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ADMS 3530 – Summer 2007 Assignment # – Solutions Question (15 marks) You issue debt of $30,000 on October 1, 2007 You decide to payback this debt by making equal payments every three months during the next years The first payment was starts on January 1, 2008 (a) If the effective annual rate (EAR) was 12%, what was the amount of each payment? (5 marks) The EAR = 12% means that the 3-month interest rate (periodic rate) is: iq= r = (1 + 12%) − = 2.8737% The debt can be seen as the present value of an annuity of 16 equal payments: 1 PV = C × − 16 r r (1 + r ) 1 $30,000 = C × − 16 2.8737% 2.8737% × (1 + 2.8737%) C = $2,365.33 Alternatively, Using your calculator: i = 2.8727%, PV = $30,000, n = x = 16, FV =0, COMP PMT Ỉ PMT = $2,365.32 (b) What is the outstanding debt just after the 6th payment? (5 marks) The outstanding debt just after the 6th payment is the present value of the remaining 10 payments: 1 PV = $2,365.33 × − 10 2.8737% 2.8737% × (1 + 2.8737%) PV = $20,307.39 Alternatively, Using your calculator: i = 2.8727%, PV = 2365.32, n = 10 (periods remaining) , FV =0, COMP PV Ỉ PV = $20,307.33 (c) If just after the 6th payment, the effective annual rate increases to 14% and you decide to keep the same payments except for the last one which has to be greater What would be the amount of the last payment? (5 marks) ADMS3530 3.0 Assignment #1 If the EAR becomes 14% just after the 6th payment, when the outstanding debt is $20,307.39, then the 3-month rate would be: Iq new = r = (1 + 14%) − = 3.3299% The outstanding debt of $20,307.39 is equal to the present value of the remaining 10 payments of $2,365.33 plus the extra payment X (smaller than the regular payment) occurring at the same time as the last payment, we have: 1 X $20,307.39 = $2,365.33 × − + 10 10 3.3299% 3.3299% × (1 + 3.3299%) (1 + 3.3299%) X = $646.86 which means that the last payment is equal to: $2,365.33 + $646.86 = $3,012.20 Alternatively, Using your calculator: Step 1) Calculate the new PV of original payments at end of 6th payment with new iq): i = 3.3299%, PMT = 2365.32, n = 10 (periods remaining) , FV =0, COMP PV Ỉ PV = $19,841.17 Step 2) Calculate the difference in original balance ($20,307,39) an PV of payments with new iq: Difference at t=6= Remaining balance at t=6 = $20,307.39 - $19,841.17 = $466.22 at t=6 But you are asked for the amount of the last payment at t=16! Remaining balance with new interest rate = $466.22 at t=6 or 466.22 x (1.033299)10 = $646.93 at t=16 -> last payment at t=16= $646.22 + 2365.33 = $3012.25 Question (15 marks) Consider the following monthly cash flows (see the diagram below): - Cash flows of an amount X are made for months 1, 3, 5, …, 17 and 19 (the 10 oddnumbered months) - Cash flows of an amount Z are made for months 2, 4, 6, …, 18 and 20 (the 10 evennumbered months) Today X Z X Z X Z 19 20 The APR is 6% and is compounded on a monthly basis Page ADMS3530 3.0 (a) Assignment #1 How much money will be accumulated 20 months from now if X = $2,000 and Z = $0? (5 marks) The monthly interest rate is 0.5% but since the X’s cash flows are made every two months, we need to calculate the 2-months equivalent interest rate: I2m = r = (1 + 0.5%) − = 1.0025% The future value 19 months from now is simply given by the annuity formula: (1 + 1.0025%)10 − 1 FV19 = $2,000 × = $20,926.80 1.0025% Alternatively, Using your calculator: I2m = 1.0025%, n=10, PMT = 2,000, PV=0, COMP FV Ỉ FV = $20,926.80 at t=19 But you need FV at t=20: Recall from above that Im = 0.50% So FV at month 20 = 20,926.80 x (1.005)1 = $21,031.43 (b) What is the present value of these cash flows today if X = $2,000 and Z = - $700? (5 marks) The present value of the Z’s cash flows is given by: 1 PVZ = −$700 × − = −$6,629.02 10 1.0025% 1.0025% × (1 + 1.0025%) Alternatively, Using your calculator: I2m = 1.0025%, n=10, PMT = -700, FV=0, COMP PV Ỉ PVz20 = -$6629.02 at t=20 (Since fist payment begins at t=2 and “i" is calculated for every month period, and last payment is at t=20) And the present value of the X’s is given by: 1 PV X = $2,000 × − × (1 + 0.5%) = $19,034.77 10 1.0025% 1.0025% × (1 + 1.0025%) The total present value is equal to: