On January 1st, 2005, Borrower Limited sold a 10%, five-year, $100 million bond when bonds of equivalent risk and maturity were yielding 5% annually. Interest payments are made on January 1st next year.

What is the Price of the Bond?

Question

On January 1st, 2005, Borrower Limited sold a 10%, five-year, $100 million bond when bonds of equivalent risk and maturity were yielding 5% annually. Interest payments are made on January 1st next year.

What is the Price of the Bond?

BlackTom AI · Accepted Answer

Question restatement: On January 1st, 2005, Borrower Limited sold a 10%, five-year, $100 million bond when bonds of equivalent risk and maturity were yielding 5% annually. Interest payments are made on January 1st next year. What is the Price of the Bond?

Option analysis step-by-step:
- First, identify the cash flows. The bond has a annual coupon rate of 10% on a $100 million face value, so annual coupon payment = 10% × $100,000,000 = $10,000,000. There are 5 years of coupons (years 1 through 5) and a redemption of $100,000,000 at the end of year 5.
- The market yield (discount rate) is 5% per year. To price the bond, we discount each coupon and the redemption at this yield.
- Present value of the coupons: The coupons form an ordinary annuity of 5 payments of $10,000,000. The present value factor for an ordinary annuity at i = 5% for n = 5 is a-angle-i-n = (1 − (1 + i)^(−n)) / i. Numerically, (1 − 1/(1.05)^5) / 0.05 ≈ (1 − 0.783526) / 0.05 ≈ 0.216474 / 0.05 ≈ 4.32948. Multiply by the annual coupon: 4.32948 × $10,000,000 ≈ $43,294,800.
- Present value of the redemption: The $100,000,000 redeemed at year 5 is discounted by (1 + i)^5 = (1.05)^5 ≈ 1.27628156. Therefore PV = $100,000,000 / 1.27628156 ≈ $78,352,640.
- Sum the present values: $43,294,800 + $78,352,640 ≈ $121,647,440. This aligns with the provided numeric result of about $121.6478 million.

Explanation of why this is the correct approach:
- The bond price equals the sum of the present values of all expected cash flows (coupons and redemption) discounted at the yield demanded by the market for bonds of similar risk and maturity.
- The stated coupon rate (10%) yields $10 million per year, which is consistent with a fixed-rate bond on a $100 million face value. Discounting at the market rate (5%) correctly reflects the investor’s required return.

Why the other (non-provided) options would be incorrect: (Note: there are no alternative options listed here, but generally the typical mistakes would include:)
- Using a different number of coupon payments (e.g., assuming only 4 coupons instead of 5) would understate the price because you’d be missing the final cash flow.
- Discounting all cash flows at a rate different from 5% (e.g., using 4% or 6%) would yield a price inconsistent with the market yield; if the rate is lower than 5%, the price would be too high, and if higher, too low.
- Treating the coupon as a perpetuity or miscomputing the annuity factor would lead to incorrect PV of coupons.
- Failing to include the redemption amount in year 5 would omit a substantial portion of the value, underpricing the bond.

Final numerical check (optional cross-verify): PV of coupons ≈ 10,000,000 × 4.32948 ≈ 43,294,800; PV of redemption ≈ 100,000,000 / 1.27628156 ≈ 78,352,640; total ≈ 121,647,440, matching the target price ~121.6478 million.

COMM_V 293 101 102 103 2025W1 Practice Quiz: Using Excel in Lockdown Browser

On January 1st, 2005, Borrower Limited sold a 10%, five-year, $100 million bond when bonds of equivalent risk and maturity were yielding 5% annually. Interest payments are made on January 1st next year. What is the Price of the Bond?

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