Question

Graham is the beneficiary of an annuity due. At an annual effective interest rate of 5%, the present value of payments is 123,000. Tyler uses the first-order Macaulay approximation to estimate the present value of Graham’s annuity due at an annual effective interest rate 5.4%. Tyler estimates the present value to be 121,212. Calculate the modified duration of Graham’s annuity at 5%.

Answer 1

Present Value (PV) at an yield of 5 % = P1 = $ 123000, Present Value (PV) at an yield of 5.4 % = P2 = $ 121212, Initial Interest Rate = 5 % and Final Interest Rate = 5.4 %

Let the Macaulay's Duration be M

Therefore, as per first order Macaulay's Approximation, we get:

P2 = P1 x [(1+0.05) / (1+0.054)]^(M)

(121212/123000) = 0.9962^(M)

0.98546 = (0.9962)^(M)

M = = 3.85118

logo.99620.98546

Modified Duration at 5 % = M / (1+Yield) = 3.85118 / (1.05) = 3.66779 ~ 3.67 years