The car Adam is considering is $35,000. The dealer has given him three payment options:

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1 Adam Rust looked at his mechanic and sighed. The mechanic had just pronounced a death sentence on his road-weary car. The car had served him well---at a cost of 500 it had lasted through four years of college with minimal repairs. ow, he desperately needs wheels. He has just graduated, and has a good job at a decent starting salary. He hopes to purchase his first new car. The car dealer seems very optimistic about his ability to afford the car payments, another first for him. The car Adam is considering is 35,000. The dealer has given him three payment options: 1. Zero percent financing. Make a 4000 down payment from his savings and finance the remainder with a 0% APR loan for months. Adam has more than enough cash for the down payment, thanks to generous graduation gifs. 2. Rebate with no money down. Receive a 4000 rebate, which he would use for the down payment (and leave his savings intact), and finance the rest with a standard -month loan, with an 8% APR. He likes this option, as he could think of many other uses for the Pay cash. Get the 4000 rebate and pay the rest with cash. While Adam doesn t have 35,000, he wants to evaluate this option. His parents always paid cash when they bought a family car; Adam wonders if this really was a good idea. Q1. What are the cash flows associated with each of Adam s three care financing options? Comments: This question is to check whether you understand the concept of amortizing loans. Amortizing loan is a loan which the borrower makes monthly payments that include interest on the loan plus some part of the loan balance (principal). The payment, C, is set so that the present value of the cash flows, evaluated using the loan interest rates, equals the original principal amount of 35,000. Please recall amortizing loan payment is annuity and thus we can use the formula: P C (1 ) r r. Just for your curiosity! There are four types of credit instruments: Simple loan Fixed payment loan: the dollar payments are the same every year so that the principal is amortized. Amortization is the process of repaying a loan s principal gradually over time. Coupon bond: A debt security that pays a regular interest payment until maturity, when face value is repaid (e.g. most corporate and government bonds) Discount bond (also known as Zero-coupon bond): a debt security with just one payment. Option 1) This loan corresponds to simple loan whose yield to maturity equals the simple interest rate. For such a loan, no interest payment accrues on the loan. So the total cash flows in year can be calculated as CF PV 1 r. Therefore, the payment each month, C 1, can be computed simply by dividing CF by. 1

2 4,000 C1 C1 C1 The cash flow associated with the option is ,000 4,000 C Option 2) The option is an amortizing loan with 8% APR. Since you received a 4,000 rebate, the present value associated with monthly payments would be 31,000. 4,000 +4,000 =0 C2 C2 C2 8% The 8% APR means % of monthly interest rates: % months. Thus, using the formula for amortizing loans (annuity), we would have ,000 31,000 31,000 C ( ) Option 3) There is no more payment required because he paid the loan at period of 0. Thus C ,000 4,000 31, = 31,000 Answers: The cash flows for three options are , , and 0 in order. Q2. Suppose that, similar to his parents, Adam had plenty of cash in the bank so that he could easily afford to pay cash for the car without running into deb now or in the foreseeable future. If his cash earns interest at a 5.4% APR (based on monthly compounding) at the bank, what would be his best purchase option for the car? Comments: This question is to check whether you understand the concept of the cost of capital (or equivalently, the investor s opportunity cost of capital). Please recall it in chapters 3 and 5. 2

3 Option 1) To see how much the opportunity cost of the Option1, we compute the present value of the sum of monthly payments at a discount rate and its initial payment. If Adam invested C every month for months after initial investment, the present value of the investment discounted at 5.4% APR would be computed by following annuity formula: PV 4, C , ,000 1, , 5.4% where 0.45% is the discount rate. ote that you need to add his initial 12months payment at period 0, which is 4,000. The opportunity cost of Option1 would be 31, Option 2) Similarly, we can compute the opportunity cost of the Option 2: PV C , , Thus, the opportunity cost of the Option2 is 32, Option 3) Since the Option3 is paying 31,000 at once, the present value of the opportunity cost would be 31,000. Answers: Among three options, the opportunity cost of Option 3 is the smallest and thus, it would be the best option. Adam s fellow graduate, Jenna Hawthorne, was lucky. Her parents gave her a car for graduation. Okay, it was a little Hyundai, and definitely not her dream car, but it was serviceable, and Jenna didn t have to worry about buying a new car. In fact, Jenna has been trying to decide how much of her new salary she could save. Adam knows that with a hefty car payment, saving for retirement would be very low on his priority list. Jenna believes she could easily set aside 3000 of her 45,000 salary. She is considering putting her savings in a stock fund. She just turned 22 and has a long way to go until retirement at age 65, and she considers this risk level reasonable. The fund she is looking at has earned an average of 9% over the past 15 years and could be expected to continue earning this amount, on average. While she has no current retirement savings, five years ago Jenna s grandparents gave her a new 30-year U.S. Treasury bond with a 10,000 face value. 3