PV = $19,034.77 − $6,629.02 = $12,405.75 Page ADMS3530 3.0 Assignment #1 Alternatively, Using your calculator: Since X begins at t=1, using your callculator for a regular annuity will give PV at t =-1: I2m = 1.0025%, n=10, PMT = 2000, FV=0, COMP PV Ỉ PVx-1 = -$18,940.07 at t= -1 (Since fist payment begins at t=1 and “i" is calculated for every month period, and last payment is at t=19, you are really calculating Pv of an annuity at t=-1) To adjust for PVx at t=0-> 18,940.07 x (1.005)1 = $19,034.77 Total PV (Z plus X) = -$6629.02 + $19,034.77 = $12,405.75 (c) If the present value of these cash flows today is $14,000, and the future value of the Z’s cash flows 20 months from now is $5000, what is the value of X? (5 marks) We know the future value of the Z’s 20 months from today, this allows us to find their present value today: PVZ = $5,000 = $4,525.31 (1 + 1.0025%)10 Alternatively, Using your calculator: I2m = 1.0025%, n=10, FVz20 = 5000, COMP PVz0 -> PVz0 = $$4,525.31 The present value of the X’s is then equal to: $14,000 - $4,525.31 = $9,474.69 The value of X can be derived from the following equation: 1 $9,474.69 = X × − × (1 + 0.5%) 10 1.0025% 1.0025% × (1 + 1.0025%) X = $995.51 Alternatively, Using your calculator: I2m = 1.0025%, n=10, PVx0 = 9,474.69, FV=0, COMP PMTx Ỵ PMTx = 1000.49 assuming “X” begins at t=2 BUT “X” begins at t=1! Ỵ PMTx = 1000.49/1.005 (monthly rate) = $995.51 Page ADMS3530 3.0 Question Assignment #1 (20 marks) You’ve decided to fund an orphanage today so that society could benefit from that service forever The orphanage is to open its operations a year from today and you are wiling to commit $5,000,000 towards this effort You’ve been informed that it would cost approximately $250,000 a year to maintain the operations (assume for now that there is no inflation) (a) Based on the above information is $5,000,000 sufficient to meet the annual operating needs of the orphanage forever? Assume the interest rate is 6% per annum Provide supporting analysis (5 marks) Answer This is a perpetuity question The formula should be of the form PV perpetuity = Annual Payment / Annual Int Rate OR Commitment = Annual Operations / Annual Int Rate Commitment Required = $250,000 / 6% = $250,000 / 0.06 = $4,167,667 Therefore $5,000,000 is sufficient to fund the annual operations based on an annual interest rate of 6% (b) If the interest rate was 5% and you committed $4,000,000 instead of $5,000,000 would it be enough to sustain the operations? If not what would be the maximum amount that would be available annually? Provide supporting analysis (5 marks) Answer Based on the answer above you would NOT be able to meet the annual needs as there is a shortfall of $4,167,667 - $4,000,000 = $167,667 With a $4,000,000 commitment only $200,000 would be available to cover the annual costs Calculated as $4,000,000 * 0.05 = $200,000 Therefore with a $4,000,000 commitment you would be able to fund a maximum of $200,000 of the annual operating costs (c) As with everything the cost of operating the orphanage is expected to go up annually The interest rate is 5% and you expect the annual costs to grow at a constant rate of 1% (inflation) starting in year Would you be able to fund this increase with your $5,000,000 commitment? If not what would be the maximum available amount? Provide supporting analysis (5 marks) Page ADMS3530 3.0 Assignment #1 Answer You are really dealing with a growing perpetuity here where the inflation rate is a negative “g” i.e PV growing perpetuity = C/ (r-g) -> Annual Exp = Commitment * (Annual Int Rate – Inflation Rate) Annual Scholarship = $5,000,000 * (0.05 – 0.01) = $200,000 With $5,000,000 you would NOT be able to fund the $250,000 in annual operating costs but would be able to fund $200,000 annually if inflation remained at 1% Alternatively, you may solve for the real interest rate where (1 = real rate) = (1 + nominal rate) / (1+ inflation rate) = 1.05/1.01 = 1.039 -> real rate = 3.90% Annual Scholarship = $5,000,000 * (0.03960) = $198,000.00 (d) What would your commitment have to be to accommodate annual operating costs of $300,000 with inflation of 1% annually? Assume the int rate is 5% Provide supporting analysis (5 