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AK/ADMS3530 3.0 Assignment #1 Solutions Winter 2007 Question (12 marks) You became incredibly wealthy one day and a huge contributing factor to your success was the education that you’d received at York/Atkinson As a way to thank the school you set up an endowment today in the amount of $5,000,000 But you don’t just hand over $5,000,000 just like that!!! You stipulate that York use the funds to provide scholarships to ADMS3530 students totaling $250,000 annually The scholarships are to commence one year from today and are to continue for as long as ADMS3530 is around (in effect for ever!!!) The market interest rate for the foreseeable future is expected to be 5% per annum compounded annually (a) Based on the above information will York be able to meet your stipulation by providing ADMS3530 students with annual scholarships totaling $250,000 for ever? Provide supporting analysis (2 marks) Answer This is a perpetuity question The formula should be of the form PV perpetuity = Annual Payment / Annual Int Rate OR Endowment = Annual Scholarship / Annual Int Rate Endowment = $250,000 / 5% = $250,000 / 0.05 = $5,000,000 Therefore your $5,000,000 endowment is sufficient to fund annual scholarships of $250,000 forever based on a prevailing annual interest rate of 5% (b) What if your endowment was $4,000,000 instead of $5,000,000 would York still be able to meet your stipulation? If not what would be the maximum annual scholarship payout? Provide supporting analysis (3 marks) Answer Based on the answer above York would NOT be able to meet you stipulation as there would be a shortfall in the endowment of $5,000,000 - $4,000,000 = $1,000,000 With a $4,000,000 endowment the Annual Scholarship = Endowment * Annual Int Rate ADMS3530 3.0 Assignment #1 Solutions Annual Scholarship = $4,000,000 * 0.05 = $200,000 Therefore with a $4,000,000 endowment York would be able to fund a maximum annual scholarship of $200,000 with prevailing annual interest rates of 5% (c) As we all know the cost of education keeps going up and up To accommodate these inflationary pressures you ask that the $250,000 scholarship grow at a constant rate of 1% annually Would York be able to fund this scholarship for ever with your $5,000,000 endowment? If not what would be the maximum annual scholarship? Provide supporting analysis (4 marks) Answer PV perpetuity = Annual Payment / (Annual Int Rate – Growth Rate of Payment) OR Endowment = Annual Scholarship / (Annual Int Rate – Growth Rate of Scholarship) Endowment = $250,000 / (0.05 – 0.01) = $6,250,000 Therefore with an endowment of $5,000,000 there would be shortfall of $6,250,000 – $5,000,000 = $1,250,000 Annual Scholarship = Endowment * (Annual Int Rate – Constant Growth Rate of Scholarship) Annual Scholarship = $5,000,000 * (0.05 – 0.01) = $200,000 With an endowment of $5,000,000 York would be able to fund an annual scholarship of $200,000 that would grow at a constant annual rate of 1% (d) What would your endowment have to be to accommodate annual scholarships of $300,000 with a constant growth rate of 1% annually? Provide supporting analysis (3 marks) Answer Endowment = Annual Scholarship / (Annual Int Rate – Growth Rate of Scholarship) Endowment = $300,000 / (0.05 – 0.01) = $7,500,000 Therefore an endowment of $5,000,000 would be required to fund an annual scholarship of $300,000 that would grow at a constant annual rate of 1% Page ADMS3530 3.0 Assignment #1 Solutions Question (9 marks) You have finally decided that this is the year that you purchase that exotic car (Aston Martin Vanquish (007’s car), Porsche Carrerra GT, Bentley Continental etc…etc.) But your spouse (Salma Hayak or Brad Pitt depending on who you are) insists that you must decide on the maximum purchase price you can afford before you ever look at your first