Consider a security with annual cash flows of $100,000 and a time to maturity of 9 years. The current interest rate is 9% p.a. What is the security’s duration?
考虑一种年现金流为 100,000 美元、到期时间为 9 年的证券。目前的利率为每年 9%。证券的期限是多久？

Question

BlackTom AI · Accepted Answer

We’re given a security with annual cash flows of 100,000 for 9 years, and the current interest rate is 9% per year. We need its duration, specifically the Macaulay duration, which for a level annuity can be computed as the present-value-weighted average of the payment times.

First, compute the discount factors and present values for each cash flow at i = 9%:
- v = 1 / 1.09 ≈ 0.917431
- PV of each 100,000 payment at time t is 100,000 * v^t
We’ll also need the sum of t * (PV of payment at time t) across t = 1..9 to get the duration numerator.

Option-by-option analysis:
Option A: 4.43
This option corresponds to the correct Macaulay duration for a level 9-year annuity with a 9% yield. When you sum t * 100,000 * v^t for t = 1 to 9 and divide by the total present value (sum of 100,000 * v^t for t = 1..9), you obtain approximately 4.43 years. The numeric sums align with D = (sum t*v^t) / (sum v^t) times 100,000, yielding about 4.43.
Option B: 4.23
This value is somewhat close but underestimates the true duration. It would imply a slightly smaller weighted average of the payment times, which would happen if you undercounted some of the later cash flows’ present value or used an approximate shortcut that undervalues later payments. The exact calculation with v^t terms shows a higher duration around 4.43.
Option C: 4.53
This is a bit high relative to the precise computation. If you overemphasize later cash flows or miscompute the weighted sum, you might land near 4.53, but the exact sum of t*v^t vs. sum v^t for t = 1..9 at i = 9% yields about 4.43 rather than 4.53.
Option D: 4.33
This option is plausible but still slightly off from the precise result. It suggests a similar but not identical balance between present values of earlier vs. later payments. The correct ratio from the exact sums lands at roughly 4.43, not 4.33.

Putting it together, the accurate Macaulay duration is obtained by:
- Calculating v = 1/1.09 and the powers v^t for t = 1..9
- Computing the present value sum: PV = 100,000 * sum_{t=1}^9 v^t
- Computing the duration numerator: N = 100,000 * sum_{t=1}^9 t * v^t
- Then D = N / PV
With the numbers above, D ≈ 4.43 years.

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