Loan at bank discount

Statement

Roberto asks for a loan lasting $8$ months at a bank discount rate of $15\%$ per year.

• (a) If the face value of the loan is $1000\ \text{€}$, how much cash does he receive?
• (b) What face value should he ask for to receive exactly $1000\ \text{€}$ in cash?

Solution

With bank discount $P = M\,(1 - d\,t)$, with $t = \dfrac{8}{12} = \dfrac{2}{3}$, so $d\,t = 0{.}15\cdot\dfrac{2}{3} = 0{.}10$. (a) $P = 1000\,(1 - 0{.}10) = \boxed{900\ \text{€}}$. (b) $M = \dfrac{P}{1 - d\,t} = \dfrac{1000}{0{.}90} \approx \boxed{1111{.}11\ \text{€}}$.