car Assume that the only source of money to finance your purchase is based on your family income The details of which follow… Your annual income is $50,000 before taxes which is taxed at a flat rate of 30% Of the after tax income you can commit 25% annually towards the purchase of the car for the next years (starting year from today) Your spouse’s annual income is $5,000,000!!! (remember it’s Salma or Brad) before taxes which is taxed at a flat rate of 50% Of the after tax income your spouse will only commit 0.5% annually to the purchase of your car for the next years (starting year from today) The prevailing rate of interest on auto loans in the market place is 7% compounded annually Assuming you can get an auto loan, can you afford one of the luxury cars which are priced today at over $250,000? Or will you have to buy a Honda Odyssey (which costs approx $50,000) mini van which is that much more practical as you have small children Show your calculations to support your decision Answer There are a number of ways to solve this question I’ve included solutions for two different approaches The question requires that you calculate the PV of cash that is available to you to finance the purchase of your car If the PV is equal to or greater than $250,000 then you can afford the exotic car If it’s greater than $50,000 but less than $250,000 then you can afford the Honda Odyssey Available money = available from my income + available from spouse’s income Money available from my income = $50,000 * (1 - 0.30) * 0.25 = $8,750 annually for yrs starting one year from today Money available from spouse’s income = $5,000,000 * (1 - 0.50) * 0.005 = $12,500 annually for yrs starting one year from today Solved using the PV formula PV = FV / (1+r)^t From my income, PV of money available for years to = [8,750 / (1 + 0.07)^1] + [8,750 / (1 + 0.07)^2 ]+[ 8,750 / (1 + 0.07)^3 ]+ Page ADMS3530 3.0 Assignment #1 Solutions [8,750 / (1 + 0.07)^4] + [8,750 / (1 + 0.07)^5] =$35,876.73 From spouse’s income, PV of money available for years to = [12,500 / (1 + 0.07)^1] + [12,500 / (1 + 0.07)^2] + [12,500 / (1 + 0.07)^3] = $32,803.95 Total PV available = $35,876.73 + $32,803.95 = $68,680.68 Therefore we cannot afford a $250,000 exotic vehicle however we can afford the $50,000 Honda Odyssey Solved using the annuity formula PV of your income PV of your spouse’s income Total PV of income available = C x [1/r - / r(1+r)^t] = = C x [1/r - / r(1+r)^t] = $68,680.68 8750*((1/0.07)-1/(0.07*(1.07^5))) = 12,500*((1/0.07)-1/(0.07*(1.07^3))) = Question (15 marks) RRSP season is coming up and Canadians will be faced with many investment opportunities Assume that you are a Canadian resident with a marginal tax rate of 40% You have decided to invest in a $1,000 RRSP Since this is tax deductible and you will receive a $400 tax refund given your marginal tax rate (40% of $1,000) You have two options available to you The first is presented here and the second appears in part (c) below In order to buy the RRSP you may borrow $1,000 and then pay down your $1,000 loan with the $400 immediate tax refund Throughout this question interest rate is assumed to be 9% compounded monthly (a) What is the monthly payment required to pay down the loan in one year? (2 marks) Answer Since $400 will be paid down immediately we need to calculate the PMT on a $600 loan balance PMT =? n = 12, i = 9% compounded monthly (i.e monthly interest rate is 0.75%), PV=$600, then: Page $35,876.73 $32,803.95 ADMS3530 3.0 Assignment #1 Solutions 1 PV of an annuity = C × − t r r (1 + r ) PV of an annuity = C × =$600 − 12 0075 0075(1 + 0075) ∴ C = $600/11.4349 = $52.47 (b) To what will the single $1,000 contribution grow over the next 20 years? (2 marks) Answer PV = $1,000, n = 20, i = 9% compounded monthly, then: FV = PV × (1 + r ) t FV = PV × (1 + 0075) FV = $6,009.15 (c) 240 