Brenda owes Cathy $\$8500$ and has signed a promissory note to repay the debt in 15 months from the signing date. The note was signed on December 6, 2009, and the maturity value of the note is $\$10043.55$. Cathy decides to sell the note to a bank on May 6, 2010. If the bank wishes to earn $r = 8\%$. What price does Cathy receive for the note?
I got the answer $$P = 10043.55 (1 + 0.08 (304/365))^{-1}$$
by using the formula provided to me which is $P = (1+r(n/365))^{-1}$ for the bank's payment using $n$ as the amount of days left until maturity, and $r$ is the bank's interest rate at discount. Is this correct?
Best Answer
The formula you got is wrong. The correct present value formula given a future value $FV$ and with compounding $m$ times per period is:
$PV_{t=0}=\cfrac{FV_t}{(1+\cfrac{r}{m})^{t\cdot m}}$
I assume $r=8\%$ is the interest rate with annual compounding, and $t$ is measured in years, so $m=1$ and the formula simplifies to:
$PV_{t=0}=\cfrac{FV_t}{(1+r)^t}=FV_t(1+r)^{-t}$
$t=\cfrac{304}{365}$ (based on actual/actual convention)
So that:
$PV_{t=0}=10043.55(1+0.08\%)^{-\frac{304}{365}}=\$9,419.97$
Alternatively, if one understands the question like this: the bank wants to make a return of 8% in the given period, i.e. 8% is not an annualized rate, but the rate the bank applies for that specific period, then you would just have:
$P=FV(1+8\%)^{-1}$
which is also not the formula you have been using.