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MGFB10H3 Principles of Finance Tutorial 1 Solutions to Exercises Future value (FV) of a cash flow 1. ( 5.2 in the textbook ) Compute the future value of $1,000 compounded annually for (i) 10 years at 5 percent, (ii) 10 years at 10 percent, (iii) 20 years at 5 percent. Why is the interest earned in part (iii) not twice the amount earned in part (i)? Solution: To find the future value of a lump sum, we use FV = PV (1 + r ) T $1 , 000 0 FV 10 FV = $1 , 000(1 . 05) 10 = $1 , 628 . 89 $1 , 000 0 FV 10 FV = $1 , 000(1 . 10) 10 = $2 , 593 . 74 $1 , 000 0 FV 20 FV = $1 , 000(1 . 05) 20 = $2 , 653 . 30 Because interest compounds on the interest already earned, the interest earned in part (iii) is more than twice the interest earned in part (i). Simple vs compound interest 2. ( 5.1 in the textbook ) First City Bank pays 8% simple interest on its savings account balances, whereas Second City Bank pays 8% interest compounded annually. If you made a $5,000 deposit in each bank, how much more money would you earn from your Second City Bank account at the end of 10 years? Solution: The time line for the cash flows is: $5 , 000 0 FV 10 The simple interest per year is $5 , 000 · 0 . 08 = $400 . So, after 10 years, you will have $400 · 10 = $4 , 000 in interest . The total balance will be $5 , 000 + $4 , 000 = $9 , 000. With compound interest, we use the future value formula: FV = PV · (1 + r ) T FV = $5 , 000 · (1 . 08) 10 = $10 , 794 . 62 . The difference is $10 , 794 . 62 - $9 , 000 = $1 , 794 . 62 . _ ■ 2722 21092923891

Present value (PV) of a cash flow 3. ( 5.7 in the textbook ) Imprudential Inc. has an unfunded pension liability of $750 million that must be paid in 20 years. To assess the value of the firm’s stock, financial analysts want to discount this liability back to the present. If the relevant discount rate is 8.2 percent, what is the PV of this liability? Solution: To find the PV of a lump sum, we use: PV = FV (1 + r ) T PV = $750 , 000 , 000 (1 . 082) 20 = $155 , 065 , 808 . 54 Basic time value of money relation: finding T, r 4. ( 5.6 in the textbook ) At 8 percent interest, how long does it take to double your money? To quadruple it? Solution: To find the length of time for money to double, triple, etc., the present value and future value are irrelevant as long as the future value is twice the present value for doubling, three times as large for tripling, etc. To answer this question, we can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that is: FV = PV · (1 + r ) T Solving for T we get: T = ln ( FV/PV ) ln(1 + r ) The length of time to double your money is: $1 0 $2 ? T = ln 2 / ln 1 . 08 = 9 . 01 years . The length of time to quadruple your money is: $1 0 $4 ? 2 _ ■ 2722 21092923891

T = ln 4 / ln 1 . 08 = 18 . 01 years . Notice that the length of time to quadruple your money is twice as long as the time needed to double your money (the difference in these answers is due to rounding). This is an important concept of time value of money. 5. ( 5.8 in the textbook ) Although appealing to more refined tastes, art as a collectible has not always performed so profitably. During 2010, Deutscher-Menzies sold Arkie under the Shower a painting by a renowned Australian painter Brett Whiteley, at auction for a price of $1,100,000. Unfortunately for the previous owner, he had purchased it three years earlier at a price $1,680,000. What is his annual rate of return on this painting? Solution: The time line is: $1 , 680 , 000 0 $1 , 100 , 000 3 To answer this question, we can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that is: FV = PV · (1 + r ) T Solving for r , we get: r = ( FV/PV ) 1 /T - 1 r = ($1 , 100 , 000 / $1 , 680 , 000) 1 / 3 - 1 = - 0 . 1317 or - 13 . 17% . Compounding frequency 6. ( 5.21 in the textbook ) What is the future value in six years of $1,000 invested in an account with a stated annual interest rate of 9 percent, (a) Compounded annually? (b) Compounded semiannually? (c) Compounded monthly? (d) Compounded continuously? Why does the future value increase as the compounding period shortens? 3 _ ■ 2722 21092923891

