# Duration-matching

Duration-matching is a strategy used to manage interest rate risk that involves matching the duration of the loan with the duration of the asset. Duration is the weighted-average maturity of the cash flows of the debt or asset. While duration-matching doesn’t eliminate the interest rate risk, it can manage the exposure for relatively minor changes in interest rates.

The value of a bond or any other debt instrument is very prone to changes in the market interest rates. If the market interest rates fall, the current value of debt increase and vice versa. This exposes companies and investors to interest rate risk. The interest rate risk is higher when the cash flows are farther away from time 0 and vice versa. Duration is a measure of the interest rate risk. There are multiple measures of duration, the most basic being the Macaulay’s duration, which is estimated using the following equation:

$$ \text{Duration}=\frac{{\rm \text{PV}} _ \text{1}}{\text{PV}}\times \text{T} _ \text{1}+\frac{{\rm \text{PV}} _ \text{2}}{\text{PV}}\times \text{T} _ \text{2}+\text{...}+\frac{{\rm \text{PV}} _ \text{n}}{\text{PV}}\times \text{T} _ \text{n} $$

Where PV_{1}, PV_{2} and PV_{n} is the present value of first, second and nth cash flow, PV is the sum of present value of all cash flows and T_{1}, T_{2} and T_{n} represent the time difference between time 0 and the relevant cash flow.

In the duration-matching strategy, you need to match the duration of the asset with the duration of the associated liability.

## Example

Your company has a pension fund which is required to make the following payments over the next 5 years:

Year | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

Cash outflows (USD in millions) | 5 | 7 | 6 | 8 | 10 |

The associated interest rate is 10%.

Your CFO has asked you to devise a strategy to hedge the associated interest rate risk.

The most effective way to hedge the interest rate risk is buy zero-coupon bonds with face value and maturity exactly matching each of the above cash flows. However, it might be hard to exactly offset the cashflows, so you might have to resort to duration-matching.

The first step in duration-matching strategy is to determine the duration of the cash flows being hedged. You need to find individual present value of each cash flow at 10%. The present value of $5 million at Year 1 work out to $4.54 million

$$ {\rm \text{PV}} _ \text{1}=\frac{\text{\$5 million}}{{(\text{1}+\text{10%})}^\text{1}}=\text{\$4.54 million} $$

In the same way you can work out present value the other cash flows. Next, you need to find the weight to be assigned to each cash flow which equals the proportion of the relevant individual present value to total present value.

The following table shows the individual present value, total present value and the associated weights::

Year | 1 | 2 | 3 | 4 | 5 | Total |
---|---|---|---|---|---|---|

Cash outflows (USD in millions) | 5 | 7 | 6 | 8 | 10 | |

PV at 10% (USD in million) | 4.55 | 5.79 | 4.51 | 5.46 | 6.21 | 26.51 |

Individual PV as a % of total PV | 17.15% | 21.82% | 17.00% | 20.61% | 23.42% |

The duration of your pension liability can now be worked out by multiplying the time duration of each cash flow with the relevant weight:

$$ \text{D}\\=\text{17.15%}\times \text{1}+\text{21.82%}\times \text{2}+\text{17.00%}\times \text{3}+\text{20.61%}\times \text{4}+\text{23.42%}\times \text{5}\\=\text{3.11} $$

To hedge the associated interest rate exposure, you should invest in investments whose duration equals 3.11. This strategy will be effective as long as the interest rate doesn’t move drastically. You might need to periodically recheck and readjust duration of your investments.

by Obaidullah Jan, ACA, CFA and last modified on

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