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Homework answers / question archive / We see the following yield curve for discount, or zero-coupon, bonds: Maturity Yield to maturity 1 year 69% 2 years 79% 3 years (a) If the price for a four year annuity paying $10o per year is $334
We see the following yield curve for discount, or zero-coupon, bonds: Maturity Yield to maturity 1 year 69% 2 years 79% 3 years (a) If the price for a four year annuity paying $10o per year is $334.57, what is the yield to maturity on a four year zero-coupon bond? (b) Suppose that you wish to lock in now (at time o) a loan rate for a loan in the amount of $20 million in year 2 to be repaid in year 3 (cash received in year 2 and interest plus principal returned in year 3). What pattern of discount bond holdings could you use to construct this loan now? That is, you will be buying and selling different bonds. How many bonds of what maturity will you buy? How many bonds of what maturity will you sell ? (c) What is the implied forward rate between year 2 and year 3 (f.,)? Problem 2. You are an investment manager for Lemon County. You are given the following information about the yields of U.S. Treasury strips (which are zero-coupon, or pure discount, bonds which pay $10o at maturity): y, = 390, y, = 49%, y., = 6.190, and y,, = 6.09, where y, is the yield-to-maturity on a j-year strip. Assume that all cash flows are riskless and we can borrow and lend at the stated rates. You buy so, ooo yo-year strips and partially finance these by issuing 5, 000 2-year strips (i.e., you are borrowing money by promising to pay $500, 000 = 5, 000x$100 in two years). (a) How much of Lemon County's money are you investing (i.e., what is the net investment)? (b) Suppose that one second after you make the above investments, the Federal Reserve Board announces that it is taking steps to raise interest rates. Assume that the yields on all of the strips immediately increase by one percentage point (i.e., y, = 40, y. = 590, y. = 7.190, and y = 7.0 %). By how much (in dollars and in percentage terms) does the value of the county's net investment change due to the unexpected announcement? Problem 3. A one-year zero-coupon bond with face value of sooo has a price of $97.50. A two-year coupon bond with annual coupon payments, a face of $1,ooo, a coupon rate of 6% and a yield to maturity of 4%. (a) What is the price of the two-year coupon bond?
answer is below, please refer below
Step-by-step explanation
Problem #1
(a)
The price of the annuity is given by:
P= 100/(1+y1)1+100/(1+y2)2+100/(1+Y3)3+100/(1+Y4)4
Then, since P = 334.57 and using the data from the table, we get
Y4 = 7.9%
(b)
Since you want to receive 20M in t = 2 , you will buy 200,000 2-years zero- coupon bonds today. The price of a 2-years zero-coupon bond is equal to
P2 = 100 /(1 + y 2 ) 2
P2 = 87.34
(c)
You want your cash flow today to be zero .You also want to replicate a loan that is repaid in year 3. Therefore you will also sell 3-years zero-coupon bonds today. Define by x the amount of 3-years zero-coupon bonds that you sell. The price of a 3-years zero-coupon bond is equal to
P3 = 100/ (1 + y 3 ) 3
= 79.38
In order to have your cash flow equal to zero at t = 0 you need:
79.38 * x = 87.34 * 200000
= ⇒ x = 220056 .2
What is your cash flow at t = 3?
CF 3 =- 100 x
=- 22 , 005 , 625
Therefore, the loan rate you lock is
r = (22 , 005 , 625/ 20 , 000 , 000)- 1
r = 10.03%
Problem #2
(a)
Net Investment made by Lemon County = Amount invested in 30-Year strips - Amount received by issuing 2-Year strips
Amount invested in 30-Year strips:
Yield = [(Future Price / Purchase Price)(1/Number of years)]-1
6% = (100/Purchase Price)(1/30) -1
Purchase Price = $17.41
Amount invested = Purchase price ($17.41) * Number of strips (50,000)
Amount invested = $870,500
Amount received by issuing 2-Year strips:
Yield = [(Future Price / Purchase Price)(1/Number of years)]-1
6% = (100/Purchase Price) ^ (1/2) -1
Purchase Price = $92.455
Amount invested = Purchase price ($92.455) * Number of strips (5,000)
Amount received= $462,275
Net Investment made by Lemon County = Amount invested in 30-Year strips ($870,500) - Amount received by issuing 2-Year strips ($462,275)
Net Investment made by Lemon County = $408,225
(b)
The increase in Yield will impact the current price of the strips, which in-turn will impact the value of the investment.
% Change in investment value = {Current value of investment - Previous value of investment}/ Previous value of investment * 100
Current Value of investment
Current value of 30-Year strips = Current price * Number of units (50,000)
Current price = $13.136 (calculated using Yield formula above)
Current value of 30-Year strips = $656,800
Current value of 2-Year strips = Current price * Number of units (5,000)
Current price = $90.7 (calculated using Yield formula above)
Current value of 30-Year strips = $453,500
Current Value of Investment: $656,800 - $453,500 = $203,300
% Change in investment value = {$203,300 - $408,225}/ $408,225 * 100
% Change in investment value = -50.20%
