Q-10-40I
Expert-verifiedQuestion: The current yield curve for default-free zero-coupon bonds is as follows:
Maturity (Years) | YTM |
1 | 10% |
2 | 11% |
3 | 12% |
a. What are the implied one-year forward rates?
b. Assume that the pure expectations hypothesis of the term structure is correct. If market expectations are accurate, what will the pure yield curve (that is, the yields to maturity on one- and two-year zero-coupon bonds) be next year?
c. If you purchase a two-year zero-coupon bond now, what is the expected total rate of return over the next year? What if you purchase a three-year zero-coupon bond?
(Hint: Compute the current and expected future prices.) Ignore taxes.
Answer
a. 12.01%, 14.03%
b. 13.02%
c. Two year zero coupon bond = 10.00% and three year zero coupon bond = 10.00%
Price for year 1 = 1000/ (1 + 10%) = $909.09
Price for year 2 = 1000/ (1 + 11%)^{2} = $811.62
Forward rate for year 2 = (1 + 11%)^{2 }/ (1 + 10%) – 1 = 0.1201 = 12.01%
Price for year 3 = 1000/ (1 + 12%)^{3} = $711.78
Forward rate for year 3 = (1 + 12%)^{3 }/ (1 + 11%)^{2} – 1 = 0.1403 = 14.03%
Therefore,
Maturity Years | YTM | Forward rates | Price |
1 | 10.00% | $909.09 | |
2 | 11.00% | 12.01% | $811.62 |
3 | 12.00% | 14.03% | $711.78 |
For this, we discount each zero’s face value at the forward rates derived in the previous part, as below:
Maturity Years | Price | Forward rates | YTM |
1 | $892.78 | 1000/1.1201 | 12.01% |
2 | $782.93 | 1000/1.1201x1.1403 | 13.02% |
According to the expectation hypothesis, this year’s upward sloping curve implies a shift upward in next year’s curve
Next year, the two-year zero will be a one-year zero, and it will therefore sell at: $1000/1.1201 = $892.78
Similarly, the current three-year zero will be a two-year zero, and it will sell for: $782.93
Expected total rate of return:
Two year bond = $892.78/$811.62 – 1 = 0.1000 = 10.00%
Three year bond = $782.93 / $711.78 – 1 = 0.1000 = 10.00%
A convertible bond has the following features:
Coupon | 5.25% |
Maturity | June 15, 2020 |
Market price of bond | $77.50 |
Market price of underlying common stock | $28.00 |
Annual Dividend | $1.20 |
Conversion ratio | 20.83 shares |
Calculate the conversion premium for this bond.
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