### The Effective Annual Interest Rate on Loan

Added on - 28 May 2020

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QUESTION `1. [6 + 6 = 12 Marks.]a)This is a two period certainty model problem.Assume that Jillian Black has a sole income from Halcyon Ltd in which she owns10% of the ordinary share capital.In its financial year 2016-17 just ended, Halcyon Ltd reported net profits after taxof $800,000, and announced its net profits after tax expectation for the nextfinancial year, 2017-18, to be 20% higher than this year’s figure. The companyoperates with a dividend payout ratio of 80%, which it plans to continue, and willpay the annual dividend for 2016-17 in mid-January, 2018, and the dividend for2017-18 in mid-January, 2019.In mid-January, 2019, Jillian wishes to spend $100,000, which will include thecost of new furniture. How much can she consume in mid-January, 2018 if thecapital market offers an interest rate of 8% per year?Answer:Income EstimationsYear2016-172017-18Net Profit800000960000Dividend Payout80%80%Dividend640000768000Equity holding of Jillian Black10%10%Dividend of Jillian Black6400076800Two period Certainty problemYearIncomeOpening AmountInterestConsumptionBalance2016-201764000057606912054176.8814943.122017-20187680014943.128256.8810000010000001

b)This question relates to the valuation of shares.Ram Shack Ltd has just paid a dividend of $3.00 a share. Investors require a13% per annum return on investments such as Ram Shack. What would a sharein Ram Shack Ltd be expected to sell for today (January, 2018) if the dividend isexpected to increase by 20% in January, 2019, 16% in January, 2020, 12% inJanuary, 2021 and thereafter by 5 per cent a year forever, from January, 2022onwards?Answer:YearDividendPVF @ 13%PV ofDividend20183.000.8852.65520193.600.7832.81920204.180.6932.89420214.680.6132.869202261.39*0.54333.319Price of Share44.555* 4.68 x (1+0.05)/(0.13-0.05)= 61.392

QUESTION 2. [(4 + 4) + (2 + 3 + 3 + 3 + 3) = 22 Marks]a)This question relates to the time value of money and deferred annuities.Colin Way is age 40 today and plane to retire on his 65thbirthday. With futureinflation, Colin estimates that he will require around $2,000,000 at age 65 toensure that he will have a comfortable life in retirement. He believes that he cancontribute $3,000 at the end of each month, starting in one months’ time andfinishing on his 65thbirthday.i)If the fund to which he contributes earns 6% per annum, compoundedmonthly (after tax), how much will he have at age 65? Will he haveachieved his targeted sum? What is the surplus or the shortfall?Answer:Total fund Balance on his 65th Birthday20,78,981.89Required Amount20,00,000.00Surplus78,981.89Hence, the surplus amount is $78981.89.(Refer Appendix)ii)Using the fund balance, Colin then wishes to commence a monthlypension payable by the fund starting one month after his 65thbirthday,and ending on his 85thbirthday, after which he expects that the fund willbe fully expended. If the fund continues to earn the above return of 6%per annum, compounded monthly, how much monthly pension will Colinreceive, if the fund balance reduces to zero as planned after the lastpension payment on his 85thbirthday?Answer:The fund balance $ 78,981.89The amount of monthly pension = $567.08(Refer appendix)b)This question relates to loan repayments and loan terms.Ray and Betty Read wish to borrow $600,000 to buy a home. The loan fromBattlers Bank requires equal monthly repayments over 20 years, and carries.aninterest rate of 4.8% per annum, compounded monthly. The first repayment isdue at the end of the first month.You are required to calculate:i)The effective annual interest rate on the above loan.Answer:Nominal Interest rate0.048Monthly Interest Rate0.004Effective Interest Rate(1+r/n)^nEffective Interest Rate4.91%ii)the amount of the monthly repayment (consisting of interest and principalrepayment components) if the same amount is to be repaid every monthover the 20 year period of the loan.Answer:3Installment=Loan Amount/(1+r)^240Loan6,00,000.00R0.004Installment=3,893.74

(Refer appendix)iii)the amount of $X, if - instead of the above - Battlers Bank agrees thatRay and Betty will repay the loan by paying the bank $3,300 per monthfor the first 12 months, then $3,750 a month for the next 12 months, andafter that $X per month for the balance of the 20 year term.Answer:The revised loan amount to be taken for calculating the amount of X willbe $ 570184.82 and interest rate will be 4.8%.Amount of installment= $ 570184.82/ Cumulative present value factor @4.8%The amount of X= 570184.82/144.45Hence X= 3,947.28(Refer Appendix)iv)how long (in years and months) it would take to repay the loan if,alternatively, Ray and Betty decide to repay $3,500 per month, with thefirst repayment again being at the end of the first month after taking theloan, and continuing until the loan was repaid.Answer:Loan amount: $600000Installment= $3500/MonthInterest= 4.8%Hence, the no. of years required to repay the loan= 24.17 yearsWe can say 24 Years and 2 months.(Refer Appendix)v)under option iv) above, the amount of the final repayment. [NOTE:Towards the end of the loan repayment period, after the final full monthlyinstalment of $3,500 is paid, a lesser amount is likely to be outstanding.That amount, plus interest to the end of the following month, is the finalloan repayment amount.]Answer:The extra amount to be paid over and above theloan amount is $ 204.23Hence, the amount of final repayment $ 6,00,204.234

