Consider a second alternative for accumulating funds to buy the new billing syst
ID: 2663593 • Letter: C
Question
Consider a second alternative for accumulating funds to buy the new billing system. In lieu of a lump sum investment, assume that five annual payments of $92,000 are made at the end of each year.A) What type of annuity is this?
B) What is the present value of this annuity if the opportunity cost rate is 10.5% annually? 10.5% compounded semiannually?
C)What is the future value of this annuity if the payments are invested in an account that pays 10.5% interest annually? 10.5% compounded semiannually?
D) What annual interest rate is required to accumulate the $600,000 needed to make the purchase, assuming a $90,000 annual payment?
E) What size annual payment would be needed to accumulate $600,000 under the annual compounding at a 10.5% interest rate?
Explanation / Answer
A) A series of cash flows that occur at the end of the each period for some fixed number of periods is called an ordinary annuity. B) Calculating the present value of annuity using formula: Annuity present value = C x [ (1 - Present value factor) / r ] Where C is the fixed payment Present value factor = [1/(1+r)^t] r is the discount rate t is the number of periods If we substitute the values in the above formula, we get the annuity present value. This annuity present value can also be calculated using excel sheet: Step1: Go to excel and click "insert" to insert the function Step2: Select the "PV" function as we are finding the present value of annuity in this case. Step3: Enter the values as Rate = 10.5%; Nper = 5; PMT = -92000; FV = 0 Step4: Click "OK" to get the desired value The value comes to "344,342.96" Therefore, the present value is $344,343 Calculating the present value for semi-annual period Discount rate = 10.5% / 2 Number of periods = 5*2 Semi-annual payment = Annual payment / 2 = $92,000 /2 = $46,000 Calculating the present value using excel sheet: Step1: Go to excel and click "insert" to insert the function Step2: Select the "PV" function as we are finding the present value of annuity in this case. Step3: Enter the values as Rate = 10.5%/2; Nper = 5*2; PMT = -46000; FV = 0 Step4: Click "OK" to get the desired value The value comes to "350,926.66" Therefore, the present value of annuity is $350,926 C) Calculating the future value of annuity for annual payment: Step1: Go to excel and click "insert" to insert the function Step2: Select the "FV" function as we are finding the future value of annuity in this case. Step3: Enter the values as Rate = 10.5%; Nper = 5; PMT = -92000; PV = 0 Step4: Click "OK" to get the desired value The value comes to "567,286.69" Therefore, the future value of annuity for annual compounding is $567,286 For semi-annual compounding, Rate = 10.5%/2 Nper = 5*2 Payment = $46,000 Step1: Go to excel and click "insert" to insert the function Step2: Select the "FV" function as we are finding the future value of annuity in this case. Step3: Enter the values as Rate = 10.5%/2; Nper = 5*2; PMT = -46000; PV = 0 Step4: Click "OK" to get the desired value The value comes to "585,379.37" Therefore, the future value of annuity for semi-annual compounding is $585,379 D) To make the purchase, the amount required is $600,000 and the annual payment made is $90,000. The number of years are same as above. Calculating the interest rate using excel sheet: Step1: Go to excel and click "insert" to insert the function Step2: Select the "Rate" function as we are finding the interest rate in this case. Step3: Enter the values as Nper = 5; PMT = 90000; PV = 0; FV = -600000 Step4: Click "OK" to get the desired value The value comes to "14.43%" Therefore, the interest rate that is needed to reach the future value is 14.43% E) Calculating the size of the annual payment using excel sheet: Step1: Go to excel and click "insert" to insert the function Step2: Select the "PMT" function as we are finding the interest payment in this case. Step3: Enter the values as Rate = 10.5%; Nper = 5; PV = 0; FV = -600000 Step4: Click "OK" to get the desired value The value comes to "$9,730.53" Therefore, the annual payment of $9,730 is needed to reach a future value of $600,000 at 10.5% interest rate.