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MGFB10H3 Principles of Finance Tutorial 2 Solutions to Exercises Perpetuity 1. ( 5.27 in the textbook ) An prestigious investment bank designed a new security that pays a quarterly dividend of $4.50 in perpetuity. The first dividend occurs one quarter from today. What is the price of the security if the stated annual interest rate is 6.5 percent, compounded quarterly? Solution: The dividend payments are made quarterly, so we must use the quarterly interest rate. The quarterly interest rate is: Quarterly rate = 0 . 065 / 4 = 0 . 01625 . Using the present value equation for a perpetuity, we find the value today of the dividends paid: PV = $4 . 50 / 0 . 01625 = $276 . 92 . Calculating annuity values 2. ( 5.54 in the textbook ) You have recently won the super jackpot in the Alberta Provincial Lottery. On reading the fine print, you discover that you have the following two options: (a) You will receive 31 annual payments of $250,000, with the first payment being delivered today. The income will be taxed at a rate of 28 percent. Taxes will be withheld when the cheques are issued. (b) You will receive $530,000 now, and you will not have to pay taxes on this amount. In addition, beginning one year from today, you will receive $200,000 each year for 30 years. The cash flows from this annuity will be taxed at 28 percent. Using a discount rate of 7 percent, which option should you select? Solution: Here, we need to compare two options. In order to do so, we must get the value of the two cash flow streams to the same time, so we will find the value of each today. We must also make sure to use the aftertax cash flows, since it is more relevant. For Option A, the aftertax cash flows are: Aftertax cash flows = Pretax cash flows · (1 - tax rate) Aftertax cash flows = $250 , 000 · (1 - 0 . 28) = $180 , 000 1 _ ■ 2722 21092923891

The aftertax cash flows from Option A start now (at time 0), so the present value of the cash flows is: PV = C + C r 1 - 1 (1 + r ) T PV = $180 , 000 + $180 , 000 0 . 07 1 - 1 (1 + 0 . 07) 30 = $2 , 413 , 627 . 41 For Option B, the aftertax cash flows are: Aftertax cash flows = Pretax cash flows · (1 - tax rate) Aftertax cash flows = $200 , 000 · (1 - 0 . 28) = $144 , 000 The aftertax cash flows from Option B are an ordinary annuity starting one year from today (at time 1), plus the $530,000 cash flow today which is not taxed, so the present value is: PV = CF 0 + C r 1 - 1 (1 + r ) T PV = $530 , 000 + $144 , 000 0 . 07 1 - 1 (1 + 0 . 07) 30 = $2 , 316 , 901 . 93 You should choose Option A because it has a higher present value on an aftertax basis. Calculating annuity payments 3. ( 5.65 in the textbook ) Your friend is celebrating her 30th birthday today and wants to start saving for her anticipated retirement at age 65. She wants to be able to withdraw $100,000 from her savings account on each birthday for 25 years following her retirement, with the first withdrawal on her 66th birthday. Your friend intends to invest her money in the local credit union, which offers 8 percent interest per year (compounded annually). She wants to make equal annual payments on each birthday into the account established at the credit union for her retirement fund. (a) If she starts making these deposits on her 31st birthday and continues to make deposits until she is 65 (the last deposit will be on the 65th birthday), what amount must she deposit annually to be able to make the desired withdrawals at retirement? (b) Suppose your friend just inherited a large sum of money. Rather than making equal annual deposits, she decides to make one lump-sum payment on her 30th birthday to cover her retirement needs. What amount does she have to deposit? (c) Suppose your friend’s employer will contribute $1,500 to the account every year as part of company’s profit-sharing plan. In addition, your friend expects a $45,000 distribution from a family trust fund on her 55th birthday, which she would also put into her retirement account. If your friend starts making deposits on her 31st birthday, what amount must she deposit annually to be able to make the desired withdrawals at retirement? 2 _ ■ 2722 21092923891

Solution: (a) Here, we are solving a two-step time value of money problem. For each savings possibility we want the FV to be equal to the PV of the retirement spending when your friend is ready to retire. The amount needed when your friend is ready to retire is: PV = $100 , 000 0 . 08 1 - (1 / 1 . 08 25 ) = $1 , 067 , 477 . 62 If your friend makes equal annual deposits into the account, this is an annuity with the FV equal to the amount needed in retirement. The required savings each year will be: FV = $1 , 067 , 477 . 62 = C (1 . 08 35 - 1) / 0 . 08 ⇒ C = $6 , 194 . 86 (b) Here we need to find a lump sum savings amount. Discounting the future value, we get: FV = $1 , 067 , 477 . 62 = PV (1 . 08) 35 PV = $72 , 198 . 36 (c) In this problem, we have a lump sum savings in addition to an annual deposit. Since we already know the value needed at retirement, we can subtract the value of these additional savings at retirement to find out how much your friend is short. Doing so gives us: FV of trust fund deposit = $45 , 000(1 . 08) 10 = $97 , 151 . 62 So, the amount your friend still needs at retirement is: FV = $1 , 067 , 477 . 62 - $97 , 151 . 62 = $970 , 326 Using the FV equation, and solving for the payment, we get: $970 , 326 = C (1 . 08 35 - 1) / 0 . 08 ⇒ C = $5 , 631 . 06 This is the total annual contribution, but your friend’s employer will contribute $1,500 per year, so your friend must contribute: Friend’s contribution = $5 , 631 . 06 - $1 , 500 = $4 , 131 . 06 Canadian mortgages 4. Your mortgage has 25 years left, and has an APR of 8% (with semiannual compound- ing) and monthly payments of $1,500. 3 _ ■ 2722 21092923891

