3530 Midterm Exam – F2010 –Solutions You are planning to establish a 20-year scholarship fund at York University that will pay $25,000 at the end of the first year and then increase by 1.5% per year The manager expects that the fund will earn a 5.75% annual rate of return How much should you donate to York today in order to maintain this scholarship? A) $50,346 B) $146,508 C) $278,225 D) $329,235 E) $588,235 Solution: D This is a growing annuity T C1  1 + g   PV of growing annuity = 1− r − g   + r   20 25,0001   + 015   PV of growing annuity = 1− 0575 − 015  1 + 0575   PV = 588,235.29 x (1 – 0.4403) = 329,235.29 (answers may vary slightly due to rounding) Compute the present value of the following set of payments for two years: $300 each quarter for the first year and $600 each quarter for the second year The effective annual rate is 6% and payments will begin at the end of the first quarter A) $3,000.56 B) $3,340.77 C) $3,514.06 D) $3,854.02 E) $3,999.14 Solution B The quarterly interest rate is (1+ 06)1/4 -1 = 01467 =1.467% Using your calculator: PV Annuity 1: PMT= 300; n=4, FV = 0; I/Y = 1.467, COMP PV PV1 = $1,157.25 PV Annuity 2: PMT= 600; n=4, FV = 0; I/Y = 1.467, COMP PV at start of year PV = 2314.50 and bring back to time zero: PV2 = 2314.50 / (1.01467)4 = 2,183.52 PV entire stream: = 1157.25 + 2183.52 = $3,340.77 How much will be in an account after 15 years if the initial deposit was $2500 and it earned interest at an APR of 6%, compounded monthly? A) $1,018.71 B) $1,242.42 C) $2,735.88 D) $5,030.49 E) $6,135.23 Solution E im = APR/m = 6%/12 = 0050 or 50% FV= 2500 x (1.005)^180 = $6,135.23 What is the effective annual rate on a deposit of $3,000 made eight years ago if the deposit is worth $4,453.52 today? The deposit pays interest semi-annually A) 2.50% B) 5.00% C) 5.06% D) 10.12% E) 12.50% Solution C Using your calculator: PV= -3,000 FV= 4,453.52 N=16 PMT =0 COMP I/Y I/Y = 2.50% this is the semi-annual rate EAR = (1+ 025)^2 -1 = 5.06% You are planning to buy a new condominium in Toronto The Condo is worth $400,000; you will put a 10% down payment and obtain a 25-year fixed rate mortgage at 6.25% (APR semi-annually compounded) for the rest You have decided to make weekly payments which will begin at the end of the first week What is your weekly payment? (Assume there are 52 weeks in one year.) A) $542.81 B) $608.42 C) $735.66 D) $808.21 E) $923.55 Solution: A Find Semi-annual rate: is = 6.25% -> 0625/2 = 03125 Find EAR: EAR = (1.03125)2 – = 06348 Find the weekly rate: iw = (1+ 0.06348)1/52 -1 = 0.001184 Find Number of payments in mortgage: n = 25yrs x 52wks/yr = 1300 weeks Find the monthly payment with a PV of $360,000 ($400,000 x 90% ) Using your calculator: PV= -360,000, n=1300, i =0.1184%, FV=0, PMT=? Answer= $542.81 You believe that you will need $1,500,000 when you retire in 35 years from today You also know that you will receive an extra payment of $100,000 from your trust fund in years time How much will you need to save at the end of each of the next 35 years to reach your retirement goal? You can earn 7% on your funds compounded annually because you manage your own money and your undergraduate major was finance A) $3,667 B) $5,344 C) $10,127 D) $10,850 E) $12,643 Solution B If you invest the $100,000 received in year until your retirement in year 35, it will grow to $100,000 × (1.07)30 = $761,225.50 Therefore, your savings plan would need to generate a future value of only $1,500,000 – $761,225.50 = $738,774.50 Now, using your calculator: FV= 738,774.50, n=35, I/Y = 7, PV=0, COMP PMT PMT = $5,344,26 You have been so successful in your finance career that you can buy a second condo today for $1.5 million You think you can sell it in five years for $2 million You will also earn $120,000 per year in rent on the property in years to (only) and the rental income occurs at the end of each year If you want to earn at least a 10% annual return on your investment, should you