Question

1. First National Bank charges 11.11 % compounded monthly on its business loans. First United Bank charges 11.81%, compounded semiannually. Calculate the EAR for First United Bank.

2. One of your customers is delinquent on his accounts payable balance. You’ve mutually agreed to a repayment schedule of $559 per month. You will charge 1.29 % per month interest on the overdue balance. If the current balance is $13709, how many years will it take for the account to be paid off?

3. A prestigious investment bank has designed a new security that pays a quarterly dividend of $1.15 in perpetuity. The first dividend occurs one quarter from today. What is the price of the security if the stated annual interest rate is 7.99%, compounded quarterly?

Answer 1

(1) First United Bank Annual Percentage Rate (APR) = Quoted Rate = 11.81 %

Semi-Annual Rate = (11.11/2) = 5.555 %

Effective Annual Rate (EAR) = (1.05555)^(2) - 1 = 0.11418 or 11.418 % ~ 11.42 %

(2) Periodic Repayments = $ 559, Current Outstanding Balance = $ 13709, Interest Rate = 1.29 % per month

Let the repayment period (in months) be T

Therefore, 13709 = 559 x (1/0.0129) x [1-{1/(1.0129)^(T)}]

(13709 / 559) x (0.0129) = 1 - { 1 / (1.0129)^(T)}

1 - [(13709/559) x (0.0129)] = 1 / (1.0129)^(T)

1- 0.31636 = 1 / (1.0129)^(T)

0.683638 = 1 / (1.0129)^(T)

(1.0129)^(T) = 1/0.683638 = 1.4627615

$\log_{1.0129}1.4627615$ = T

T = 29.6724 months = (29.6724/12) ~ 2.47 years

(3) Quarterly Perpetual Dividends = $ 1.15, Annual Interest Rate = 7.99% or Quarterly Interest Rate = (7.99 / 4) = 1.9975 %

Security Price = Quarterly Dividend / Quarterly Interest Rate = 1.15 / 0.019975 = $ 57.571965 ~ $ 57.57