Question

The borrower is in a $238,000 loan makes interest payments at the end of each six months for eight years. These are computed using an annual effective discount rate of 6.5%. Each time he makes an interest payment, the borrower also makes a deposit into a sinking fund earning a nominal interest rate of 4.2% convertible monthly. The amount of each sinking und deposit is D in the first three years and 2D in the remaning five years, and the sinking fund balance at the end of the eight years is equal to the loan amount. Find D.

Answer 1

loan = $ 238000

annual discount rate = 6.5%

Since the borrower is making deposits semiannually,
the effective semiannual rate by computing 4.2%/12=0.35% effective monthly, and (1.0035)^6-1=2.118461% effective semiannually.

Since the deposits are made at the end of every six months, the first deposit of D will accumulate interest 15 times before the end of eight years, the second deposit 14 times, and so on until the sixth deposit, which is the last deposit of amount D and which accumulates 10 times. Then the borrower deposits 2D which accumulates 9 times, on until his last deposit of 2D which accumulates no interest (it's at the end of the eight years).

The simplest way to treat this calculation is to break it into deposits of D for eight years and additional deposits of D for five years. Then:

D*((1.02118461)^16-1)/(1.02118461-1)+D*((1.02118461)^10-1)/(1.02118461-1)=238000

Which gives D=7980.98