buy the condo? A) Yes, the present value of all the cash inflows is greater than the purchase price B) Yes, the future value of the rental income is greater than the selling price C) No, the present value of all the cash inflows is less than the purchase price D) No, the future value of the rental income is less than the selling price E) You are indifferent as the cash inflows equal the cash outflows Solution A First, find the PV of the rental payments (delayed annuity) and then add PV of selling price Using your calculator: PMT = 120,000; n= 4; I/Y = 10%; FV = , COMP PV PV ordinary annuity = 380,383.85 PV Delayed Annuity = 380,383.85 /(1.10) = 345,803.50 PV of selling price = 2,000,000/(1.10)5 = 1,241,842.65 Total PV = 345,803.50 + 1,241,842.65 = 1,587,646.15 Since Total PV cash inflows > Purchase price Yes, you should buy the condo You currently have $10,000 in the bank earning 0.50% per month You need $35,000 to make a down payment on a house You can save an additional $500 at the end of each month How long will it take for you to accumulate the $35,000? A) Less than year B) Between and years C) Between and years D) Between and years E) More than years Solution D: Your savings goal is -35,000 = FV You currently have $10,000 in the bank = PV PMT = $500 and I/Y = 0.50% Comp “n” = 41.06 months or between and years You are comparing a perpetuity having annual payments of $5,000, with a 20-year annuity having the same payments If interest rates are 7% annually and the payments begin immediately, which of the following statements are true? A) The present value of the perpetuity payments is $71,428 B) The present value of the annuity stream is $52,970 C) The perpetuity is worth over $19,000 more than the annuity stream D) Both A) and B) are true E) None of the above statements is true Solution C The perpetuity due is worth $5000/.07 + $5,000 = $76,428.57 The PV of the twenty-year annuity due: Use your financial calculator and set the BGN function: FV =0, PMT = 5,000; I/Y = 7%; n= 20, COMP PV PV = $56,677.98 The difference between the perpetuity and annuity is $19,750.59 10 If the EAR is known to be 14.75% on a debt that has quarterly payments, what is the APR on the debt? A) 3.50% B) 3.69% C) 14.00% D) 14.73% E) 15.60% Solution C APR/m (or im) = (1 + EAR)1/m -1 The quarterly interest rate = (1.1475)1/4-1 = 3.50% APR = im x m So the APR = 0.0350 × = 14.00% 11 Joshua Inc wants to sell bonds to finance its business expansion To accomplish this, they plan to sell 20-year, $6.2 million face value, zero-coupon bonds The bonds will be priced based on a yield to maturity of 9.5% How many bonds must they sell to raise the required capital? A) 38,079 B) 42,500 C) 54,500 D) 57,500 E) 61,333 Solution: A The bond price is the present value of all future cash flows discounted at the required rate of return (i.e YTM) Note that there are no coupons in this case Bond Price (PV) = $1,000 / (1 + 0.095)20 = $162.82 for bond ($1,000 face value) We have $6,200,000 (face value) bonds therefore the number of bonds required can be calculated as $6,200,000 / $162.82 = 38,079 12 Julia Limited sells a 12-year semi-annual coupon bond that has an effective annual return of 4.6529% What is the amount of each interest payment if the bond issue price is the same as the face value of the bond? A) B) C) D) E) 22.50 22.75 23.00 46.00 45.53 Solution: C First we need to calculate the YTM of the bond as an APR on the basis semi-annual compounding YTM (APR) = m x [(1 + EAR)1/m – 1]= x [(1 + 0.046529) ½ - 1] = 4.6% compounded semi-annually Given that the bond is selling at face value The annual coupon rate =YTM (4.6%) Therefore the semi - annual coupon = ($1,000 x 4.6%)/ = $23 13 Ashley Inc has a 20 year bond outstanding that is currently selling at a 4.5625% premium with a 4.75% coupon rate If the bond pays interest semi-annually what is the yield to maturity? A) B) C) D) E) 4.40% 4.45% 4.54% 4.61% 4.89% Solution: A PV -1,045.625 