Problem Set #2
4.
(a) Assuming that all the bonds make only annual payments, what spot rates are imbedded in these prices? First, we need to find the Discount Factors:
Bond A: 〖DF〗_1*$100=$93.46
Bond B: 〖DF〗_1*$4+〖DF〗_2*$104=$94.92
Bond C: 〖DF〗_1*$8+〖DF〗_2*$8+〖DF〗_3*$108=$103.64
〖DF〗_1=93.46/100=0.9346
〖DF〗_2=((94.92-(0.9346*4)))/104=0.8767
〖DF〗_3=((103.64-(0.9346*8)-(0.8767*8)))/108=0.8255
Since 〖 DF〗_t=1/〖(1+r_t)〗^t , we have r_t=(1/〖DF〗_t )^(1⁄t)-1. Then solving for r1, r2 and r3 we get: r_1=0.069976≅0.070 r_2=0.068008≅0.068 r_3=0.066009≅0.066 So the spot rates are: Bond A = 7%, Bond B = 6.8% and Bond C = 6.6%.
(b) What forward rates are embedded in these prices?
The formula for forward rates is:
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Neutralizing year 3:
Bond X pays out $104 (face value plus coupon) in year 3. Since bonds A and B only run one or two years, only C can neutralize this. C’s cash flow would be $108 in year 3. Therefore: $104=$108*C C=$104/$108 C=0.96296
This means that bond C should take up 96.296% of the portfolio. Calculating Year 1 and 2 the same way, we have:
Year 2: $4=$104*B+$8*C B=($4-$8*0.96296)/$104 B=-0.03561
Year 1: $4=$100*A+$4*B+$8*C A=($4-$4*(-0.03561)-$8*0.96296)/$100 A=-0.03561
The negative outcomes in share of our portfolio suggest that we sell bonds A and B.
All of this results in the following table:
Cash Flows (in $)
Operations start year year 1 year 2 year 3
Sell bond D 95.00 -4.00 -4.00 -104.00
Buy 0.96269 of bond C -99.80 7.70 7.70 104.00
Sell 0.03561 of bond B 3.38 -0.14 -4.00
Sell 0.03561 of bond A 3.33 -3.56
Total 1.91 0.00 0.00 0.00
$1.91 is the arbitrage we could earn here.
Problem Set #3
1.
A company holds $ 2.6 million in cash.
(a) What is the interest rate?
According to the capital markets line $ 4 million are worth $ 5 million tomorrow. So the interest rate needs to be: r=5/4-1=0.25=25%
(b) How much should the company invest in plant and machinery?
The
For Investment B: (40 – 5)/ 30= 1.16 standard units= close to 88% to get the 40 million in
* Interest rate had a 125bp spread over the current yield on 10-year US Treasury bonds (=4.25%).
2. Now, regardless of your answer to Question 1, assume that the 5-year bond selling for $800.00, the 15-year bond is selling for $865.49, and the 25-year bond is selling for $1,320.00.
You have been making payments for the last 25 years and have finally paid off your mortgage. Your original mortgage was for $345,000 and the interest rate was 5% per year compounded semi-annually for the entire 25 year period. How much interest have you paid over the last 5 years of the mortgage?
Question 6. The mean return for the Vanguard Total Stock Index is 20.8 while the mean return for the Vanguard Balanced Index is 12.9 (with bonds). Based on this data you would conclude that bonds do not reduce the overall risk of an investment portfolio since the mean return was actually less when the porfolio has bonds in
risk free bonds. According to the Law of One Price, what must be the price of Bond C
∆P/P = –D*(∆y) D* = D/(1 + y) = 7/1.073 = 6.52 ∆P/P = –D*(∆y) = –6.52(–0.09%) = .59% New price = $1,073(1.0059) = $1,079.33 Learning Objective: 11-02 Compute the duration of bonds; and use duration to measure interest rate sensitivity.
a. What would Mrs. Beach have to deposit if she were to use high quality corporate bonds an earned an average rate of return of 7%.
This figure represents SIMP’s Total interest at the end of 10 years at a 5% interest rate
The bonds have 20 years to maturity, pay interest at 9.3%, have a par value of $1,000 and are currently selling for $890.
I take the “Long –Term U.S. Government Bond Returns from 1926-1987” as appropriate risk free rate from Exhibit 4. The risk-free rate represents the interest an investor would expect from an absolutely risk-free investment over a specified period of time. “Long –Term U.S. Government Bond Returns from 1926-1987” are include the whole lifetime of Marriot and are more or less risk free.
b. Generate a graph or table showing how the bond’s present value changes for semi-annually compounded interest rates between 1% and 15%.
* We assume a risk-free rate of 5.09%. This number comes from the current yield of the 30 year T-bond as shown in Exhibit 5.