Recall the approximate formulas for change in present value when interest rate changes.

In terms of the “modified” quantities M(i) = −P′(i)/P(i) , C(i) = P′′(i)/P(i) there is a straightforward Taylor approximation

1 2 P(i)≈P(i0) 1−M(i0)(i−i0)+2C(i0)(i−i0) .

(a)(10 points) If the cash flow is a single payment of F = $1000.00 at time t = 10, then P(i)=$1000(1+i)−10. Leti0 =10%=.1andcalculateP(.1),M(.1),C(.1).

(b)(10 points) Use the quantities you just calculated in (a) to calculate the first and second order approximations to P(.09). Compare with the exact P(.09) = $1000(1.09)−10.

In terms of the Macaulay duration D there’s a more accurate first order formula P(i) = P(i0)(1 + i0)D

(1+i)D

. In fact this is exact if the cash flow consists of a single payment at future time t = D.

But what if the cash flow consists of two payments (the next simplest case)? For simplicity assume a payment of $500 / (1.1) at time t = 9 and $500(1.1) at t = 11. I’m happy to tell you that P(.1) is the same as you calculated in part (a), and the Macaulay duration D = 10.

(c)(20 points) Calculate P(.09) exactly, and the Macaulay first order approximation. How close are they? (You may have to keep a lot of digits of accuracy!)