Ec 1745 Fall 2008 Problem Set 2

Tarun Singh Worked with Cliff Beltzer and Rohan Parsad Ec 1745 Problem Set 2 1. a) NPV=C0+(C1/(1+r))+…+(Ct/((1+r)t)) NPV(Red Pig)=-400+(250/(1.09))+(300/(1.092))=$81.86 NPV(Caffeine Delight)=-200+(140/(1.09))+(179/(1.092))=$79.10 Following the NPV rule, Coca Cola should produce Red Pig b) The IRR rule tells us that the project with the higher IRR can stand a higher cost of capital and still be a positive NPV project and that we should invest in a project if the IRR > cost of capital. However, in this case both projects have an IRR that is greater than the cost of capital (9%), so we can’t simply use this rule to tell us which project to invest in. What we can do is create another synthetic project that measures the differences between the two projects and then calculate the IRR for this synthetic project to see if one should invest in it, if one should, then the result is the same as part a). Project Synthetic Payoffs: Year 0 Year 1 Year 2 -400-(-200)=-200 250-140=110 300-179=121 NPV=-200+(110/(1+r))+(121/(1+r)2)=0 r=.1 so IRR=10% Since the IRR for the synthetic project is greater than the cost of capital, you should take on both the synthetic project and the Caffeine Delight project, which is the same thing as taking on the Red Pig Project. Thus, by using this method we achieve the same result as in part a). 2. Let F=Face Value Let C=Coupon Rate Let M= value at maturity n n Price=PV=(FC)((1-(1/(1+r) ))/r)+(M/((1+r) )) PV=(1000(.10))((1-(1/(1.08)20))/.08)+(1000/((1.08)20)) =$ 1196.36 when C=.1 and r=.08 PV=(1000(.04))((1-(1/(1.08)20))/.08)+(1000/((1.08)20))=$607.27 when C=.04 and r=.08 PV=(1000(.1))((1-(1/(1.12)20))/.12)+(1000/((1.12)20))=$850.61 when C=.1 and r=.12 PV=(1000(.04))((1-(1/(1.12)20))/.12)+(1000/((1.12)20))=$402.45 when C=.04 and r=.12 The price of the bond increases as the coupon rate increases. The price of the bond decreases as interest rate in the economy increases. 3. Let F=Face Value Let C=Coupon Rate Let M= value at maturity Let F=1000 PV=(FC/r)(1-(1/((1+r)5)))+(F/((1+r)5)) PV=((1000(.06))/.04)(1-(1/((1.04)5)))+(1000/((1.04)5))=$1089.04 when C=.06 and r=.04

Semiannual coupon payments r=4%: effective rate=((1.04)(1/2))-1=1.98%: PV=((1000(.06))/.0198)(1-(1/((1.0198)(5x2))))+(1000/((1.0198)(5x2)))=$1361.44 when C=.06 and r=.04 PV=((1000(.06))/.03)(1-(1/((1.03)5)))+(1000/((1.03)5))=$1137.39 when C=.06 and r=.03 Semiannual coupon payments=((1.03)(1/2))-1=1.49% PV=((1000(.06))/.0149)(1-(1/((1.0149)(5x2))))+(1000/((1.0149)(5x2)))=$1416.14 when C=.06 and r=.03 The bond value would increase if the interest rate fell from 4% to 3%. 4. a) ((dP/P)/(dr/(1+r)) → (dP/dr)((1+r)/P) = (-(C1/((1+r)2))-( 2C2/((1+r)3))-… (tCt/((1+r)(t+1))))((1+r)/P) = (-1/(1+r))((1+r)/P)((C1/(1+r))+( 2C2/((1+r)2))+… +(tCt/((1+r)t))) =-(1/P)((C1/(1+r))+( 2C2/((1+r)2))+… +(tCt/((1+r)t))) =-D This shows that the Macaulay duration is the negative of the discount-rate elasticity of the bond price b) Part a) shows that the Macaulay Duration= -((dP/P)/(dr/(1+r))= -D Price volatility of the bond= -(dP/P) -(dP/P)= -D(dr/(1+r))= -dr(D/(1+r)) The Macaulay Duration discounted by an additional time period and then multiplied by the negative change in the discount rate is equal to the price volatility of the bond.