A ten-year, zero-coupon bond with a yield to maturity of 4% has a face value of $1000. An investor purchases the bond when it is initially traded, and then sells it four years later. What is the rate of return of this investment, assuming the yield to maturity does not change?

Question

BlackTom AI · Accepted Answer

{
  "reasoning": "We are given a ten-year zero-coupon bond with a yield to maturity (YTM) of 4% and face value $1000. The investor buys when first issued and holds for 4 years, assuming YTM stays at 4% throughout.

First, recall how zero-coupon bonds are priced: price = face value / (1 + YTM)^{years to maturity}.
- At issuance (time 0), the bond has 10 years to maturity, so the purchase price P0 = 1000 / (1.04)^{10}.
- After holding for 4 years, the bond has 6 years left to maturity, so the sale price P4 = 1000 / (1.04)^{6}.

The holding-period return (HPR) over the 4-year interval is the total return from P0 to P4: HPR_total = (P4 / P0) − 1.
Compute the ratio: P4 / P0 = [1000 / (1.04)^{6}] / [1000 / (1.04)^{10}] = (1.04)^{10} / (1.04)^{6} = (1.04)^{4}.
Thus HPR_total = (1.04)^{4} − 1 ≈ 1.16986 − 1 = 0.16986, i.e., about 16.99% over the 4-year holding period.

To express the investment’s return in a common annual metric, convert the total return to an annualized rate: annualized return = (1 + HPR_total)^{1/4} − 1 = (1.04)^{1} − 1 ≈ 0.04, i.e., about 4.0% per year.

Now evaluate the multiple-choice options:
- 2.40%: This is far lower than the annualized return derived from the math above; it would imply a much smaller overall growth than the 16.99% total gain over 4 years, so this is incorrect.
- 4.00%: Matches the calculated annualized return of about 4.0% per year. This aligns with the idea that a constant 4% YTM implies a 4% per-year return when held over any horizon, adjusted for the changing time to maturity.
- 2.00%: This underestimates the return; the math shows a higher annualized rate, so this is incorrect.
- 3.20%: This is also not supported by the calculation; the exact annualized value from the given YTM is 4%, not 3.2%.

In summary, the correct interpretation yields an annualized return of about 4.0% per year over the 4-year holding period, given the YTM remains at 4% throughout." }

A ten-year, zero-coupon bond with a yield to maturity of 4% has a face value of $1000. An investor purchases the bond when it is initially traded, and then sells it four years later. What is the rate of return of this investment, assuming the yield to maturity does not change?单项选择题

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