Suppose that you purchase a bond that matures in four years and pays a 13% coupon rate annually. The bond is priced to yield 10%. Assume the par value of $1,000. What is Macaulay’s Duration? If the market yield decreases by 50 basis points, what would be a percent change in the bond price?

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Question

Suppose that you purchase a bond that matures in four years and pays a 13% coupon rate annually.  The bond is priced to yield 10%.  Assume the par value of $1,000.  What is Macaulay’s Duration?  If the market yield decreases by 50 basis points, what would be a percent change in the bond price?

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BlackTom AI · Accepted Answer

We begin by identifying the bond’s cash flows and the given yield to maturity, then compute Macaulay duration and the approximate price change for a yield shift.

First, lay out the bond details: a four-year bond with a 13% annual coupon on a $1,000 par value, so each year t=1 to t=4 pays a coupon of 130, and at t=4 you receive 1,000 par plus the final coupon, i.e., 1,130. The yield to maturity is 10% (0.10).

Now compute the present value of each cash flow to obtain price and the Macaulay duration numerator:
- PV of CF1 = 130 / (1.10)^1 = 118.18
- PV of CF2 = 130 / (1.10)^2 = 107.44
- PV of CF3 = 130 / (1.10)^3 = 97.63
- PV of CF4 = 1,130 / (1.10)^4 = 772.69
Sum of PVs (price) ≈ 118.18 + 107.44 + 97.63 + 772.69 = 1,095.94

Next, compute the Macaulay duration numerator by multiplying each PV of cash flow by the time period t and summing:
- t=1: 1 × 118.18 = 118.18
- t=2: 2 × 107.44 = 214.88
- t=3: 3 × 97.63 = 292.89
- t=4: 4 × 772.69 = 3,090.76
Sum = 118.18 + 214.88 + 292.89 + 3,090.76 ≈ 3,716.71

Macaulay duration D_M ≈ 3,716.71 / 1,095.94 ≈ 3.39 years. This matches the 3.39-year figure across the options.

Now address the second part: a 50 basis point drop in yield (Δy = -0.005). For small yield changes, the approximate percentage price change is given by: ΔP / P ≈ -MD × Δy, where MD is the modified duration, MD = D_M / (1 + y).
Compute MD: MD = 3.39 / (1 + 0.10) ≈ 3.39 / 1.10 ≈ 3.082... (about 3.08).
Then ΔP / P ≈ -3.08 × (-0.005) ≈ +0.0154, i.e., +1.54%.
So, a fall in yield by 50 bps should increase the price by roughly 1.54%.

Now evaluate each answer option one by one:
- Option A: 3.32 years; -3.01%. The duration, 3.32 years, is a bit off from the calculated ~3.39 years, and the price change sign is incorrect given a yield decrease; a yield drop would not produce a negative price change of about 3%. This option is therefore inconsistent both in duration and in the direction/size of price change.
- Option B: 3.54 years; -2.06%. The duration 3.54 years is higher than the computed 3.39 years, suggesting a longer time to cash flow recovery than actual. The price change is negative, which would imply price falls when yield falls, a clear contradiction to standard convex/duration behavior for typical fixed-rate bonds. This makes it incorrect.
- Option C: 3.39 years; +1.54%. The duration matches the calculated Macaulay duration of about 3.39 years. The price change is positive, matching the expectation that a decrease in yield increases price, with magnitude around 1.54% using the duration approximation.
- Option D: 3.32 years; +3.01%. The duration is off (3.32 vs ~3.39), and the magnitude of the price change (about +3.01%) would require a much larger yield move or a different sensitivity than computed; this does not align with the calculated figures.
- Option E: 3.39 years; -1.54%. The duration equals the computed Macaulay duration, but the price change sign is wrong for a yield decrease; it implies price would fall when yields drop, which is inconsistent with standard bond behavior.

In summary, among the provided choices, the combination of a duration of 3.39 years with a positive price change of about 1.54% best aligns with the calculations for Macaulay duration and the estimated price impact of a 50-basis-point decrease in yield. The other options either misstate the duration, misstate the price-change direction, or both, reflecting common misconceptions about duration, convexity, and the relationship between yields and bond prices.

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