Question

What is the Macaulay duration of a 4-year annuity that makes annual payments of $1000 if the appropriate discount rate for this annuity is 8% and the first payment is made exactly 1 year from today?

a) 2.5 years
b) 3 years
c) 3.5 years
d) 4 years

Answer 1

The correct answer is option a. The Macaulay duration of a 4-year annuity with annual payments of $1000 and a discount rate of 8% is approximately 2.387 years, which means the closest given answer is 2.5 years (option a).

Step-by-step explanation:

The Macaulay duration of an annuity can be calculated by taking the sum of the present values of all cash flows, multiplied by the time until each payment is received, and then dividing by the total present value of the annuity. To find the present value (PV) of each cash flow, use the formula PV = C / (1 + r)^t, where C is the cash flow, r is the discount rate, and t is the time period.

For a 4-year annuity with annual payments of $1000 and a discount rate of 8%, the cash flows are $1000 at the end of each year for 4 years. Let's calculate the present value for each year and then the duration:

1. PV1 = $1,000 / (1 + 0.08)^1 = $925.93
2. PV2 = $1,000 / (1 + 0.08)^2 = $857.34
3. PV3 = $1,000 / (1 + 0.08)^3 = $793.46
4. PV4 = $1,000 / (1 + 0.08)^4 = $734.64

Total PV of annuity = PV1 + PV2 + PV3 + PV4 = $3,311.36

Now calculate duration:

Duration = (1*$925.93 + 2*$857.34 + 3*$793.46 + 4*$734.64) / $3,311.36 = $7,909.37 / $3,311.36 = 2.387 years

The closest answer to 2.387 years is 2.5 years, which would be answer (a).