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(a) What is the outstanding balance? How much will you pay in interest, and how much will you pay in principal, during the next year? (b) Suppose you can increase the payment in order to pay off the mortgage earlier than in 25 years. If the APR is still 8% (with semiannual compounding), what monthly payments will ensure that you pay off the mortgage in 10 years (the mortgage with outstanding balance you computed in (a))? (c) Instead of making monthly payments of $1,500, you can make half the payment every two weeks (so that you will make 52/2=26 payments per year). How long (in years) will it take you to pay off the original mortgage if the APR remains the same at 8% with semiannual compounding? Solution: (a) The outstanding balance today can be computed as follows PV 0 = C r 1 - 1 (1 + r ) T , where we use C = 1 , 500, T = 25 × 12 = 300, and the effective monthly interest is r monthly = (1 + 0 . 08 2 ) 2 / 12 - 1 ≈ 0 . 0066 and therefore PV = $196 , 537 . 42 (or $195,688.55 if you used the rounded interest rate). To find out how much interest and how much principal repayment will be paid over the next year we need to compute the outstanding balance a year from now with the same formula as before but using T = 24 × 12 = 288. We then obtain PV 1 = $193 , 911 . 20 (or $193,094.26 if you used the rounded interest rate). The principal repayment is then obtained as the difference between the two balances, PV 1 - PV 0 . Whereas interest payment is the difference between the total payments for the year 12 × 1 , 500 and the principal repayment. (b) Here we need to compute the payment C using the outstanding balance and interest rate for our original mortgage, but with T = 120. PV 0 = C r 1 - 1 (1 + r ) T ⇒ C = 2360 . 80 (c) This question is about backing out T from PV = C r 1 - 1 (1 + r ) T , where C = 750, the effective bi-weekly interest rate is r bi-weekly = (1 + 0 . 08 2 ) 2 / 26 - 1 ≈ 0 . 0030 4 _ ■ 2722 21092923891

and T is measured in two-week periods. Rearranging and taking logs on both sides gives PV × r/C = 1 - 1 (1 + r ) T ⇒ 1 - PV × r/C = (1 + r ) - T ⇒ ln(1 - PV × r/C ) = - T ln(1 + r ) ⇒ T = - ln(1 - PV × r/C ) / ln(1 + r ) You should find that T/ 26 is between 19 and 20 years, depending on your rounding of the inputs. Delayed perpetuity 5. ( 5.26 in the textbook ) Mark Weinstein has been working on an advanced technology in laser eye surgery. His technology will be available in the near term. He anticipates his first annual cash flow from the technology to be $215,000, received two years from today. Subsequent annual cash flows will grow at 4 percent in perpetuity. What is the PV of the technology if the discount rate is 10 percent? Solution: The present value of this growing perpetuity is: PV = C r – g PV = $215 , 000 0 . 10 - 0 . 04 = $3 , 583 , 333 . 33 . It is important to remember that when dealing with annuities or perpetuities, the present value equation calculates the present value one period before the first payment . In this case, since the first payment is in two years, we have calculated the present value one year from now. To find the value today, we simply discount this value as a lump sum. Doing so, we find the value of the cash flow stream today is: PV = FV 1 + r PV = $3 , 583 , 333 . 33 1 + 0 . 10 = $3 , 257 , 575 . 76 . Delayed annuity 6. Let’s rewind a few years back. You are 18 and just graduated from high school. You are facing two alternatives: go straight to work, or go to college. Interest rate is 5.8269% continuously compounded. If you go straight to work, you will be able to put away $5 , 000 in a year into your savings account and grow this deposit by 2% every year thereafter. Your last deposit will be in 47 years, when you turn 65. 5 _ ■ 2722 21092923891

If you decide to go to the university, you will have to pay tuition for four years while you are studying. You would go to an expensive school and pay $50 , 000 a year for tuition with the first payment in one year. After you graduate in four years, you’ll start working at a higher paying job that will allow you to save $15 , 000 in five years (when you are 23 years old), and this amount will grow by 3% thereafter. Your last deposit will be in 47 years, when you turn 65. Is going to the university worth it? Solution: First, we need to find the discount rate. EAR of a continuously com- pounded rate is r = e 0 . 058269 - 1 = 6% . To figure out which alternative is better (high school vs. university education), we need to calculate the present value of cash flows under these two alternatives. Let’s first start with the high school (HS) education. Cash flows from saving for 47 years represent a growing annuity. Using the formula, we get that the present value of HS savings is PV HS = 5 , 000 0 . 06 - 0 . 02 · 1 - 1 . 02 1 . 06 47 ! = 104 , 501 Next, present value of cash flows associated with going to university (U) is the sum of two components: tuition and savings. PV Tuition = - 50 , 000 0 . 06 · 1 - 1 1 . 06 4 ! = - 173 , 255 PV U Savings = 15 , 000 0 . 06 - 0 . 03 · 1 - 1 . 03 1 . 06 43 ! · 1 1 . 06 4 = 280 , 809 Present value of savings less tuition is PV U = PV Tuition + PV U Savings = - 173 , 255 + 280 , 809 = 107 , 554 Clearly, PV U > PV HS , so you are better off going to the university. 6 _ ■ 2722 21092923891

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