marks) Answer Commitment = Annual Exp / (Annual Int Rate – Inflation Rate) Committment = $300,000 / (0.05 – 0.01) = $7,500,000 Alternatively, using the real rate calculated in “d”, Committment = $300,000 / (0.03960) = $7,575,758 Therefore $7,500,000 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 ADMS 3530 – Assignment #1 – SOLUTIONS Question : Ch.4 Mortgages (20 marks) You have recently graduated from York University and have landed a terrific job near Barrie You have been reading up on the potential growth of the Barrie area and have decided to buy a home before prices increase further You are looking at a house worth $250,000 and you are able (with the help from a rich aunt) to make a down-payment of 10% The posted mortgage rate from ING Direct is 6.5% for a year-term and a 25 year amortization period a) What will be your monthly mortgage payment? (5 marks) b) You earned an “A” in your ADMS 3530 course and have decided to go with a 20 year (instead of the regular 25 year) amortization period What will be the total interest savings over the life of the mortgage by reducing the amortization period to 20 years from 25 years? (7 marks) c) Suppose in 2010 (3 years from today),you decided to re-finance your mortgage as posted mortgage rates for a year term have decreased to 5.75% Also, you would like to have a weekly payment plan instead of monthly What will be your weekly payment on the mortgage in 2010? (assume there are 52 weeks in a year and you originally went with the 20 year amortization period in 2007) (8 marks) Solution: a) Recalling from p 113 of your text, mortgage rates in Canada are quoted as APRs, compounded semi-annually: Step 1: Find the semi-annual period rate: Ö Semi-annual Period rate = 6.5% / = 0325 = 3.25% Step 2: Find EAR: EAR = (1 + 0325) -1 = 0.066056 or 6.6056% Step 3: Find monthly (or period) rate Im = (1 + 0.066056) 1/12 - = 0.005345 or 0.5345% per month Step 4: Calculate the number of months in 25 years n = 12 * 25 = 300 months or periods Step 5: Calculate monthly payment: Note: We are looking at a PV Annuity problem where we are asked for the monthly PMT variable: Page ADMS3530 3.0 Assignment #1 Using your calculator: n= 300 (25 years x 12 months/yr) i = 0.5345% per month PV = 90% x $250,00 = 225,000 FV = COMP PMT -> PMT = 1507.15 Thus, the monthly payment on the mortgage is $1507.15 per month b) Total interest paid using the 25 year amortization period: = $1507.15 x 300 payments - $225,000 = $452,145 - $225,000 = $227,145 Total interest paid, using the 20 year amortization period: First – find new monthly payment: Using your calculator: n= 240 (20 years x 12 months/yr) i = 0.5345% per month PV = 90% x $250,00 = 225,000 FV = COMP PMT -> PMT = $1666.17 = $1666.17 x 240 payments - $225,000 = $399,881 - $225,000 = $174,881 Ö Total interest savings by paying off the mortgage in 20 years, instead of 25 years, is $227,145 - $174,881 = $52,264! c) The remaining balance of the mortgage loan (or remaining principal) after years will be (with 17 years remaining, or 204 months) Using your calculator: n= 204 i = 0.5345% per month PMT = $1666.17 FV = COMP PV -> PV =$ 206,653.06 First – find weekly payment Note we have the same EAR from step above but we must find the weekly interest rate i.e Find weekly interest rate First, EAR = (1 + 0575/2)2 -1 = 0583266 Iw = (1 + 0.0583266) 1/52 - Page ADMS3530 3.0 Assignment #1 = 0.00109077 or 0.1091% per month next: Calculate the number of weeks in 17 years n = 52 * 20 = 884 weeks next: Calculate weekly payment: Using your calculator: n= 884 weeks i = 0.1091% per week! PV = $206,653.06 (remaining balance after years) FV = COMP PMT -> PMT =$ 364.46 Your new weekly payment starting in 2010 will be $364.46 Question 5: Ch.5 Bonds (20 marks) In 1998, Algoma Inc issued 20-year bonds at par value with a coupon rate of 9% In 2004, Algoma was granted court protection under the CCAA due to a looming liquidity crisis and the yield to maturity of the bonds increased to 19% The bonds pay interest semi-annually (a) What happened to the price of the bonds in 2004? (4 marks) (b) Compute the current yield in 1998 and in 2004 Why is the current yield not an appropriate measure of the return for the