Level 1 CFA® Exam:
Duration of Bond Portfolio

In the previous lessons, we talked a lot about the interest rate risk of bonds. We saw how to compute the measures of interest rate risk for a bond, namely duration and convexity. Of course, if we can calculate a single bond duration, we should also be able to calculate the duration of a bond portfolio.

In this lesson, you will learn the most popular and easy way to compute the duration of a bond portfolio.

We will begin with 2 formulas.

The first one is the formula for Macaulay duration:

\(MacD=\Sigma^{n}_{i=1}\frac{P_{i}}{\Sigma^{n}_{j=1}P_j}\times{MacD_i}\)

As you can see, the Macaulay duration of a portfolio is the weighted average of the Macaulay durations of the bonds included in the portfolio.

The weight for each bond is equal to the current value of the bond divided by the total value of the bond portfolio.

The second formula is very similar and it is the formula for modified duration:

\(ModD=\Sigma^{n}_{i=1}\frac{P_{i}}{\Sigma^{n}_{j=1}P_j}\times{ModD_i}\)

As you can see, the modified duration of a portfolio is the weighted average of the modified durations of the bonds included in the portfolio.

As in the case of the first formula, the weight for each bond is equal to the current value of the bond divided by the total value of the bond portfolio.

Note, however, that if we use these two formulas, the relation that we talked about in the previous lessons, namely that the modified duration equals Macaulay duration divided by 1 plus yield, will usually not hold. This is because usually, the yield-to-maturity is different for different bonds, so even if we compute the yield-to-maturity for the whole bond portfolio, the modified duration won’t be exactly equal to the Macaulay duration divided by 1 plus yield-to-maturity of the portfolio.

We will use the formula for the modified duration of a bond portfolio to measure the interest rate risk of the portfolio. Before we solve an example, let’s enumerate the pros and cons of the formula.

Pros:

• The formula is easily applicable.
• It is a good measure of a bond portfolio interest rate risk if one particular condition is met (described below in the cons section).
• The formula can be used even if bonds have options embedded in them. For this kind of bonds, we will use the effective duration instead of the modified duration.

Cons:

• The formula gives only an approximation of the change in the bond portfolio value. The most accurate solution to compute the modified duration of the portfolio would be to imagine that the whole portfolio is ONE bond with cash flows of different values occurring in different periods. For this ONE bond, we should calculate the yield-to-maturity and then calculate the modified duration of the portfolio. However, even though it may seem tempting, analysts usually avoid this solution because it is more complicated and has its drawbacks.
• It works fine only if the yields of all bonds change by the same amount in the same direction at the same time. In other words, it works fine if there is a parallel shift in the yield curve. If the changes in yields are different for different bonds, the change in the bond portfolio value obtained using this formula can be not accurate enough.