Par values of both the 1-Year zero and the 2-Year zero are $1,000. Suppose that

Question

Par values of both the 1-Year zero and the 2-Year zero are $1,000. Suppose that the short rate today is 6 percent and the expected short rate next year is 9 percent. A. Find the price of the 2-Year zero under the Expectations Hypothesis. B. Find the price at which the 2-Year zero can be sold after holding it for one year under the Expectations Hypothesis. C. Suppose a short term-investor (one that is interested only in 1-Year investments) is willing to pay $800 for the 2-Year zero. Find the expected holding period return given this price that the investor is willing to pay. D. Find the risk-premium implied by the expected holding period return. E. Find the yield to maturity of the 2-Year zero given the par value and the price that the investor demands. F. Find the forward rate implied by this yield to maturity. G. Find the Liquidity Premium resulting from this forward rate.

Explanation / Answer

Par Value of 1 Year Zero coupon bond = $1,000

Par Value of 2 Year Zero coupon bond = $1,000

Rate for year 1 = 6%

Expected Rates for Year 2 = 9%

a) Under the Expectation Hypothesis, the rate for the 2 Year coupon bond = (rate for year 1 + expected rate for year 2) / 2

So the rate for 2 year zero coupon bond = (6% + 9%) / 2 = 7.5% = 0.075

Now the present value of 2 year zero coupon bond

= Par value of 2 year zero coupon bond / (1+ rate of 2 year zero coupon bond) 2

= 1000 / (1+0.075)2

= $865.3326

b) Expected Rates for Year 2 = 9%

Par Value of 2 Year Zero coupon bond = $1,000

So the expected price of 2 Year Zero coupon bond after one year

= Par value of 2 year zero coupon bond / (1+ Expected Rates for Year 2)

= 1000 / (1+ 0.09)

= $917.4312

c) The investor holding period = 1 year

Present value of 2 year zero coupon bond = $800

Par value of 2 year zero coupon bond = $1,000

Under the Expectation Hypothesis, the rate for the 2 Year coupon bond = (rate for year 1 + expected rate for year 2) / 2

So the rate for 2 year zero coupon bond = (6% + 9%) / 2 = 7.5% = 0.075

The return of holding period = Present value of 2 year zero coupon bond x (1 + rate for 2 year zero coupon bond)

                                                          = 800 x (1+0.075) = 860

d) Risk premium = Rate of expected return of 2 year zero coupon bond – rate of current 1 year bond

                                      = 7.5 % - 6% = 2.5%

e)

Yield to maturity = y

Par value of 2 year zero coupon bond = $1,000

Price of 2 year zero coupon bond = $800

So Par value of 2 year zero coupon bond = Price of 2 year zero coupon bond x (1 + y) 2

=> 1000 = 800 x (1 + y) 2

=> (1+y) 2 = 1000/800

=> 1 + y = (1.25)1/2

=> y = 1.1180 – 1

=> y = 0.1180 = 11.80%

f) (1 + Yield to maturity) = (1 + 1 year rate) x (1 + forward rate)

=> (1 + 0.1180) = (1 + 0.06) x (1 + Forward rate)

=> (1 + Forward rate) = 1.1180 / 1.06

=> Forward rate = 1.054717 – 1

=> Forward rate = 0.054717 = 5.47%

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