Algoma bonds? (6 marks) (c) In 2007 merger mania continued in Canada and Algoma became the target of a takeover bid by India-based Essar Global Ltd for $56 per share The yield to maturity of the bonds decreased to 7% What was a bondholder’s annual rate of return from 1998 to 2007? (Assume the coupons are not reinvested.) (6 marks) (d) What is the rate of return for a bondholder who purchased the bond in 2004 and sold it in 2007? (Assume the coupons are not reinvested.) (4 marks) a) Solution Price of bonds in 1998: $1,000 (i.e they were issued at face value) Coupon rate of bonds = 9% (Issue Price = Face Value = Par Value) Maturity date of bonds = 2018 Price of bond in 2004: Using your calculator: I = 19/2 = 9.5 n = 14 years x2 = 28 semi-annual periods PMT = 45 FV = 1000 COMP PV -> PV = $515.15 Price of bonds dropped to $515.15 in 2004 Page ADMS3530 3.0 Assignment #1 Solution b) Current Yield in 1998 = $90/$1,000 = 9% (Since bonds are issued at face value, Current Yield = YTM) Current Yield in 2004 = $90/$515.15 = 17.47% • The current yield may not be an appropriate measure of the return for Algoma bonds because current yield calculation ignores any potential capital gains or losses • If Algoma is successful in its restructuring efforts, the bondholder who purchased the bond in 2004 may reap a large capital gain at the maturity date of the bond in 2008 (i.e $1,000 - $515.15 = $484.85) Conversely if Algoma goes bankrupt and cannot meet its bond obligation in 2018, the bondholder who purchased the bond in 2004 (or in 1998) may lose all of his/her capital investment in the bond • c) Solution: Price of bond in 2007: Using your calculator: I - 07/2 = 3.5 N= 11 years x = 22 periods remaining PMT = 45 FV = 1000 COMP PV -> PV = $1151.67 Price of bond in 2007 was $1151.67 Rate of return on Investment = Coupon Income + Price Change Original Investment = $45 x 18 payments + (1151.67 – 1000) $1000 = 96.167% over years Annual rate of return = (1.96167) 1/9 – = 7.774% d) Rate of return on Investment = Coupon Income + Price Change Original Investment = $45 x payments + (1151.67 – 515.15) $515.15 = 175.97% over years Annual rate of return = (2.7597) 1/3 – = 40.27% Page 10 ADMS3530 3.0 Assignment #1 Question Bonds (10 marks) This question has the following three parts, (a), (b), and (c) (a) Compute the yield to maturity on a 20-year bond that was issued one year ago and has a coupon rate of 8%, a face value of $1,000, and currently sells at $1,104.21 The bond pays coupons semi-annually (3 marks) (b) Suppose the real rate of return on the bond in part (a) is expected to be 3.25% annually What is the implied annual inflation rate, given the nominal yield to maturity in part (a)? (2 marks) (c) Sometimes bonds may be issued with a face value other than $1,000 Suppose a 20-year 7% coupon bond with semi-annual coupon payments is currently priced at $901.95, yielding 9% annually What is the face value of this bond? Suppose you are trying to raise $1.5 million with this bond, how many bonds you need to issue? (5 marks) Solution a) Using your calculator: n = 19 years x = 38 periods PMT = 40 PV = - $1104.21 FV = $1000 COMP “I” -> I = 3.50% +> YTM = 3.50% x = 7.00% b) The inflation rate is equal to ((1+7.0%) / (1+3.25%)) -1 = 3.63196% c) Using your calculator: n = 20 years x = 40 periods PMT = 35 PV = - $901.95 I = 09/2 =4.5% COMP FV => FV = $1500.00 (note: FV = par value = maturity value) Number of bonds you need to issue: = Dollar amount to be raised / Current price per bond = $1,500,000 / $901.95 = 1663 bonds Page 11 ... PMT -> PMT = $1666.17 = $1666.17 x 240 payments - $225,000 = $399,881 - $225,000 = $174,881 Ö Total interest savings by paying off the mortgage in 20 years, instead of 25 years, is $227,145 - $174,881... calculating Pv of an annuity at t =-1 ) To adjust for PVx at t= 0-> 18,940.07 x (1.005)1 = $19,034.77 Total PV (Z plus X) = -$ 6629.02 + $19,034.77 = $12,405.75 (c) If the present value of these cash... $20,307.39, then the 3-month rate would be: Iq new = r = (1 + 14%) − = 3.3299% The outstanding debt of $20,307.39 is equal to the present value of the remaining 10 payments of $2,365.33 plus the
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