b. Suppose that investors believe that Castles can make good on the promised coupon payments but that the company will go bankrupt when the bond matures and the principal comes due. The expectation is that investors will receive only 82% of face value at maturity. If they buy the bond today, what yield to maturity do they expect to receive?
Explanation
Some values below may show as rounded for display purposes, though unrounded numbers should be used for actual calculations.
Since the bonds were issued at par, the coupon rate had to match the yield to maturity at the time of issuance. Thus, the coupon rate is 7.4%.
a.
Since the bond has semiannual payments, the coupon payment, the interest rate, and the number of periods must all be expressed in semiannual terms:
| Bond price | = | PV of coupon payments + PV of face value |
| = | C × ((1 / r) – {1 / [r(1 + r)t]}) + FV / (1 + r)t | |
| = | [(.0740 × $1,000) / 2] × [[1 / (.1200 / 2)] – (1 / {(.1200 / 2)[1 + (.1200 / 2)](8 × 2)})] + $1,000 / [1 + (.1200 / 2)](8 × 2) | |
| = | $767.56 |
b.
| Bond price | = | PV of coupon payments + PV of face value |
| $767.56 | = | C × ((1 / r) – {1 / [r(1 + r)t]}) + FV / (1 + r)t |
| = | [(.0740 × $1,000) / 2] × [[1 / (r / 2)] – (1 / {(r / 2)[1 + (r / 2)](8 × 2)})] + (.8200 × $1,000) / [1 + (r / 2)](8 × 2) |
To compute r, use trial-and-error, a financial calculator, or a computer. See the calculator solution below.
Calculator computations:
a.
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Enter
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8 × 2
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12 / 2
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–74 / 2
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–1,000
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N
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I/Y
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PV
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PMT
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FV
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Solve for
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767.56
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b.
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Enter
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8 × 2
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12 / 2
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74 / 2
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820
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N
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I/Y
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PV
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PMT
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FV
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Solve for
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–768
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YTM = 5.11% × 2 = 10.21%
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