portfolio in relation to the changes in the duration of the index. A benchmark can be a peer group where the duration of the benchmark cannot be measured, but can be estimated through market movements. We will cover this issue in more detail in Chapter 19.
Before we calculate the duration of a portfolio, we introduce the concept of the value of a basis point (VBP) or dollar value of a basis point (DV01), which is the change in the market value of a portfolio resulting from the change of 1 basis point in the level of interest rates:
1.19
where Mj and Dj are the market value and duration of a bond j in the portfolio. Consider a portfolio of N securities, each with multiple cash flows. The total market value of the portfolio can be written as
1.20
where pj is the price of security j, wj is the weight or face amount of security j, cij is the ith cash flow of security j and tij is the time to that cash flow. For a small uniform change in the yield of all bonds in the portfolio, it can easily be shown that the change in the market value will be
1.21
Alternatively,
1.22
If D is the duration of the portfolio, the change in market value for a change of
1.23
The duration of the portfolio is
1.24
Thus, the duration of a portfolio is the market value weighted sum of the duration of all bonds in the portfolio. Alternatively, the duration of a portfolio is the sum of all the VBPs divided by the market value and multiplied by 10, 000.
To estimate the yield of the portfolio, we note that the market value of the portfolio can be calculated by discounting all bonds in the portfolio by their respective yields or by discounting all the cash flows in the portfolio by the portfolio yield:
1.25
where y is the overall yield of the portfolio and yj is the yield of bond j. Subtracting the summations and expanding the resulting difference by Taylor series and retaining only the first two components leads to
1.26
The portfolio yield can now be calculated:
The yield of a portfolio as calculated by (1.27) can be significantly different from the market value weighted yield in a non-flat yield curve environment.
The conventional duration of a portfolio also requires some adjustments in a non-flat yield curve environment. If M and D are the market value and duration of a portfolio respectively, and Dj and Xi are the duration and convexity of a security, then
1.28
Expanding the summation using Taylor series and keeping only the first two components leads to
For a portfolio of two zero coupon bonds, (1.29) simplifies to
Table 1.1 shows an example of the yield and duration of a portfolio of two zero coupon bonds. The market value weighted yield of the portfolio is 2.5 % (line 3) which is significantly below the actual yield (3.714 %). The actual yield is calculated by iteratively finding the yield that correctly reproduces the market value of the portfolio. The duration weighted yield on line 4 (3.7 %) is very close to the actual yield of the portfolio in line 5.
Table 1.1 Yield and duration of a portfolio
The correct method for calculating the duration of A + B as one security is by discounting each cash flow by its respective discount yield and then summing the contributions from all cash flows, resulting in a duration of 10 years. Calculating the duration of the combined securities by discounting all the cash flows by a yield of 3.714 % on line 5 will result in a duration measurement that is off by about 0.42 years from the correct duration. The weighted duration calculated from equation (1.30), shown on line 4, is very close to the market convention duration of the combined securities.
When we combine two securities into one, in a steep yield curve environment, the yield of the combined security will be lower than the yield of the longer duration security. Therefore, the market value of the longer duration security will be higher than if it was discounted by its correct yield. On the other hand, the market value of the shorter duration security will be lower than its actual market value, resulting in the market value of the combined security being equal to the sum of individual securities. In the combined security, the calculated duration is overestimated in a steep yield curve environment and underestimated in an inverted yield curve environment.
This example suggests that the duration of a 20-year coupon bond, in a steep yield curve environment, will be higher than the duration of a portfolio of zero coupon bonds with exactly the same cash flows. The correct method for calculating the duration of a portfolio is to discount all cash flows by their actual discount yield. In our example, the correct duration for the portfolio is the market value weighted average, which is equal to 10 years.
Since in a steep yield curve environment the durations of all bonds are overestimated, a portfolio manager who structures a portfolio using zero coupon bonds is likely to construct a portfolio that is longer than he desires for the respective index. Portfolios composed of coupon bonds will also overestimate the duration and the two overestimates offset each other. In 2012, the duration of the Barclays Treasury Bond Index was overestimated by about 0.09 years. This is a much larger uncertainty than most portfolio managers are willing to accept and is the extra duration that a portfolio manager who uses zero coupon bonds will have in his portfolio. For example, if the duration of the benchmark is 5.6 years and the portfolio manager desires to be long 0.20 years, he will target a duration of 5.8 years and rebalance at the end of the month when the index changes or when a transaction takes place to maintain 0.20 years' relative duration.
The mismatch in duration measurement can be significantly larger for credit securities, in particular high yield and emerging markets and at times when the spreads widen. We will see in Section
