Money, Banking, and International FinanceHowever, if it is daily, subsequently, this rate is terrible. Borrower took money from a loan
shark. For this book, we define all interest rates in annual terms, unless otherwise stated.
Banks and finance companies usually calculate interest payments and deposits monthly.
Thus, we adjust the present value formula for different time units. If you refer to Equation 11,
we add a new variable, m, the compounding frequency while APR is the interest rate in annual
terms. In the monthly case, m equals 12 because a year has 12 months.
ܸܨ்=ܸܲቀ 1 +
ோ
ቁ
∙்
( 11 )
For example, you deposit $10 in your bank account for 20 years that earns 8% interest
(APR), compounded monthly. Consequently, we calculate your savings grow into $49.27 in
Equation 12: If your bank compounded your account annually, then you would have $46.61.
ܸܨ்=ܸܲቀ 1 +
ோ
ቁ
∙்
=$10ቀ 1 +
.଼
ଵଶቁ
ଵଶ∙ଶ
=$49. 27 ( 12 )
Although the compounding frequency is usually 12 months, we could use semi-annually
(two payments per year, or m equals 2), or quarterly (four payments per year, or m equals 4).
We can convert any compounding frequency into an APR equivalent interest rate, called the
effective annual rate (EFF). From the previous example, we convert the 8% APR interest rate,
compounded monthly into an annual rate without compounding, yielding 8.3%. We show the
calculation in Equation 13. The EFF is the standard compounding formula removing the years
and the present value terms.
ܨܨܧ=ቀ 1 +
ோ
ቁ
− 1 =ቀ 1 +
.଼
ଵଶቁ
ଵଶ
− 1 = 0. 083 ( 13 )
If you deposited $10 in your bank account for 20 years that earn 8.3% APR with no
compounding (or m equals 1), then your savings would grow into $49.27, which is the identical
to an interest rate of 8% that is compounded monthly. We calculate this in Equation 14.
ܸܨ்=ܸܲቀ 1 +
ோ
ቁ
∙்
=$10ቀ 1 +
.଼ଷ
ଵ ቁ
ଵ∙ଶ
=$49. 27 ( 14 )
We can adapt the present value formula for frequency compounding. For example, what is
the present value if you receive $50 in a month, $100 in six months, and $75 in 13 months at
10% APR? We must adjust our time units to months, which the smallest time unit.
We expressed the interest rate in APR, so divide it by 12 to obtain the monthly interest
rate, yielding 0.8333% in our case. All time subscripts are monthly.