1st-Order Macaulay Approximation

Study guide: Exam FM-2

The 1st-order Macaulay approximation estimates the price $P(i)$ of a bond or other cash-flow stream at a new yield $i$ from its price $P(i_0)$ and its Macaulay Duration $D_{Mac}$ at the current yield $i_0$ , by treating the whole stream as if it were a single payment due at time $D_{Mac}$ .

$P(i) \approx P(i_0)\left(\frac{1+i_0}{1+i}\right)^{D_{Mac}}$

$i_0$ and $i$ are effective rates per period, and $D_{Mac} = D_{Mac}(i_0)$ is measured in those same periods.
Where it comes from. With $\delta = \ln(1+i)$ the Force of Interest, $\frac{d \ln P}{d\delta} = -D_{Mac}$ . Holding that slope fixed gives $\ln P(i) \approx \ln P(i_0) - D_{Mac}\,(\delta - \delta_0)$ , and exponentiating gives the formula. It is therefore exact for a single cash flow (a zero-coupon bond) and approximate for anything with several payment dates.
Contrast with the 1st-Order Linear Approximation, which uses Modified Duration and is a straight tangent line: $P(i) \approx P(i_0)\left[1 - (i - i_0)\,D_{Mod}\right]$ . The two agree in level and slope at $i_0$ , but the Macaulay version is a curve, so it never goes negative and picks up part of the Convexity.
Ordering. For a stream of positive cash flows, at every yield: linear estimate $\le$ Macaulay estimate $\le$ exact price. Both first-order methods understate the price, and the Macaulay one is always the closer — whether yields rise or fall.
Exam use: given only $P(i_0)$ and $D_{Mac}$ , estimate the new price or present value of an asset or liability; used in Immunization and Duration Matching to compare how assets and liabilities move.

Repricing a Coupon Bond After a Rate Rise A 5-year $1,000 bond pays annual coupons of $60 and is priced at a yield of $i_0 = 5\%$ . Find its price and Macaulay duration, then estimate the price at $i = 7\%$ with the 1st-order Macaulay and 1st-order linear approximations, and compare with the exact price.
Answer Present values of the cash flows at 5%: $57.14,\ 54.42,\ 51.83,\ 49.36,\ 830.54$ .
 $\begin{align*} P(0.05) &= 1{,}043.29 \\ D_{Mac} &= \frac{1(57.14) + 2(54.42) + 3(51.83) + 4(49.36) + 5(830.54)}{1{,}043.29} \\ &= \frac{4{,}671.61}{1{,}043.29} \\ &= 4.4778 \end{align*}$ 
Macaulay approximation:
 $\begin{align*} P(0.07) &\approx 1{,}043.29\left(\frac{1.05}{1.07}\right)^{4.4778} \\ &= 1{,}043.29 \times 0.91898 \\ &= 958.76 \end{align*}$ 
Linear approximation, with $D_{Mod} = 4.4778/1.05 = 4.2645$ :
 $\begin{align*} P(0.07) &\approx 1{,}043.29\left[1 - (0.02)(4.2645)\right] \\ &= 954.31 \end{align*}$ 
The exact price at 7% is $959.00. The Macaulay estimate misses by $0.24, the linear one by $4.69 — both are below the true price, as they must be.

Present Value of a Claims Liability When Rates Move An insurer's run-off liabilities have a present value of $10,000,000 at an effective rate of 4% and a Macaulay duration of 6.2 years. Estimate the present value if the rate rises to 5%.
Answer
 $\begin{align*} PV(0.05) &\approx 10{,}000{,}000\left(\frac{1.04}{1.05}\right)^{6.2} \\ &= 9{,}423{,}952 \end{align*}$ 
The liability's value falls by about $576,000. The linear approximation, $10{,}000{,}000\,[1 - 0.01(6.2/1.04)] = 9{,}403{,}846$ , gives a slightly larger fall; the true value lies above both.