Week 7

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Simon Fraser University
Business Administration
BUS 315
George Blazenko

Business 312: Assignment #7 Each of the following questions is a former quiz or exam problem. Cutting, pasting, and printing the suggested solution is not sufficient effort for a mark on this assignment. 1. Today, you purchase a financial asset. This financial asset offers a rate of return of i percent per annum compounded monthly. This rate is not expected to change over the life of this particular financial asset. Equal quarterly payments of $C each are expected on this financial asset indefinitely into the future (i.e., in perpetuity) with the first payment exactly one quarter from today. There are no other cash flows or payments on this financial asset. You reinvest the payments when received at 8% per annum compounded monthly. The annualized holding period rate of return compounded monthlyon your investment between today and immediately after the receipt of the 20’th payment is 8.05 percent. Required: Determine i the per annum rate of return offered on the financial asset (compounded monthly). (Hint: you cannot determine the amount $C in this problem. That is, in representing the HPRR to calculate the effective quarterly rate, the $C cancels from the numerator and denominator.) Solution We need to solve this question by working backwards. Let C be the quarterly payment. Let r be the effective quarterly rate of return on the financial asset. PP=C/r SP=C/r The reinvestment rate for payments is (1+0.08/12) -1=2.013363%. The FV of reinvested payments at the time of the 20’th payment is: 20 C*[1.02013363 -1]/0.02013363 = 24.3297*C. 60 HPRR =1+ 0.0805 ÷ −1 =0.49355  12  SP+FVRP−PP C/ r +24.3297*C−C/ r Now, 0.49355= = ( )= = 24.3297*r. PP C/ r Notice that the “C” cancels in this equation, so that we do not need to know the amount of the quarterly payment. Solving, r=0.0202859, which is an effective quarterly rate. Finally, we need to transform the effective quarterly rate into a per annum rate compounded monthly. That is, i=r*m=12*[(1+0.0202859) -1] = 8.060%, where this “r” is the effective rate/month. 2. Part A: The value of services provided by your house are expected to be $9,500 at the end of this year (measure services at ends of years). The value of services is expected to decline at a rate of 3% thereafter (i.e., depreciation which is a negative growth factor) until your house is demolished exactly twenty years from now (i.e., no further services rendered by the house after this time). If current interest rates are 10% per annum and the value of your house is calculated as the present value of future services, how much can you sell your house for today? Assume that the purchaser uses the same theory of value as you do. Part B: Use the information from part A above to answer the following question. If you enter into a contract to sell your house exactly 12 years from today but you ask for payment immediately, how much should you demand? Assume the purchaser uses the same theory of value and think carefullyabout the future services to be received by the purchaser. Solution The following diagram might be helpful to solve this question. Part a: Use the formula for the PV of a growing (in this case, declining) annuity. The value of  20  9,500 1 −  0.97   0.10 + 0.03   1.1   your home is = $67,170.00 The number 0.97 is calculated as 1+g = 1-0.03 =0.97. Notice also that r-g = 0.10-(-0.03)=0.1+0.03. Part b: If you take payment immediately but deliver your home in twelve years, you should be looking for about: 8 9,500 * 0.97 12   0.97   0.10 + 0.03 1 −  1.1     /(1.1)12 = $10,248.89. What this calculation means is that the purchaser buys the services of the house from year 13 to year 20 (that is 8 years), but pays immediately. Notice that 9500*0.97^12 is the value of services in the 13’th year. Discounting by 1.1^12 arises because the declining annuity is deferred (recall the power is one less than the time of the first cash-flow). 12 9,500  0.97   1−    Alternatively, value = $67,167.00 - 0.10 + 0.03   1.1   = 10,248.89. This calculation is the value of the house today (part a) less the value of service from years 1 to 12. 3. You borrowed from the Bank of Montreal in order to purchase your Burnaby home. The 15 year mortgage calls for equal monthlypayments. The first payment is one month after you receive the borrowed funds. The contract rate of interest on the mortgage is 8.5% per annum compounded monthly. The interest portion of the 39’th payment is $1,750.00. How much did you originallyborrow? Solution The interest portion of a payment is the payment less principal reduction at that payment. Payment less principal portion of the 39’th payment is $1,750. Use the formula for principal reduction at the k’th payment, then, payment - payment/(1+0.085/12)^(180-39+1) = $1,750. So, the payments are $2,764.78. The original amount borrowed in the prospective approach to the determination of the outstanding balance is the PV of remaining payments at the contract rate (over the appropriate payment interval), 2,764.78  1  1− 180 = $280,762.56 0.085/12  (1+0.085/12)  4. When you purchase a financial asset today, for $5,000, it offers a per annum rate of return of 5.7 percent compounded semi-annually. The financial asset offers 27 semi-annual payments where the first payment is two months from today (thereafter, each payment is received in six month intervals). The first payment is $A and each subsequent payment is 1% greater than the previous. You reinvest the payments at an interest rate of 4.25% per annum compounded monthly. You sell the financial asset for $B immediately after you receive the thirteenth payment. At this time the financial asset offers a per annum rate of return of 5.3% compounded semi- annually. Required: Find your annualized holding period rate of return, compounded quarterly, between your purchase and your sale of the financial asset. Solution First, find the amount $A. A   1.01 27 4 /6 $5,000 = 1−  ÷ (1.0285) 0.0285−0.01   1.0285   Solve this equation to find, A=$234.33 13 The fourteenth payment is 234.33*(1.01) = $266.69.  14 $B = 266.69 1−  1.01  =$3,280.63. 0.0265−0.01   1.0265     The semi-annual rate for
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