#### Subject 4. Bond Portfolio Duration

There are two ways to calculate the duration of a bond portfolio:

• The weighted average of the time to receipt of aggregate cash flows. This method is based on the cash flow yield, which is the internal rate of return on the aggregate cash flows.

Limitations: This method cannot be used for bonds with embedded options or for floating-rate notes due to uncertain future cash flows. The cash flow yield is not commonly calculated. The change in cash flow yield is not necessarily the same as the change in the yields-to-maturity on the individual bonds. Interest rate risk is not usually expressed as a change in the cash flow yield.

• The weighted average of the durations of individual bonds that compose the portfolio. The weight is the proportion of the portfolio that a bond comprises.

Portfolio Duration = w1D1 + w2D2 + w3D3 + ... + wkDk

wi = the market value of bond i / market value of the portfolio
Di = the duration of bond i
k = the number of bonds in the portfolio

This method is simpler to use and quite accurate when the yield curve is flat. Its main limitation is that it assumes a parallel shift in the yield curve.

To illustrate this calculation, consider the following three-bond portfolio in which all three bonds are option-free:

• 10% 5-year 100.0000 10 \$4 million \$4,000,000 3.861
• 8% 15-year 84.6275 10 \$5 million \$4,231,375 8.047
• 14% 30-year 137.8586 10 \$1 million \$1,378,586 9.168

In this illustration, it is assumed that the next coupon payment for each bond is exactly six months from now (i.e., there is no accrued interest). The market value for the portfolio is \$9,609,961. Since each bond is option-free, modified duration can be used.

• w1 = \$4,000,000/\$9,609,961 = 0.416, D1 = 3.861
• w2 = \$4,231,375/\$9,609,961 = 0.440, D2 = 8.047
• w3 = \$1,378,586/\$9,609,961 = 0.144, D3 = 9.168

The portfolio's duration is: 0.416 (3.861) + 0.440 (8.047) + 0.144 (9.168) = 6.47.

A portfolio duration of 6.47 means that for a 100 basis point change in the yield for each of the three bonds, the market value of the portfolio will change by approximately 6.47%. Keep in mind that the yield for each of the three bonds must change by 100 basis points for the duration measure to be useful. This is a critical assumption and its importance cannot be overemphasized.