Let’s say that you opted rather than borrow the $1,000 and immediately pay down the $400 and then pay out the loan over a one year period (as in part (a) above) that you agreed last year to make monthly contributions to a savings plan in order to realize a $600 payout by the end of the 12th month (i.e today) and then borrow the remaining $400 to purchase the $1,000 RRSP now, how much would you need to save on a monthly basis? You will immediately pay down the $400 by the end of the first month that it is due (3 marks) Answer FV = $600, n = 12, i = 9% compounded monthly, then: (1 + r ) t − 1 FV of an annuity = C × r (1 + 0075)12 − 1 $600 = C × 0075 C = $600/12.5076 = $47.97 (d) Assuming that you would like to continue your RRSP investment in part (a) or part (c) annually for the next 20 years until your retirement Page ADMS3530 3.0 Assignment #1 Solutions Compare the two RRSP plans by calculating the PV of both options (4 marks) Answer The actual present value of both will be the same The difference will lie in the net expense of carrying the annuity each year for 20 years PMT = $1,000, n = 20, i = 9% compounded monthly, then: the monthly interest rate (rmon) = i/12 = 0.75%, and the EAR = (1 + rmon ) m − = (1 + 0.75%)12 − = 9.38069% 1 PV of annuity = PMT × [ − ] EAR EAR (1 + EAR ) n 1 = $1,000 × [ − ] 0.0938069 0.0938069(1.0938069 20 ) = $1,000 × 8.8862 = $8,886.20 (e) What is the benefit of one plan over the other assuming that the interest rate remains at 9% monthly compounded throughout the next 20 years? (4 marks) Answer The difference will lie in the actual cash flow savings and payments generated by the savings plan versus the interest which needs to be paid on the loans (ignoring the tax expense deduction for now) From (a) we can calculate that the annual interest payments as equal to ($52.47)(12) - $600 = $29.64 This should be a negative cash flow per year from the perspective of the RRSP investor From (c) we can calculate the positive cash flow of $24.36 (= $600 – $575.64) each year This is beneficial to the RRSP investor So the savings plan in (c) is preferred to the borrowing plan in (a) Question (13 marks) Another Canadian investor is planning his retirement information please help him with his calculations (a) Given the following What will be the amount in an RRSP after 25 years, at which time he will retire and live off the proceeds, if contributions of $3,000 are made at each year-end for its first seven years and month-end contributions of $500 are Page ADMS3530 3.0 Assignment #1 Solutions made for the subsequent 18 years? Assume that the plan earns 8% compounded quarterly for the first 12 years, and 7% compounded semiannually for the subsequent 13 years (4 marks) Answer For the first years, i = n = 1(7) = 7, c = 8% = 2%, PMT = $3,000, = 4, and EAR = (1 + i ) − = (1.02)4 – = 0.082432160 c Amount in the RRSP after years will be 1.08243216 − = $26,968.51 0.08243216 FV = $3,000 For the next years, PMT = $500, n = 12(5) = 60, and / 12 c rmon = (1 + EAR ) − = (1.08243216) – = 0.00662271 Amount in the RRSP after 12 years will be (1 + i )n − 1 i FV = PV (1 + i )n + PMT 1.00662271 60 − = $26,968.51 (1.02 )20 + $500 0.00662271 = $76,761.75 For the last 13 years, PMT = $500, n = 12(13) = 156, i = 72% = 3.5%, and EAR = (1.035) − = 0.071225 rmon = (1 + EAR ) − = (1.071225) / 12 c – = 0.00575 Amount in the RRSP after 25 years will be (1 + i )n − 1 i FV = PV (1 + i )n + PMT 1.00575156 − = $76,761.75 (1.035 )26 + $500 0.00575 = $313,490.28 (b) Your investor would like to set up a Scholarship Fund in his name at Atkinson College with a $500 annual award to a deserving applicant The first award will be made at the end of the 15th year of his retirement Subsequent awards will be made at the end of each year perpetually Page ADMS3530 3.0 Assignment #1 Solutions Given the RRSP amount in part (a) above, how much can your investor expect to withdraw at the end of each month, starting the first month after his retirement, and still be able to set up the Scholarship Fund? (Assume the interest rate is 10% compounded monthly throughout his retirement.) (6 marks) Answer i First we need to calculate how much will be required to set aside at Year 15 for the scholarship: i = 1012% = 83333%, C = $500, c = 121 = 12, and i2 = (1 + i )c − = (1.0083333) – = 0.104709 12 PV of a perpetuity = = C r $500 0.104709 = $4,775.14 ii Then we need to calculate the PV of this amount and deduct it from the fund at the start of the retirement $313,490.28 PV of $4,775.14 for 15 years at 10% compounded monthly is $1,072.18 Subtract this from the fund: $313,490.28 - $1,072.18 = $312,418.10 (Alternately by using the calculator you could find the answer by allowing for a residual FV at the beginning of Year 15 of $4,775.14.) iii Calculate the monthly payments on the PV of $312,418.10 at 10% compounded monthly: 1 PV of an annuity = C × − t r r (1 + r ) 1 − 180 008333 008333(1 + 008333) $312,418.10 = C × C = $3,357.96 Page ADMS3530 3.0 (c) Assignment #1 Solutions What complications might occur in achieving the plan in part (b)? (3 marks) Answer The main purpose of the question is to have you discuss the fact that upon retirement no one can really plan when they will die There are two concerns to be addressed: What will happen if the investor dies before the beginning of the 15th year? A simple solution would include a will that ensures that the annuity is taken over by a 3rd party, either as an inheritance or as an asset in the estate The real issue is what if the investor survives beyond the 15th year; there is no more money except for the small amount that has been set aside as a scholarship The investor either becomes a ward of the family or the state Question (14 marks) Throughout this question consider the following bond: face value of $1,000, coupon rate is 8%, semi-annual coupon payments, years of maturity, and a purchase price of $1,055.69 (a) Calculate the current yield and yield to maturity on the bond as of the date of purchase (3 marks) Answer $80 annual coupon 0.08 × $1,000 = = = 7.58% bond price $1,055.69 $1,055.69 If you use your financial calculator, you will find the yield to maturity (YTM) to be 6.4% Alternatively, if you use the approximate formula, the YTM is: Current yield = Page ADMS3530 3.0 Assignment #1 Solutions (face value − current price) maturity YTM = (face value + current price) ($1,000 − $1,055.69) $80 + = = 6.4287% ($1,000 + $1,055.69) annual coupon + (b) Calculate the current yield and bond price on each anniversary date of the bond purchase until maturity Suppose on each of these dates the yield to maturity on the bond is 7%, 6.6%, 6.2%, and 6.36%, respectively (6 marks) Answer To illustrate, let’s compute the current yield and bond price on the first anniversary The calculations for the other anniversaries follow suit face value 1 Bond price = coupon × [ − ]+ t r r (1 + r ) (1 + r ) t $1,000 1 − ]+ = $1,026.64 0.035 0.035(1 + 0.035) 1.035 annual coupon 0.08 × $1,000 $80 Current yield = = = = 7.79% bond price $1,026.64 $1,026.64 = $40 × [ For the second and third anniversaries and the maturity date, the bond price and current yield (in parentheses) is: $1,025.83 (7.8%), $1,017.2 (7.86%), and $1,000 (8%) (c) Assume instead of holding the bond until maturity, you sell the bond on the second anniversary of its purchase (right after you receive the last coupon) Based on your results in part (b) above, what is your total rate of return over this 2-year holding period? What is your annual rate of return over the same period? Assume you can reinvest the previous coupons at an APR of 10% quarterly compounded (5 marks) Answer First, we need to compute the semi-annual interest rate on