Question

A borrower is repaying a loan with 10 annual installments of $2000. Half of the loan is repaid by the amortization method at an effective rate of i = .06. The other half of the loan is repaid by the sinking fund method in which the lender receives i = .06 and the sinking fund accumulates at i = .05. Find the amount of the loan to the nearest dollar.

Answer 1

Number of annual instalments=10

Interest rate=6%=0.06

Uniform Series Present Worth Factor(P/A, i, N)=PWF:

(((1+i)^N)-1)/(i*((1+i)^N))

i=Interest Rate=0.06

N=Number of instalments=10

PWF= ((1.06^10)-1)/(0.06*(1.06^10))= 7.3601

Assume the loan amount=X

Half the amount is repaid by amortization method

Assume amount of annual repayment for half the loan=Y

X/2=Y*7.3601

X=14.7202*Y……………….(Equation 1)

Total annual repayment=$2000

Amount repaid by amortization method per year=Y

Amount deposited annually in sinking fund at 5%=2000-Y

Present value of amount deposited in sinking fund=X/2(half the loan amount)

Uniform Series Present Worth Factor of sinking Fund=PWF(1):

(((1+i)^N)-1)/(i*((1+i)^N))

i=Interest Rate=0.05

N=Number of instalments=10

PWF(1)= ((1.05^10)-1)/(0.05*(1.05^10))=7.7217

Present value of sinking fund payments=(2000-Y)*7.7217

Present value of sinking fund payments=half the amount of loan amount =X/2

(2000-Y)*7.7217=X/2…………….(Equarion2)

(From Equation 1) X=14.7202Y

(2000-Y)*7.7217=(14.7202/2)*Y=7.3601*Y

2000*7.7217-7.7217*Y=7.3601Y

Y*(7.3601+7.7217)=2000*7.7217

Y=(2000*7.7217)/(7.3601+7.7217)= 1023.979

Amount of repayment for amortization method=$1023.979

Amount of annual payment for sinking fund=(2000-1023.98)= $   976.02

Amount of Loan =X=14.7202*1023.979=15073.18

Rounding off to nearest dollar:

Amount of loan=$15073

Amount of loan

$        15,073