###### Matrix Pricing

When fixed-rate bonds are not actively traded, or there is no market price... **Read More**

Like equities, bonds are typically held in a portfolio. Therefore, bond portfolio managers need to measure the whole portfolio duration.

There are two methods for calculating the duration of a bond portfolio:

- the weighted average of time to receipt of the aggregate cash flows; and
- the weighted average of the individual bond durations comprising the portfolio.

The first method is the theoretically correct approach which is, nevertheless, practically hard to implement. The second method is commonly used by fixed-income portfolio managers but has its own limitations.

Assume that an investor holds the following portfolio of two zero-coupon bonds.

$$

\begin{array}{c|c|c|c|c|c|c|c|c}

\textbf{Bond} & \textbf{Maturity} & \textbf{Price} & \textbf{Yield} & \textbf{Mac. Dur} & \textbf{Mod. Dur} & \textbf{Par Value} & \textbf{Market Value} & \textbf{Weight} \\ \hline

\text{X} & \text{1 year} & \text{98.00} & \text{2.0408%} & \text{1.000} & \text{0.980} & \text{\$1,000,000} & \text{\$980,000} & \text{16.81%} \\

\text{Y} & \text{2 years} & \text{97.00} & \text{1.5346%} & \text{2.000} & \text{0.984} & \text{\$5,000,000} & \text{\$4,850,000} & \text{83.19%} \\

\end{array}

$$

The prices are per 100 of par value. The yields-to-maturity are effective annual rates. The total market value for the portfolio is 5,830,000. The portfolio is unevenly weighted in terms of market value between the 2 bonds (with weights of 16.81% and 83.19% for X and Y, respectively).

The first approach views the portfolio as a series of aggregated cash flows, in which case the aggregate cash flow yield is 12.366% (the solution to the following equation).

$$5,830,000=\frac{1,000,000}{(1+r)^1} +\frac{5,000,000}{(1+r)^2}; \quad r = 1.5811\%$$

There are just 2 future cash flows in the portfolio, which are the redemption of the principal of the 2 zero-coupon bonds. However, in more complex portfolios, a series of coupon and principal payments may occur on various dates.

Using a discount rate of 1.5811%, we can, therefore, find the portfolio’s Macaulay duration as the weighted average of time to receipt of aggregated cash flow.

$$MacDur=\frac{\frac{1×1,000,000}{1.015811^1} +\frac{2×5,000,000}{1.015811^2}}{\frac{1,000,000}{1.015811^1}+\frac{5,000,000}{1.015811^2 }}=\frac{8,810,040}{4,850,000}=1.8311$$

The modified duration for the portfolio is 1.8026

$$ModDur=\frac{1.8165}{1+0.12366}=1.8026$$

The modified duration indicates the percentage change in the market value given a change in the cash flow yield. For example, if the cash flow yield increases or decreases by 100 bps, the portfolio’s market value is expected to increase or decrease by about 1.8026%.

Although this approach is theoretically correct, it is difficult to use in practice. To begin with, the cash flow yield is not commonly calculated for bond portfolios. Aside from this, the amount and timing of future cash payments are uncertain. Lastly, interest rate risk is not usually expressed as a change in benchmark interest rates. In practice, the second approach to portfolio duration is commonly used, and there is a stronger likelihood you will have to use the second formula in your CFA level 1 exam.

Using this approach:

$$\text{Portfolio duration} =w_1 D_1+w_2 D_2+⋯+w_n D_n$$

Where:

*w _{i}* = market value of bond

*D _{i}* = duration of bond

*n* = number of bonds in portfolio

Using the above data, we can calculate portfolio duration as follows:

$$\text{Portfolio Macaulay duration} =0.1681×1+0.8319×2=1.8319$$

$$\text{Modified duration} =\frac{1.8319}{1+0.12366}=1.6303$$

One of the advantages of this approach is that it can be an effective measure of interest rate risk. For instance, if the yields-to-maturity on the bonds in the portfolio increase by 100 bps, the estimated drop in the portfolio value is 1.6303%. However, its main disadvantage is that it assumes a parallel shift in the yield curve where all rates change by the same amount in the same direction. As is usually observed, interest rates change in an irregular manner, resulting in a steeper or flatter yield curve.

Question 1When using the weighted average of time to receipt of the aggregate cash flows method (method 1), you would

most likelyuse:

- The price of the bonds calculate the weights.
- The par value of the bonds calculate the weights.
- The market value of the bonds calculate the weights.

SolutionThe correct answer is

B.According to method 1, you would use the par values of the bonds to come up with the weights used to calculate the aggregated cash flow yield. After that, you would compute the Macaulay duration and finally, the modified duration.

Question 2A bond portfolio consists of the following three fixed-rate bonds. Assume annual coupon payments and no accrued interest on the bonds. Prices are per 100 of par value.

$$

\begin{array}{c|c|c|c|c|c|c}

\textbf{Bond} & \textbf{Maturity} & \textbf{Price} & \textbf{Yield} & \textbf{Coupon} & \textbf{Modified Duration} & \textbf{Market Value} \\

\hline

\text{1} & \text{6 years} & \text{85.00} & \text{4.95%} & \text{2.00%} & \text{5.42} & \text{170,000} \\

\text{2} & \text{10 years} & \text{80.00} & \text{4.99%} & \text{2.40%} & \text{8.44} & \text{850,000} \\

\text{3} & \text{9 years} & \text{85.78} & \text{5.00%} & \text{3.00%} & \text{7.54} & \text{180,000} \\

\end{array}

$$The bond portfolio’s modified duration is

closestto:

- 6.55
- 7.54
- 7.88

SolutionThe correct answer is

C.The portfolio’s modified duration is closest to 7.62. Portfolio duration is commonly estimated as the market-value-weighted average of the yield durations of the individual bonds in the portfolio.

The total market value of the bond portfolio is 170,000 + 850,000 + 180,000 = 1,200,000.

The portfolio duration is 5.42 × (170,000/1,200,000) + 8.44 × (850,000/1,200,000) + 7.54 × (180,000/1,200,000) = 7.88