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Kramer borrowed $4000 from George at an interest rate of 7% compounded semiannually. The loan is to be repaid by three payments. The first payment, $1000, is due two years after the date of the loan. The second and third payments are due three and five years, respectively, after the initial loan. Calculate the amounts of the second and third payments if the second payment is to be twice the size of the third payment.

Correct Answer: 1339.33

Given:j=7% compounded semiannually making m=2 and i = j/m= 7%/2 = 3.5%Let x represent the third payment. Then the second payment must be 2x.PV1,PV2, andPV3 represent the present values of the first, second, and third payments.

Since the sum of the present values of all payments equals the original loan, thenPV1 + PV2 +PV3 =$4000 -------(1)PV1   =FV/(1 + i)^n =$1000/(1.035)^4= $871.44

At first, we may be stumped as to how to proceed forPV2 and PV3. Let’s think about the third payment of x dollars. We can compute the present value of just $1 from the x dollars

pv=1/(1.035)^10=0.7089188

PV2   =2x * 0.7089188 = 1.6270013xPV3   =x * 0.7089188=0.7089188xNow substitute these values into equation ➀ and solve for x.$871.442 + 1.6270013x + 0.7089188x =$4000

2.3359201x =$3128.558

x=$1339.326Kramer’s second payment will be 2($1339.326) =$2678.65, and the third payment will be $1339.33

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