In the previous section, we saw how to calculate the duration for an individual bond. What if a portfolio consists of a number of bonds: how will its duration be calculated?
There are two ways to calculate the duration of a bond portfolio:
The key points related to each method are outlined in the table below.
|Weighted Average Time to Receipt of Aggregate Cash Flows||Weighted Average of Individual Bond Durations in Portfolio|
|Theoretically correct, but difficult to use in practice.||Commonly used in practice.|
|Cash flow yield not commonly used. Cash flow yield is the IRR on a series of cash flows.||Easy to use as a measure of interest rate risk.|
|Amount and timing of cash flows might not be known because some of these bonds may be MBS, or with call options.||More accurate as difference in YTMs of bonds in portfolio become smaller.|
|Interest rate risk is usually expressed as a change in benchmark interest rates, not as a change in the cash flow yield.||Assumes parallel shifts in the yield curve, i.e. all rates change by the same amount in the same direction that seldom happens in reality.|
|Change in the cash flow yield is not necessarily the same amount as the change in yields-to-maturity on the individual bonds.|
Example 12: Calculating portfolio duration
An investment fund owns the following portfolio of three fixed-rate government bonds:
|Bond A||Bond B||Bond C|
The total market value of the portfolio is 84,216,775. Each bond is on a coupon date so that there is no accrued interest. The market values are the full prices given the par value.
Solution to 1:
The average modified duration of the portfolio is 6.23.
(14,769,542/84,216,775) × 4.328 + (25,650,379/84,216,775) × 5.643 + (43,796,854/84,216,775) × 7.210 = 6.23.
Solution to 2:
The estimated decline in market value if each yield rises by 10 bps is 0.623%. -6.23 × 0.001 = -0.00623.