Question

A company took a loan which requires a payment of $40 million plus interest two years after the loan's date of issue. The interest rate on the $40 million face value is 9.6% compounded quarterly. Before the maturity date, the lender sold the loan contract to a pension fund for $43 million. The sale price was based on a discount rate of 8.5% compounded semi-annually from the date of sale. The sale took place months before the maturity date of the loan. Round to the nearest month.

Answer 1

Loan amount (P)= $40 Million

Rate of interest on loan = 9.6% compounded quarterly. Hence rate per period of compounding (r)= 9.6%/4= 0.024

Period of loan= 2 years. Compounded quarterly . Hence number of times compounding (n) = 2*4 = 8

Repayment on maturity (F)= P*(1+r)^n = $40 Million*(1+0.024)^8 = $40 Million* 1.208926 = $48.35703 Million.

When this loan is sold,

Discounted value (V) = $43 Million

Discount rate= 8.5%. Frequency- Semiannual. Hence discount rate per period (d)= 5.5%/2= 0.0425

Discounted value (V)= F/(1+d)^n Where n= number half years before maturity

$43 Million= $48.35703/(1+0.0425)^n

43= 48.35703/1.0425^n. Or, 1.0425^n = 48.35703/43 = 1.124582

Using natural logarithm,

In (1.0425^n) = In (1.124582)

n* In(1.0425) = In (1.124582)

Therefore, n= In(1.124582/In(1.0425) = 0.1174114 / 0.041621674 = 2.82092 half years. Or, 2.82092*6= 16.92552 months

Therefore, the sale took place 17 months before maturity of the loan.