Question

Discuss the effects of doubling a payment will have on the term of the compounded loan?

Answer 1

Let initially payment be pmt per period, r bw the rate of interest per period, n the number of periods it takes to repay the loan

Then, Loan=pmt/r*(1-1/(1+r)^n)

Now if we increase the payment per period to 2pmt and number of periods it take now is x

Then, loan=2pmt/(1-1/(1+r)^x)

Dividing 1st by 2nd we get

1=(1-1/(1+r)^x)/(2*(1-1/(1+r)^n)

=>1-2/(1+r)^n=-1/(1+r)^x

So, (1+r)^x=((1+r)^n)/((1+r)^n-2))

x=log(1/(1-2/(1+r)^n))/log(1+r)

Thus we can say that just doubling the payment does not reduce the term by half rather there is a logarithmic relationship of the new term with old term and interest rate