FV 1000 N X 20 =40 PMT 47.50/2 = 23.75 I 2.2022 per mo YTM = 2.2022 X2 = 4.40% 14 Kamal Ltd has a 10 year bond outstanding that is quoted at a price of $1,002.19, pays interest semi-annually and offers a yield to maturity of 5.97060% What is the coupon rate on this bond? A) B) C) D) E) 3.00% 3.25% 6.00% 6.04% 6.50% Solution: C PV -1,002.19 FV 1000 N X 10 = 20 I 5.97060/2 = 2.98530% per mo CPT PMT 30.00 X = $60 p.a., which is 6.00% 15 A 15 year, 6% coupon bond pays interest annually The bond has a face value of $1,000 What is the change in the price of this bond if the yield to maturity rises from 6.25% to 6.5%? A) B) C) D) E) decreased by 2.37% decreased by 2.43% increased by 2.37% decreased by 2.50% increased by 2.43% Solution: A Initial bond price is the present value of all future cash flows discounted at the yield to maturity Bond Price = PV of coupons + PV of the face value = $60 x PVIFA(6.25%, 15) + $1,000 / (1 + 0.0625)15 = $573.33 + $402.78 = $976.11 If the yield to maturity increases to the 6.5% then the new bond price= $60 x PVIFA(6.5%, 15) + $1,000 / (1 + 0.065)15 = $564.16 + $388.83 = $952.99 Based on the above the change in bond price = (Initial Bond Price – New Bond Price) / Initial Bond Price = ($976.11 - $952.99) / $976.11 or that the bond price declined by 2.37% 16 What is the most likely value of the PVGO for a stock with current price of $50, expected earnings of $6 per share, and a required return of 20 percent? A) B) C) D) E) $10 $20 $25 $30 $40 Solution B Cash Cow Value = EPS / r = / 0.2 = $30 PVGO = Price – Cash Cow Value = 50 – 30 = $20 17 What is the expected, constant growth rate of dividends for a stock with a current price of $100, expected dividend payment of $10 per share, and a required return of 16 percent? A) B) C) D) E) 6.00 percent 6.25 percent 8.00 percent 10.00 percent 12.00 percent Solution A g = r – D1/P0 g = 0.16 – / 100 = 0.06 18 What is the value of the expected dividend per share for a stock that has a required return of 16 percent, a price of $45, and a constant growth rate of 10 percent? A) B) C) D) E) $2.70 $3.60 $4.50 $7.20 $9.60 Solution A D1 = P0 (r – g) D1 = 45 (0.16 – 0.10) = $2.70 19 What should be the stock value one year from today for a stock that currently sells for $35, has a required return of 15 percent, an expected dividend of $2.80, and a constant dividend growth rate of percent? A) B) C) D) E) $37.45 $37.80 $40.25 $43.05 $45.03 Solution A P1 = P0 (1 + g) P1 = 35 (1 + 07) = $37.45 20 ABC companies earnings and dividends are expected to grow at a rate of 10% in year 1, 10% in year 2, 8% in the third year, and at 6% thereafter If current dividend paid was $2.00 and the required rate of return on its common stock is 12% How much should you pay today for one share of ABC? A) B) C) D) E) $19.31 $24.66 $38.62 $51.93 $59.13 Solution C DIV0 = $2.00 DIV1 = $2.20 DIV2 = $2.42 DIV3 = $2.6136 DIV4 = $2.7704 P3 = DIV4 /(r-g) = 2.7704/.12-.06) = $46.1736 P0 = DIV1 / (1 + r) + DIV2 / (1+r )2 + DIV3 / (1 + r)3 + P3 / (1 + r)3 = 2.20 / (1.12) + 2.42 / (1.12)2 + 2.6136 / (1.12)3 + 47.1736 / (1.12)3 = $38.62 Conceptual Questions (2 marks each) 21 Which of the following statements is/are accurate with respect to time value of money? All else being the same I present values increase as the discount rate increases II present values increase as the future value is further away in terms of time III present values are always smaller than future values when both “r” and “t” are positive A) B) C) D) E) I only I and II only II only III only II and III only Solution: D 22 Monika has $6,000 in her investment account She wants to withdraw her funds when her account reaches $10,000 A decrease in the rate of return on her investment over this period will: A) B) C) D) E) Increase the value of her account faster Cause her to wait longer before withdrawing her money Cause the present value of her account to decrease Allow her to withdraw more money sooner