The (zero coupon) U.S. Treasury strip maturing in two years is selling at an ann

Question

The (zero coupon) U.S. Treasury strip maturing in two years is selling at an annualized yield to maturity of 3.60%, which is equivalent to a price of 93.272 % of par (face) value.  The (zero coupon) U.S. Treasury strip maturing in three years is priced at 88.389 % of par value, with an annualized yield to maturity of 4.2 %. The interest payments for coupon-paying bonds occur annually (once per year) at the end of each year and bond yields are quoted as annualized interest rates (i.e., don't worry about semiannual compounding). Assuming that the forward interest rate for a one-year period that begins in one year is 4.92 %,

a. determine the yield to maturity for a risk-free zero coupon bond having a par value of $1000 that matures in one year

b. determine the yield to maturity for a risk free US Government bond having 3 years to maturity, a par value of $1000 and a coupon rate of 12.5%

c. assuming that investors determine the prices of risky bonds by discounting their expected payoffs using the discount rate for risk-free U.S. Treasury Strips having the same time to maturity, and that the probability that there is 15% probabililty that Tesla Motors will default on its outstanding issue of zero coupon bonds having a par value of $1000 and one year to maturity, determine the promised yield for Tesla's one year zero coupon bonds.

Explanation / Answer

a. Let the yield to maturity for a risk-free zero coupon bond having a par value of $1000 that matures in one year be X%.

Then (1+X)*(1+ 1yr forward interest rate) = (1+ 2yr yield)*(1+ 2yr yield)

So (1+X)*(1+4.92%) = (1+3.6%)*(1+3.6%), which means X= 2.3%

b. The cashflows are $125 in year 1, $125 in year 2, $125 in year 3, and the par value of $1000 in year 3, i.e. cashflows of 125 in year 1, 125 in year 2, 1125 in year 3.

These need to be discounted at the appropriate zero coupon yields.

So bond price = 125/(1+2.3%) + 125/(1+3.6%)^2 + 1125/(1+4.2%)^3 = $1,233.03

If Yield to maturity for this bond is X, then 125/(1+X) + 125/(1+X)^2 + 1125/(1+X)^3 = $1,233.03

This leads to X=4.09%

c. Payment after 1 year = par value = 1000

Accounting for the 15% default probability, the expected payoff = (1-15%)*1000 = 850

1 year US Treasury yield (as determined in part a) = 2.3%

So bond price = 850/(1+2.3%) = 830.89

So yield = par value/bond price -1 = 1000/830.89 -1 = 20.35%

Hope this helped ! Let me know in case of any queries.

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