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Solution: To find the FV of a lump sum with discrete compounding, we use: FV = PV (1 + r/m ) m · T (a) FV = $1 , 000(1 . 09) 6 = $1 , 677 . 10 (b) FV = $1 , 000(1 + 0 . 09 / 2) 12 = $1 , 695 . 88 (c) FV = $1 , 000(1 + 0 . 09 / 12) 72 = $1 , 712 . 55 (d) To find the future value with continuous compounding, we use the equation: FV = PV e r · T FV = $1 , 000 e 0 . 09 · 6 = $1 , 716 . 01 (e) The future value increases when the compounding period is shorter because inter- est is earned on previously accrued interest. The shorter the compounding period, the more frequently interest is earned, and the greater the future value, assuming the same stated interest rate. Interest rates: APR vs. EAR 7. A payday loan (also called a payday advance, salary loan, payroll loan, short term, or cash advance loan) is a small, short-term unsecured loan. These loans tend to have high interest rates and short repayment periods, which means frequent compounding. Ontario enacted the Payday Loans Act, 2008 to limit the fees charged on loans in Ontario to $21 per $100 borrowed for a period of two weeks. (a) If the lender charges exactly the limit fee, what is the APR (the stated rate) the he/she needs to disclose for such a loan? What is the EAR? (b) Effective January 1, 2018, Ontario has reduced to maximum rate for payday loans down to $15 per $100 loaned. What is the EAR for such loan? Solution: (a) Recall that the annual percentage rate, APR, indicates the amount of simple interest earned in one year, that is, the amount of interest earned without com- pounding even though compounding may occur. In our example, $21 per $100 borrowed for a period of two weeks means that interest rate per two weeks is 21%. Assuming that there are 26 two week periods in one year this gives APR = 0 . 21 × 26 = 5 . 46 = 546% Because it does not include the effect of compounding, the APR quote is typically less than the actual amount of interest that you will pay (or earn). To compute the actual amount of interest paid in one year, the APR must be converted to an effective annual rate, EAR. EAR = (1 + 0 . 21) 26 - 1 = 141 . 04 = 14 , 104% 4 _ ■ 2722 21092923891

(b) Under the new regulation, the EAR for such a loan is EAR = (1 + 0 . 15) 26 - 1 = 36 . 86 = 3 , 686% Interest rates: effective rate for any period 8. You want to save for a special vacation that you will take in six years. Given a rate of 6% per year with monthly compounding for your savings account at CIBC, you wish to know what will be the future value of an annuity of deposits you will make to your account. This information (together with annuity formulas we will learn in Lecture 2) will help you decide the best way to save. You are considering the following payment schemes over a six year time frame. Compute effective rate per period corresponding to the cashflows for each of the cases. (a) Monthly deposits. (b) Semiannual deposits. (c) Annual deposits. (d) Bi-annual deposits (every two years). Solution: To compute the effective rate per period we use: r k = 1 + APR m m/k - 1 In this question the stated interest rate is quoted with monthly compounding, so m = 12. However, in each of the cases the payment frequency ( k in our lecture notation) is different. (a) The payment frequency is monthly, so k = 12. The effective monthly rate is r monthly = (1 + 0 . 06 12 ) 12 / 12 - 1 = 0 . 06 12 = 0 . 005 = 0 . 5% (b) The payment frequency is semi-annual, so k = 2. The effective semi-annual rate is r semi-annual = (1 + 0 . 06 12 ) 12 / 2 - 1 = 0 . 030378 = 3 . 0378% (c) The payment frequency is annual, so k = 1. The effective annual rate (EAR) is r yearly = (1 + 0 . 06 12 ) 12 / 1 - 1 = 0 . 061678 = 6 . 1678% (d) The payment frequency is bi-annual, so k = 1 / 2. The effective bi-annual rate (per two-year period) is r bi-annual = (1 + 0 . 06 12 ) 12 / 0 . 5 - 1 = 0 . 12716 = 12 . 7160% 5 _ ■ 2722 21092923891

Interest rates: real vs. nominal 9. In 2000, short-term interest rates in Canada were about 5.8% and the rate of inflation was about 3%. In 2003, interest rates where about 2.7% and inflation was about 3.1%. What was the real interest rate in 2000 and 2003? Solution: To compute the real interest rate we use r real = r nominal - i 1 + i . The real interest rate in 2000 was (0 . 058 - 0 . 03) / 1 . 03 = 0 . 0272 or 2.72% (which is approximately equal to the difference between the nominal rate and inflation: 5 . 8% - 3% = 2 . 8%). In 2003, the real interest rate was (0 . 027 - 0 . 031) / 1 . 031 = - 0 . 0039 = - 0 . 39%. Note that the real interest rate was negative, indicating that interest rates were insufficient to keep up with inflation. Interest rates: before-tax vs. after-tax 10. Suppose you have a credit card with 19.9% APR with daily compounding, a bank savings account paying 5% EAR, and a car loan with a 4.8% APR with monthly compounding. Your income tax rate is 40%. The interest on the savings account is taxable, and the interest on the credit card and on the car loan is not tax-deductible. What is the effective after-tax interest rate of each instrument, expressed as an EAR? What should your priorities be in terms of your financial situation? Solution: The savings account has EAR of 5%. The EAR of the credit card is (1 + 0 . 199 / 365) 365 - 1 = 22 . 01% and the EAR of the car loan is (1 + 0 . 048 / 12) 12 - 1 = 4 . 91%. It looks like you should pay down the credit card first, because it carries a high interest rate. Then you should save because your savings account earns more interest than the interest charged on your car loan. However, taking into account tax effects can change these conclusions. Because credit card interest and the card loan interest are not tax deductible, the after-tax rate is the same as the pre-tax interest rate. However, the savings account’s interest is taxed. The after tax interest rate you will earn on the savings account is 5% × (1 - 0 . 40) = 3%. Taking taxes into account, it is clear that your priorities should be to first pay off the credit card, and second, pay down the car loan. Then you can put the leftover money in your savings account. 6 _ ■ 2722 21092923891

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