Cause the compounding effect to increase Solution: B 23 Suppose you are evaluating two annuities They are identical in every way, except that one is an ordinary annuity and one is an annuity due Assuming an interest rate of 10% per annum, which of the following is true? A) B) C) D) E) The ordinary annuity must have a higher present value than the annuity due The annuity due must have the same present value as the ordinary annuity The ordinary annuity must have a lower future value than the annuity due The present value of annuities will differ by the one annuity payment Answers C & D are correct Solution: C 24 Ashley has a credit card that charges interest every month to her account balance In this case, Ashley is paying an effective annual interest rate that: A) B) C) D) E) Equals the rate stated on her billing statement as the APR Is equal to the APR compounded continuously Is greater than the APR shown on her billing statement Is equal to the annual percentage rate as required by the government Will decline automatically as her account balance declines Solution: C 25 Johan wants to save $1,200 a year In order to maximize his future savings, he should: A) B) C) D) E) Deposit $1,200 into his savings account on the last day of each year Deposit $100 each month as an annuity due Deposit $100 each month as an ordinary annuity Deposit $300 into his account at the end of each quarter Deposit $600 into his account at the end of every six month period Solution: B 26 Coop Products, Inc just issued 10-year, 8% coupon bonds at par Outstanding Elkana Corp bonds, which have a maturity of 10 years, sell at a premium and are viewed by investors as having the same risk as the Coop bonds Therefore, it must be true that the: A) B) C) D) E) Coupon rate on the Elkana bonds is equal to that on the Coop bonds Coupon rate on the Elkana bonds is higher than that on the Coop bonds Coupon payment on the Elkana bonds is lower than that on the Coop bonds Yield to maturity on the Elkana bonds is higher than the YTM on Coop bonds Elkana bonds pay coupons more often than twice a year Solution: B 27 Jimmy wants to compute the present value of a six year semi-annual 8% coupon bond that has a 9% yield to maturity Which one of the following is correct? A) B) C) D) E) The number of interest payments is twelve The present value is assumed to be $1,000 The amount of each interest payment is $80 The bond is selling at a premium Both A & C are correct Solution: A 28 Which of the following is least assured for firms that plowback a portion of earnings into the firm? A) Growth in earnings per share B) C) D) E) Growth in dividends per share Growth in book value of equity Growth in stock price Growth in return on equity Solution D 29 To justify a high P/E ratio, the market must believe one of the following about a firm: A) It has low growth opportunities B) It will have constant dividends forever C) It has high growth opportunities D) It will use low depreciation to increase earnings E) It will increase dividends gradually Solution C 30 A corporation’s board of directors: A) is selected by and can be removed by management B) can be voted out of power by the shareholders C) has a lifetime appointment to the board D) is also elected by customers of the corporation E) is responsible for the day-to-day operations of the corporation Solution B ... E) $923.55 Solution: A Find Semi-annual rate: is = 6.25% -> 0625/2 = 03125 Find EAR: EAR = (1.03125)2 – = 06348 Find the weekly rate: iw = (1+ 0.06348)1/52 -1 = 0.001184 Find Number of payments... $1,002.19, pays interest semi-annually and offers a yield to maturity of 5.97060% What is the coupon rate on this bond? A) B) C) D) E) 3.00% 3.25% 6.00% 6.04% 6.50% Solution: C PV -1 ,002.19 FV 1000 N... pay coupons more often than twice a year Solution: B 27 Jimmy wants to compute the present value of a six year semi-annual 8% coupon bond that has a 9% yield to maturity Which one of the following