Consider a bond with maturity 3 years, par value $1000, and annual coupon rate 8%. If its yield to maturity is 5%, its duration, expressed in years, is equal to

Question

BlackTom AI · Accepted Answer

We start by outlining the bond details and the cash flows to be used in duration calculations.
- Par value: 1000
- Annual coupon rate: 8% of par, so annual coupon = 80
- Maturity: 3 years
- Yield to maturity (discount rate): 5%

Compute the present value of each cash flow to obtain the price and the weighted present value for duration.
- Discount factor for year 1: 1 / 1.05 = 0.95238
- Discount factor for year 2: (1 / 1.05)^2 = 0.90703
- Discount factor for year 3: (1 / 1.05)^3 = 0.86384

Present value of cash flows:
- Year 1 coupon: 80 × 0.95238 = 76.1904
- Year 2 coupon: 80 × 0.90703 = 72.5623
- Year 3 coupon plus principal: (80 + 1000) × 0.86384 = 1080 × 0.86384 = 932.9434
- Bond price (sum of PVs): 76.1904 + 72.5623 + 932.9434 = 1081.6961 (approx)

Next, compute the Macaulay duration, which is the weighted average of the times to cash flows, using the PVs as weights:
- Numerator: 1 × 76.1904 + 2 × 72.5623 + 3 × 932.9434
             = 76.1904 + 145.1246 + 2,798.8302
             = 3,020.1452 (approx)
- Denominator: bond price = 1081.6961 (approx)

Duration (in years) = 3,020.1452 / 1081.6961 ≈ 2.79 years.

Now we review the answer options:
- 2.96: This is too high relative to the computed weighted-average timing of cash flows; it would require a larger concentration of value in earlier cash flows or a lower price. The calculated value is clearly closer to 2.79, not 2.96.
- 2.46: This underestimates the average timing of cash flows. Given three years of coupons plus principal and the discounting at 5%, the duration should be around 2.8, not as low as 2.46.
- 2.79: This matches the computed Macaulay duration from the cash-flow timing and present values above. The arithmetic aligns with the price and weighted cash-flow timing at a 5% yield.
- 2.90: This is slightly above the computed value; while close, it does not align with the precise calculation shown, which yields about 2.79.

Summary reasoning: by discounting each cash flow at the yield, summing the PVs gives the price, and weighting each PV by its year yields the Macaulay duration. The resulting value is approximately 2.79 years, which best matches option 2.79 and diverges from the other options due to the precise balance of earlier and later cash flows at a 5% yield.

COMM_V 371 101-107 2025W1 Practice Quiz 4

Consider a bond with maturity 3 years, par value $1000, and annual coupon rate 8%. If its yield to maturity is 5%, its duration, expressed in years, is equal to

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