Econ 134A-100 (3:30 pm) Winter 2013, Test 1 Solution sketches

Solve each of the following (a) George is quoted a price for a bond of $800. This bond has a face value of $900. Two coupons of 10% each will be paid. The first coupon will be paid later today and the second coupon will be paid one year from today. If the bond matures one year from today, what is the yield on this bond (expressed as an effective annual discount rate)? 800=90+ 90 1+𝑟 + 900 1+𝑟 710 1+𝑟 =990 𝑟=0.394366

Solve each of the following (cont.) (b) Cui will receive 3 payments. Each payment will be $80 every six months, starting six months from today. The stated annual interest rate is 7%, and interest is compounded every three months. What is the total present value of the 3 payments? First, we find the semi-annual discount rate: 1.0175 2−1=0.03530625 80 1.03530625 + 80 1.03530625 2 + 80 1.03530625 3 =77.27+74.64+72.09=$224.00

Projects A and B For the following problem, assume that discount rates can only be non-negative. There are only two projects to consider. Project A has a cost of $1,800 now, and has a benefit of $2,300 two years from now. Project B has a cost of $900 now and a benefit of $1,200 one year from now. For what discount rates will Project A have the higher net present value? −1800+ 2300 1+𝑟 2 >−900+ 1200 1+𝑟 0>900 𝑟 2 +3000𝑟−200 𝑟= −3000± 3000 2 −4×900×−200 2×900 = −3000±3117.69 1800 Ignoring the negative discount rate, 𝑟=0.06538414 Then pick an interest rate above and below 0.06538414 and you will find that the inequality holds only if 𝑟<0.06538414

Caesar Caesar makes monthly deposits of $500 into a bank account, starting today. How many of these monthly deposits will Caesar have to make in order reach an account balance of $30,611.30? Assume a stated annual interest rate of 12%, compounded monthly. (Note: Include today’s deposit in your answer.) 30611.30 1.01 𝑇 =500+ 500 0.01 1− 1 1.01 𝑇 , where 1% is the monthly SAIR. Note that T does not account for today’s deposit Solving for T: 1.59626= 1.01 𝑇 T= log 1.59626 log 1.01 =47 monthly payments after today, or 48 payments total.

Aragon Aragon will be receiving a $50,000 loan from the Spaniard National Bank today. The loan will last for 12 years but have a 30-year amortization schedule. (In other words, all payments will be made over the next 12 years, but the payment schedule will be made to mimic a 30-year payback for the loan.) The effective annual interest rate is 17%. The regular payments will be made yearly, starting one year from today.  (a) (7 points) How much will each regular payment be? 50000= 𝑐 0.17 1− 1 1.17 30 𝑐=$8577.23

Aragon (cont.) (b) (5 points) How much will the balloon payment have to be 12 years from today in order to completely pay back the loan? PV of 12 payments: 8577.23 0.17 1− 1 1.17 12 =$42786.55 FV of balloon payment: 1.17 12 50000−42786.55 =$47465.01

Junk bond You are considering buying a junk bond that promises 5 coupon payments of $30 each. The payments are promised to occur every year starting later today. You decide that for this bond, the effective annual discount rate for the first three years (i.e. from today to three years from today) is 29%. The effective annual discount rate after the first three years (i.e. from three years from today onward) is 39%. Based on these assumptions, what is the present value of the 5 coupon payments? 30+ 30 1.29 + 30 1.29 2 + 30 1.29 3 + 30 1.29 3 ×1.39 =30+23.26+18.03+13.98+10.05=95.32

Prestigious Paula Paula currently works for Prestigious Plastics. She is analyzing two ways to produce a new superplastic. Method A requires $50,000 to be invested today and results in $125,000 in additional profits for Prestigious Plastics two years from now. Method B requires $100,000 to be invested today and results in $250,000 in additional profits three years from now. The effective annual discount rate for the new superplastic project is 17%. (Note: Assume that any money not invested in one of these projects is invested with a net present value of zero.) What is the internal rate of return for each method that could be used? −50000+ 125000 1+ 𝜌 𝑎 2 =0 1+ 𝜌 𝑎 2 =2.5 𝜌 𝑎 =0.581139 −100000+ 250000 1+ 𝜌 𝑏 3 =0 1+ 𝜌 𝑏 3 =2.5 𝜌 𝑏 =0.357209

Prestigious Paula (cont.) (b) Since only one method is needed to produce the superplastic, which method should be used? Justify your answer in 30 words or less. 𝑁𝑃𝑉 𝑎 =−50000+ 125000 1.17 2 =41314.19 𝑁𝑃𝑉 𝑏 =−100000+ 250000 1.17 3 =56092.64 Choose method b, because it has a higher NPV.