# Finance Hw Ps2-4

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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:
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