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The following problem is what I am working on and I cannot solve it.

Under the current market conditions Bond 1 has a price (per 100 of face amount) of $P_1=88.35$ and a Macaulay duration of $D_1=12.7$, and Bond 2 has a price (per 100 of face amount) of $P_2=130.49$ and Macaulay duration of $D_2=14.6$. A portfolio is created with a combination of face amount $F_1$ of Bond 1 and face amount $F_2$ of Bond 2. The combined face amount of the portfolio is $F_1+F_2=100$ and the Macaulay duration of the portfolio is $D_0=13.5$. Find the portfolio value.

There are two things that I am not understanding.

a), What is a portfolio value?

I am thinking that the portfolio value represents the present value of the combined present value of Bond 1 and Bond 2 where $F_1 \ \text{and} \ F_2$ is how much of each bond have been purchased. From the problem I assumed that the coupon rate, interest rate etc. is irrelevant.

b), How do we set up an eqn.?

I understand that given the present values of each cash flows and their duration, the duration of the portfolio can be found as

$$\frac{P_1D_1+P_2D_2}{P_1+P_2}=D$$

where in this case, $F_1=F_2=100$.

but of course $F_1 \ne F_2$ and the total present value of the portfolio (which I think we are looking for and I will denote it as $P$) I am not quite sure how to set up an equation without using extra variables.

I have a feeling that this problem should not be as complicated as I am making it, but I am stuck. Can I ask for some help?

Thank you.

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As stated, $P_1$ and $P_2$ are bond prices per $100$ of face amount. The amount invested in each bond, given face amounts $F_1$ and $F_2$, is

$$V_1 = F_1\frac{P_1}{100}, \\V_2 = F_2\frac{P_2}{100}.$$

The portfolio value is the total amount invested:

$$V = V_1 + V_2 = F_1\frac{P_1}{100} + F_2\frac{P_2}{100}.$$

In order to find the portfolio value, we must find $F_1$ and $F_2$. We are given that the combined face amount is $100$. Hence, we have one equation

$$F_1 + F_2 = 100. \tag{1}$$

The second equation for $F_1$ and $F_2$ comes from the specification of portfolio duration, $P_0 = 13.5$

The Macaulay duration of a bond is a weighted average term of the cash flows. The weight of the term of each cash flow is the present value of the cash flow divided by the price of the bond.

For Bonds $1$ and $2$, the sums of each cash flow term times the discounted cash flow per $100$ face amount are $P_1D_1$ and $P_2D_2$, respectively. The corresponding sums for actual cash flows, given face amounts $F_1$ and $F_2$, are $F_1P_1D_1/100$ and $F_2P_2D_2/100$, respectively.

For the portfolio, the combined cash flow stream is the sum of the streams for each bond, and the portfolio Macaulay duration is

$$D_0 = \frac{F_1P_1D_1/100 + F_2P_2D_2/100}{F_1P_1/100 + F_2P_2/100}.$$

Whence,

$$\frac{F_1P_1D_1 + F_2P_2D_2}{F_1P_1 + F_2P_2}= D_0.$$

Rearranging this equation, we get

$$F_1P_1D_1 + F_2P_2D_2 = F_1P_1D_0 + F_2P_2D_0,$$

and,

$$F_2 = \frac{P_1(D_0-D_1) }{P_2(D_2-D_0)}F_1. \tag{2}$$

Solving (1) and (2) simultaneously we get

$$F_1 = 67.01, F_2 = 32.99.$$

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  • $\begingroup$ That was very helpful! Thank you! $\endgroup$
    – hyg17
    Dec 18 '14 at 8:43
  • $\begingroup$ You're welcome. In short, $(V_1D_1 +V_2D_2)/(V_1 + V_2) = D_0$. The duration of a bond portfolio is always the weighted average of the durations -- where the each weight is the fraction of wealth invested in the corresponding bond. $\endgroup$
    – RRL
    Dec 18 '14 at 9:04

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