4 Q3. Suppose Jenna s Treasury bond has a coupon interest rate of 6.5%, paid semiannually, while current Treasury bonds with the same maturity date have a yield to maturity of 5.45% (expressed as an APR with semiannual compounding). If she has just received the bond s 10 th coupon, for how much can Jenna sell her treasury bond? Comments: For coupon bond, we use the following formula: C C C F P, where C CP(Coupon Payment). n 1 r 1r 1r 1r The coupon payment will be: 2 n CP Coupon Rate Face Value , umber of Coupon Payment per year 2 The yield to maturity is 5.45% APR with semiannual compounding, thus r / Jenna s U.S. Treasury bond has 30-year maturity, and she is assumed to just receive its 10 th coupon. This means that the rest of period would be 60 months 10 months = 50 months, which implies n 50. Face value is 1,000. The price Jenna can sell her treasury bond would be: C C C F C 1 F P 1 1 r , , , , r r r r r r Answers: She can sell her treasury bond at 11,4.01. n n n n Jenna wants to know her retirement income if she both (1) sells her Treasury bond at its current market value and invests the proceeds in the stock fund and (2) saves an additional 3000 at the end of each year in the stock fund from now until she turns 65. Once she retires, Jenna wants those savings to last for 25 years until she is 90. Q4. Suppose Jenna sells the bond, reinvests the proceeds, and then saves as she planned. If, indeed, Jenna earns a 9% annual return on her savings, how much could she withdraw each year in retirement? (Assume she begins withdrawing the money from the account in equal amounts at the end of each year once her retirement begins.) Comments: This problem is related to future value of an annuity in chapter 4. First, she would invest the proceeds in the stock fund, and the present value of the cash flows would be 11,4.01 from Q3. At age 65, the value would be 11, Second, she planned to save an additional 3,000 at the end of each year in the stock fund. Since she would save 3,000 for years 4

5 (=65-22) at 9%, C 3, 000, r 9% 0.09, and n. So the future value of annuity at 65 from regular savings would be: 1 1 FV PV 1r C 1 1r r 1 r ,4.01 3,000 3,000 3,000 The total amount she will get at her retirement would be 1,787, , , , , , ,4.013, , , , , , ,787, From then, she wants to withdraw equal amount for 25 at the end of each year ,787, C C C Therefore, this problem is back to compute each payment with the formula for annuity: PV C 1 1, 787, Thus, 1,787, , C 181,

6 Answers: She can withdraw 181, at the end of each year for 25 years. 5. Jenna expects her salary to grow regularly. While there are no guarantees, she believes an increase of 4% a year is reasonable. She plans to save 3000 the first year, and then increase the amount she saves by 4% each year as her salary grows. Unfortunately, prices will also grow due to inflation. Suppose Jenna assumes there will be 3% inflation every year. In retirement, she will need to increase her withdrawals each year to keep up with inflation. In this case, how much can she withdraw at the end of the first year of her retirement? What amount does this correspond to in today s dollars? (Hint: Build a spreadsheet in which you track the amount in her retirement account each year) Comment (1): At age 65, the value from selling the U.S. Treasury bond would be 11, annuity.. In addition, her savings amount would be future value of the growing ,4.01 3,000 3,000*(1.04) Since the future value of a growing annuity from 3,000: FV PV 1 r C 1 1 r r g 1 r , , , , ,116, g 30 3, The total amount she will get at her retirement would be 2,581, : 465, , ,116, ,116, ,581, You can think it as the present value first, and then you can compute it as the future value by multiplying it by , 000 6

7 Without the formula, you can compute the retirement income this messy way PV 11,4.013,000 3,000 3,000 3, ,4.01 3,000 3,000 3,000 3, ,4.01 3,000 3, , , ,4.01 3,000 3,000 3,000 3, , ,4.01 3, , ,4.01 3, , FV 63, ,581, Answer (1): Thus, the total amount she will get at her retirement would be 2,581,628. Comment (2): From the net income at her 65, she will withdraw some amount, C, at the end of the year for 25 years. However, because of the inflation, her nominal savings would grow by the inflation rate. Therefore, you need to use the growing annuity formula to compute the amount of her withdrawal: 25 C PV65 2,581, C C Thus, her withdrawal would be 204, Since the question asks you to compute the present value today, you will consider the inflation rate as a discount rate. ote that is increased by one year because the withdrawal would be implemented at her , , Answer (2): Thus, the present value of her first withdrawal would be 55,

8 6. Should Jenna sell her Treasury bond and invest the proceeds in the stock fund? Give at least one reason for and against this plan. If she didn t sell the bond, the future value at its maturity---when she is 47 years old---would be, : n C C C F C 1 F PV 1r 1 2 1r n n n n 1 r 1 r 1 r 1 r r 1 r 1 r , , If she sold the bond and reinvest the proceeds in the retirement, she would have 352, under the assumption that she deposits 3,000 every year and she can earn 9% return from the account with zero inflation rate. (Please see the Excel sheet 4). In such a case, it would be better not to sell the bond. However, if she sold the bond and reinvest the proceeds in the retirement, she would have 352, at 47 under the assumption that her deposit grows at 4%, and the return from the account is 9% with 4% of inflation rate, she must have 456,031 at her 47. Thus, in such a case it is better for her to sell the bond and invest the proceeds in the stock (retirement) fund. Therefore, the scenarios depend on what kinds of assumptions are made with regard to the return, interest rates, and the inflation rate. n 8