QUESTION 3. [(2 + 2 + 3 + 3 + 3 + 3 = 16 marks]This question relates to alternative investment choice techniquesLaurel Hardy is considering the following cash flows for two mutually exclusiveprojects.Year Cash Flows, Investment X ($) Cash Flows, Investment Y ($)0 -42,000 -42,0001 12,000 18,0002 18,000 18,0003 27,000 18,000You are required to answer the following questions:i)If the cash flows after year 0 occur evenly over each year, what is thepayback period for each project, and on this basis, which project wouldyou prefer?YearCash Flows XCash Flows Y0-42000-42000112000180002180001800032700018000As given in the question the cash flows occur evenly. So, we take the averageof cash flows for project X and project Y cash flows are already even.Average =(12000+18000+27000)/357000Average Cash flows19000PaybackPeriod(Years)2.212.33Hence, project X will be preferred over project Y.IN THE REMAINING PARTS, ASSUME THAT ALL CASH FLOWSOCCUR AT THE END OF EACH YEAR.ii)Would the payback periods then be any different to your answer in i)? Ifso, what would the payback periods be?YearCash FlowsXCumulative CashFlowsCash FlowsY0-42000-42000-42000112000-3000018000218000-12000180003270001500018000PaybackPeriod(Years)2.442.335

In this case the cash flows are even and hence, the payback period ofproject X increased.iii)Sketch freehand the net present value (NPV) profiles for each investmenton the same graph. Label both axes and the NPV profile for eachinvestment.Di0%2%4%6%8%10%12%14%16%18%20%22%24%26%28%30%(15,000...(10,000...(5,000...-5,000...10,000...15,000...20,000...NPV(X)NPV(Y)iv)Calculate the internal rate of return (IRR) for each project and indicatethem on the graph. [NOTE: It is satisfactory if the approximate IRR is6Discounting RateNPV(X)NPV(Y)0%15,000.0012,000.002%12,508.459,909.904%10,183.387,951.646%8,010.416,114.228%5,976.682,763.3410%4,070.621,232.9612%2,281.841,232.9614%600.96-210.6216%-980.48-1,573.9918%-2,470.16-2,863.0920%-3,875.00-4,083.3322%-5,201.33-5,239.6524%-6,454.87-6,336.5426%-7,640.86-7,378.1128%-8,764.07-8,368.1030%-9,828.86-9,309.97

calculated for Investment X by trial and error, and stated as a percentagecorrect to the nearer whole number. The IRR for Investment Y should becalculated as a percentage exactly, correct to 1 decimal place.]Project XYearCashFlowsPVF @14%PVPVF @15%PVPVF @14.75%PV0-420001.000-42000.001.000-42000.001.000-42000.001120000.87710526.320.87010434.780.87110457.892180000.76913850.420.75613610.590.75913670.943270000.67518224.230.65817752.940.66217871.16NPV600.962-201.6930.00IRR=LDR+NPV atLDRx (UDR-LDR)NPV atLDR- NPVat UDRIRR=14.75%Project YYearCashFlowsPVF @ 13%PVPVF@14%PVPVF@13.7%PV0-420001.000-42000.001.000-420001.000-420001180000.88515929.200.87715789.4740.88015831.132180000.78314096.640.76913850.4160.77413923.63180000.69312474.900.67512149.4870.68012245.91NPV500.75-210.62351IRR=LDR+NPV at LDRx (UDR-LDR)NPV at LDR- NPV at UDRIRR=13.7%7

IRR TableIRRNPV(X)NPV(Y)13%1428.464500.74713.70%846.5580.64714%600.962-210.62414.75%0.000-728.10715%-201.693-901.948IRR13%13.70%14%14.75%15%-1500.000-1000.000-500.0000.000500.0001000.0001500.0002000.0002500.000IRR GraphNPV(Y)NPV(X)v)Calculate the exact crossover point and indicate it on the above graph.Answer:Calculation of Crossover PointYearProject X CashFlowsProject Y Cash FlowsDifference0-42000-42000011200018000-6000218000180000327000180009000Crossover Point22.47%8

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