reinvestment Semi − annual rate = (1 + quarterly rate) − APR = (1 + ) − = 5.0625% Page 10 ADMS3530 3.0 Assignment #1 Solutions Since the income from coupons can be regarded as a 4-period annuity, we apply the future value formula for an annuity: (1 + r ) t − Income = FV of an annuity = coupon × [ ] r (1 + 0.050625) − = $40 × [ ] = $172.57 0.050625 The total rate of return over the two-year holding period is: coupon income + capital loss Rate of return = initial price $172.57 + ($1,025.83 − $1,055.69) = = 13.5182% $1,055.69 The annual rate of return is calculated as: (1 + total rate of return )1 / m − = (1 + 0.135182)1 / − = 6.54% Question (8 marks) This question has two parts, (a) and (b) (a) Do you think it is necessary for bond prices to fluctuate in response to changing interest rates? Why? Please limit your answer to one paragraph (4 marks) Answer Yes Bonds are long-term debt instruments that are issued with fixed coupon rates At time of issue, it is likely that the coupon rate was approximately equal to the expected interest rate for a bond of this rating However, as interest rates change during the period before maturity, investors may be unwilling to purchase the bond in the secondary market unless its price changes such that the investor can obtain a then-current yield to maturity on the investment Thus, if bond prices were fixed, there may not be any opportunities for sale of the bond prior to maturity and investors would be required to make long-term, potentially disadvantageous investment decisions (b) Why should investors be cautious when replying on yield to maturity as a measure of the return on their bond investment? Please limit your answer to one paragraph (4 marks) Answer Many bond investors not intend or, even if they intend, will not be Page 11 ADMS3530 3.0 Assignment #1 Solutions able to, to hold their bonds until maturity In this case it is worthwhile to remember that a yield to maturity is promised only if: 1) the bond is held until maturity, and 2) the issuer does not default Many investors might be better off in calculating or forecasting a total return based upon the time period they expect to hold the bond While this calculation is obviously made with risk, it reminds investors that bond prices can change dramatically and, if you need to raise cash quickly, the once-calculated yield to maturity can be no longer valid Question (15 marks) Rite Bite Enterprises historically has not been a dividend paying company but in its latest board meeting the company decided to start paying a fixed dividend of $1.5 for the next year After the first dividend, they expect that the dividend will grow at 5% over the following three years and it will maintain a constant 6% annual growth rate thereafter (Please draw a timeline to show your work.) a What are the expected dividends in years and 10? (2 marks) b If the discount rate for the stock is 10%, at what price will the stock sell today? (5 marks) c What is the expected stock price three years from now? (3 marks) d If your annual required rate of return is 12% for the next years and 8% thereafter how much will you pay for a share of Rite Bite’s stocks today? (5 marks) Answer: 7…… I I I I I I I I I D0 D1 D2 D3 D4 g=5% D5 D6 g=6% P4 a D0 = $0, D1 = $1.5 D2 = $1.5*1.05 = $1.575 D3 = $1.575*1.05 = $1.6538 D4 = $1.6538*1.05 = $1.7365 D5 = $1.7365*1.06 = $1.8407 D10=$1.8407*(1.06)^5 = $2.4633 Page 12 ADMS3530 3.0 Assignment #1 Solutions b P4 = $1.8407/(0.10-0.06) = $46.02 PV of Dividends and P4 PV of D1= $1.5*(PVIF10%,1) = $1.3636 PV of D2 = $1.575*(PVIF10%,2) = $1.3017 PV of D3 = $1.6538*(PVIF10%,3) = $1.2425 PV of D4 = $1.7365*(PVIF10%,4) = $1.1861 PV of P4 = $46.02*(PVIF10%,4) = $31.4323 or or or or or $1.5/1.10 $1.575/1.102 $1.6538/1.103 $1.7365/(1.10)4 $46.02/(1.10)4 P0 = $1.3636+$1.3017+$1.2425+$1.1861+$31.4323 = $36.53 c P3 =$1.7365*(PVIF10%,1) + $46.02*(PVIF10%,1) = $1.5786+$41.8364=$43.42 d 7…… I I I I I I I I I D0 D1 D2 D3 D4 g=5% r=12% D5 D6 g=6% r=8% P4 D1 = $1.5 D0 = $0, D2 = $1.5*1.05 = $1.575 D3 = $1.575*1.05 = $1.6538 D4 = $1.6538*1.05 = $1.7365 D5 = $1.7365*1.06 = $1.8407 P4 = $1.8407/(0.08-0.06) = $92.04 PV of Dividends and P4 PV of D1= $1.5*(PVIF12%,1) = $1.3393 PV of D2 = $1.575*(PVIF12%,2) = $1.2556 PV of D3 = $1.6538*(PVIF12%,3) = $1.1771 PV of D4 = $1.7365*(PVIF12%,4) = $1.1036 PV of P4 = $92.04*(PVIF12%,4) = $58.4931 or or or or or $1.5/1.12 $1.575/1.122 $1.6538/1.123 $1.7365/(1.12)4 $92.04/(1.12)4 P0 = $1.3393+$1.2556+$1.1771+$1.1036+$58.4931 = $63.37 Question (14 marks) This question has two parts, (a) and (b) (a) The common stock of Veritas Ltd is currently trading at $ 48 on the TSX You also know the following information about the Veritas stock: its P/E ratio is 16, Page 13 ADMS3530 3.0 Assignment #1 Solutions Veritas Ltd has a required rate of return of 10% on its common equity, and the company’s dividends are expected to grow at 6% forever What is Veritas’ payout ratio today? What is Veritas’ plowback ratio today? (6 marks) Answer By use of the constant growth dividend discount model, we can compute the dividend paid by VERITAS Ltd today, D0, as: D (1 + g ) D (1 + 6%) D0 (1.06) P0 = ⇒ $48 = = r−g 10% − 6% 0.04 $48 × 0.04 ∴ D0 = = $1.811 1.06 We know that this company has an earning per share (EPS) of $48/16 = $3 Using the definition of payout ratio, we can compute VERITAS’ payout ratio today as $1.811/$3 = 60.37% Finally, VERITAS’ plowback ratio today is (1 – payout ratio) = 39.63% (b) Butterfly Tractors Corp has experienced an outstanding growth in recent years, which is expected to continue in the future In particular, earnings and dividends are forecasted to grow at a rate of 5% during the next years, and at a constant rate of 6% thereafter Butterfly just paid a dividend of $2 and the required rate of return on its stock is 12% Suppose that you buy one share of Butterfly stock today and sell it at the end of Year Compute the 1-year dividend yield, 1-year capital gains yield, and 1-year total rate of return on your investment for each of Years 1, 2, 3, and Assume that previous dividends are not reinvested (8 Marks) Answer: We first calculate the stock price today, P0: D0 = $2 D1 = $2*1.05=$2.1 D2 = $2.1*1.05 = $2.205 D3 = $2.205*1.05 = $2.3153 D4 = $2.3153*1.06 = $2.4542 P0 = $2.1 $2.205 $2.3153 $2.4542 + + +[ ] /(1.12) 3 − 1.12 1.12 12 % % 1.12 = 1.875 + 1.7578 + 1.648 + (40.9033 / 1.4049) = $34.396 Likewise, we can compute Butterfly’s stock price at the end of Year 1, 2, 3, and as: Page 14 ADMS3530 3.0 Assignment #1 Solutions P1 = $2.205 $2.3153 + 40.9033 + = $36.4224 1.12 (1.12) P2 = $2.3153 + $40.9033 = $38.588 1.12 P3 = $2.4542 12% − 6% P4 = $2.4542 × (1 + 6%) =$43.3583 12% − 6% = $40.9033 Therefore, we have the following 1-year returns: Year Dividend Yield 2.1/34.396=6.11% 2.205/36.4224=6.05% 2.3153/38.588=6% 2.4542/40.9033=6% Capital Gains Yield (36.4224-34.396)/34.396=5.89% (38.5880-36.4224)/ 36.4224=5.95% (40.9033-38.5880)/ 38.5880=6% (43.3583-40.9033)/ 40.9033=6% Total Rate of Return 12% 12% 12% 12% Page 15 ... PV of your income PV of your spouse’s income Total PV of income available = C x [1/r - / r(1+r)^t] = = C x [1/r - / r(1+r)^t] = $68,680.68 8750*((1/0.07 )-1 /(0.07*(1.07^5))) = 12,500*((1/0.07 )-1 /(0.07*(1.07^3)))... = $1.8407/(0.1 0-0 .06) = $46.02 PV of Dividends and P4 PV of D1= $1.5*(PVIF10%,1) = $1.3636 PV of D2 = $1.575*(PVIF10%,2) = $1.3017 PV of D3 = $1.6538*(PVIF10%,3) = $1.2425 PV of D4 = $1.7365*(PVIF10%,4)... = $1.8407/(0.0 8-0 .06) = $92.04 PV of Dividends and P4 PV of D1= $1.5*(PVIF12%,1) = $1.3393 PV of D2 = $1.575*(PVIF12%,2) = $1.2556 PV of D3 = $1.6538*(PVIF12%,3) = $1.1771 PV of D4 = $1.7365*(PVIF12%,4)
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