternative is to use the coupon-paying bonds that have higher yields. To implement cash flow matching with coupon-paying bonds, you need to start with the longest-maturity liability and work back to the shortest. The steps aren’t difficult, but do require that you pay attention to all of the cash flows. The key to understanding this approach is to recall that the final payment on a coupon-paying bond is par plus coupon, so the par value you need is obtained by dividing the cash flow that you need by (1 + coupon rate):

\begin{align}par\ +\ coupon\ &=\ liability\\

\\

par\ +\ \left(par\ ×\ coupon\ rate\right)\ &=\ liability\\

\\

par\ ×\ \left(1\ +\ coupon\ rate\right)\ &=\ liability\\

\\

par\ &=\ \frac{liability}{1\ +\ coupon\ rate}

\end{align}

- Buy €10,576,923 (= €11,000,000 ÷ 1.04) par of the 4-year, 4% coupon bonds. The final cash flow on these bonds will be par plus the coupon or €10,576,923 + €423,077 = €11,000,000, which will cover the liability in 4 years. Note that these bonds will pay annual coupons of €423,077 one year, two years, and three years from today.
- As you already have a cash flow of €423,077 at year 3, you need only €6,576,923 (= €7,000,000 − €423,077) more; buy €6,385,362 (= €6,576,923 ÷ 1.03) par of the 3-year, 3% coupon bonds. The final cash flow on these bonds will be par plus the coupon or €6,385,362 + €191,561 = €6,576,923, so the total cash flow at year 3 will be €7,000,000 (= €6,385,362 + €191,561 + €423,077), which will cover the liability in 3 years. Note that these bonds will pay annual coupons of €191,561 one year and two years from today.
- As you already have cash flows of €423,077 and €191,561 at year 2, you need only €3,385,362 (= €4,000,000 − €423,077 − €191,561) more; buy €3,318,983 (= €3,385,362 ÷ 1.02) par of the 3-year, 2% coupon bonds. The final cash flow on these bonds will be par plus the coupon or €3,318,983 + €66,380 = €3,385,362, so the total cash flow at year 2 will be €4,000,000 (= €3,318,983 + €66,380 + €191,561 + €423,077), which will cover the liability in 2 years. Note that these bonds will pay annual coupons of €66,380 one year from today.
- As you already have cash flows of €423,077, €191,561, and €66,380 at year 1, you need only €1,318,982 (= €2,000,000 − €423,077 − €191,561 − €66,380) more; buy €1,305,923 (= €1,318,982 ÷ 1.01) par of the 1-year, 1% coupon bonds. The final cash flow on these bonds will be par plus the coupon or €1,305,923 + €13,059 = €1,318,982, so the total cash flow at year 1 will be €2,000,000 (= €1,305,923 + €13,059 + €66,380 + €191,561 + €423,077), which will cover the liability in 1 year.

Here are the cash flows in a table, which is probably easier to read:

Cash Flow by Year |
||||

Bond |
Year 1 |
Year 2 |
Year 3 |
Year 4 |

4-year |
€423,077 | €423,077 | €423,077 | €423,077 + €10,576,923 |

3-year |
€191,561 | €191,561 | €191,561 + €6,385,362 | |

2-year |
€66,380 | €66,380 + €3,318,983 | ||

1-year |
€13,059 + €1,305,923 | |||

Total CF |
€2,000,000 | €4,000,000 | €7,000,000 | €11,000,000 |

The total cost for this approach is €21,587,191, or €46,520 less than using the zero-coupon bonds.

The reason that this approach is less expensive than using only zero-coupon bonds is that the longer maturity bonds have higher yields; so, for example, the €2,000,000 cash flow in year 1 is funded partly by a bond with a 1% yield, and partly by bonds having yields of 2%, 3%, and 4%; using only zero-coupon bonds, the entire €2,000,000 is funded by a bond with a 1% yield. (However, see the Note, below.)

The reason that cash flow matching may be difficult to implement in practice is perfectly illustrated by this example: as most euro-denominated bonds have par values of €1,000, it will be impossible to purchase exactly €10,576,923 of 4-year bonds; you would have to decide between purchasing €10,576,000 par (and falling slightly short of your cash flow need) and purchasing €10,577,000 par (and spending too much). The same is true for the par values required at years 3, 2, and 1.

Note: in the example given the yields on the zero-coupon bonds and on the coupon-paying bonds of the same maturity are equal. That, of course, shouldn’t happen with an upward-sloping yield curve (lest there be an arbitrage opportunity), but that’s the sort of example that you see in the curriculum. In fact, if the zero-coupon and coupon-paying bonds are priced fairly, there should be no difference in the cost of cash flow matching with zeros and cash flow matching with coupon-paying bonds; once again, if there were a cost difference, there would be an arbitrage opportunity.

Sigh.