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Finance Hw Ps2-4

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Submitted By emsti
Words 1673
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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: f_t=〖(1+r_t)〗^t/〖(1+r_(t-1))〗^(t-1) -1=〖DF〗_(t-1)/〖DF〗_t -1
Plugging in the discount factors from (a) we get the forward rates: f_1=1/〖DF〗_1 -1=1/0.9346-1=0.069976≅7.00% f_2=〖DF〗_1/〖DF〗_2 -1=0.9346/0.8767-1=0.066043≅6.60% f_3=〖DF〗_2/〖DF〗_3 -1=0.8767/0.8255-1=0.062023≅6.20%
(c) What should the price of a 3-year bond with a face value of $100 and a 6% annual coupon be?
We calculate the bond price by discounting the annual cash flows by the discount factors from (a): P=〖DF〗_1*〖CF〗_1+〖DF〗_2*〖CF〗_2+〖DF〗_3*〖CF〗_3 P=0.9346*$6+0.8767*$6+0.8255*106 P=$98.37

(d) A 3-year bond with a face value of $100 and a 4% annual coupon is trading at $95.00. Show that this bond is mispriced by showing how you would take advantage of its price. In doing so, make sure that your arbitrage profits are realized today.
In order to find arbitrage opportunities, we need to calculate the fair price of the bond first:
P=〖DF〗_1*〖CF〗_1+〖DF〗_2*〖CF〗_2+〖DF〗_3*〖CF〗_3
P=0.9346*$4+0.8767*$4+0.8255*$104 P=$93.09
Since the fair price is $1.91 lower than the current price, we could sell the bond for $95 now to realize the arbitrage opportunity

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