Finance Hw Ps2-4

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Problem Set #2
(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

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
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?

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