Citibank: basic of corporate finance
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ANSWER KEY
1. If you borrow $300,000 for two years at a 9% simple annual interest rate, how
much interest will you pay?
$54,000
I = P x R x T
I = $300,000 x 0.09 x 2
I = $54,000
2. If you invest $270,000 for one-half year at a 9.5% simple annual interest rate, how
much will you earn on the investment?
$12,825
I = P x R x T T = 1/2 or 0.5 or (180 days/360 days)
I = $270,000 x 0.095 x 0.5
I = $12,825
3. If you borrow $500,000 for one year and have to pay back $540,000 at the end of
the year ($500,000 is the return of the principal; $40,000 is the interest), what is
the simple interest rate you are being charged for the loan?
8.0%
R = I / (P x T)
R = $40,000 / ($500,000 x 1)
R = 0.08 or 8.0%
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PRACTICE EXERCISE 3.1
(Continued)
4. If you borrow $350,000 for two years and have to pay back $413,000 in two years,
what is the simple interest rate you are being charged for the loan per year?
______%
5. If the rate is 12% per annum (p.a.), what is the simple interest accrued over a 20 day
period within the same month on a $100 investment?
$_____________________
6. If the rate is 7.5% p.a. on a $1,000 investment, what is the simple interest accrued
over the period 13/08/91 to 20/09/91 (dd/mm/yy)?
$_____________________
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ANSWER KEY
4. If you borrow $350,000 for two years and have to pay back $413,000 in two years,
what is the simple interest rate you are being charged for the loan per year?
9.0%
R = I / (P x T)
R = $63,000 / ($350,000 x 2)
R = 0.09 or 9.0%
5. If the rate is 12% per annum (p.a.), what is the simple interest accrued over a 20 day
period within the same month on a $100 investment?
$0.67
I = P x R x (days/360)
I = $100 x 0.12 x (20/360)
I = $0.67
6. If the rate is 7.5% p.a. on a $1,000 investment, what is the simple interest accrued
over the period 13/08/91 to 20/09/91 (dd/mm/yy)?
$7.92
There are 38 days from Aug. 13,1991 to Sep. 20, 1991.
I = P x R x (days/360)
I = $1,000 x 0.075 x (38/360)
I = $7.92
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Compound Interest
Reinvestment
of interest
payment
It is common for investments to receive (pay) multiple interest
payments rather than one payment at the end of the investment period.
Usually, each interest payment is reinvested at the same rate as the
original principal.
For example, interest is calculated on a savings account balance at the
end of some period and the interest payment is credited to the
account. At the end of the next period, interest is calculated on the
new balance (the original principal plus the last interest payment) and
credited to the account. The process of compounding interest
continues for each period over the life of the investment. There are
two ways to calculate compound interest: discretely and
continuously.
Discrete Compounding for Annual Periods
Interest paid at
regular
intervals
Interest calculated at specified regular intervals is said to be
discretely compounded. An example is a savings account on which
interest is calculated monthly and deposited in the account. The
formula for calculating discretely compounded interest is:
I = P (1 + R)
T
- P
Where:
I = Interest earned or paid
P = Principal
R = Stated (nominal) interest rate for
each interest paying period
T = Number of interest paying periods
Let's use the formula to calculate the interest on an investment.
Suppose that you invest $400,000 at 6% per annum (p.a.) for two
years, compounded annually. At the end of the two years, the
investment earns $49,440 in interest.
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I = P (1 + R)
T
- P
I = ($400,000) (1 + 0.06)
2
- ($400,000)
I = ($400,000) (1.1236) - ($400,000)
I = ($449,440) - ($400,000)
I = $49,440
Future value of
principal
By adding the interest and principal, you can see that at the end of the
two year period the investment will be worth $449,440. This is the
future value of the $400,000 investment. Future value is the worth of
an investment at a specified rate of interest at the end of a stated
period of time. We can convert the interest payment formula to a
future value formula.
FV
T
= P (1 + R)
T
Where:
FV
T
= Future value after T paying periods
P = Principal
R = Stated interest rate for each interest paying
period
T = Number of interest paying periods
Future value
interest factor
The expression (I + R)
T
is referred to as the future value interest
factor. The future value is the principal multiplied by the future value
interest factor.
Example
To illustrate, suppose that we invest $500 in an account earning 12%
annual interest. At the end of one year, the future value of our
investment is:
FV
T
= P (1 + R)
T
FV
1
= ($500) (1 + 0.12)
1
FV
1
= ($500) (1.12)
FV
1
= $560
If we leave the $560 dollars in the account for another year, at the
same rate, the future value is:
FV
1
= ($560) (1 + 0.12)
1
FV
1
= ($560) (1.12)
FV
1
= $627.20
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We can arrive at the same result by using the future value formula
with a two year time frame.
FV
2
= ($500) (1 + 0.12)
2
FV
2
= ($500) (1.2544)
FV
2
= $627.20
The interest earned in the second year is $67.20: $60 dollars on the
$500 principal, and $7.20 on the $60 of interest earned during the
first year. Compounded interest is interest earned on interest.
Discrete Compounding for Nonannual Periods
Adjust for
number of
periods
Many investments are compounded for periods other than annually.
Some bonds are compounded semiannually or quarterly; many savings
accounts are compounded monthly, weekly, or even daily.
We use the future value equation to calculate discrete compounding
for nonannual periods with two slight modifications. First, we adjust
the interest rate by dividing the annual rate (R) by the number of
compounding periods during the year (M). For semiannual
compounding, M is 2; for monthly compounding, M is 12. We also
adjust the number of compounding periods by multiplying the time in
years (T) by the number of compounding periods during the year (M).
Our new future value formula then becomes:
FV
T
= P (1 + R/M)
T x M
Where:
FV
T
= Future value after T years
P = Principal
R = Stated annual interest rate
M = Number of compounding periods within a
year
T = Number of years
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Example
To illustrate the use of this formula, suppose that you invest $10,000
in an account that earns 8% annual interest compounded quarterly for
five years. Assuming no other transactions, how much will be in your
account at the end of five years? Using the future value formula, P =
$10,000, R = 0.08, M = 4, and T = 5:
FV
T
= P (1 + R/M)
T x M
FV
5
= ($10,000)(1 + 0.08/4)
5 x 4
FV
5
= ($10,000)(1.485947)
FV
5
= $14,859.47
One more example: What is the future value of the $10,000 if it is
invested for one year and compounded monthly at an annual rate of
8%?
FV
T
= P (1 + R/M)
T x M
FV
1
= ($10,000)(1 + 0.08/12)
1 x 12
FV
1
= ($10,000)(1.082995)
FV
1
= $10,829.95
Continuous Compounding
Infinite number
of periods
Continuous compounding is a special case of nonannual
compounding. It extends the notion of compounding periods to a
point where the number of periods becomes infinitely large and the
length of each period is correspondingly small. In other words,
interest is always being calculated and compounded. The future value
interest factor for continuous compounding is represented by the
term e
RT
. For continuous compounding, the future value formula is:
FV
T
= P x e
RT
Where:
FV
T
= Future value after T periods
P = Principal
e = Base of the natural logarithm (2.718282)
R = Stated (nominal) annual interest rate for
each period
T = Number of periods
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Example
For example, let's calculate the future value of a $10,000 investment
for one year at an 8% annual rate compounded continuously. We will
use the values P = $10,000, R = 0.08, and T = 1.
FV
T
= P x e
RT
FV
1
= ($10,000)(2.718282)
(0.08)(1)
FV
1
= ($10,000)(1.083287)
FV
1
= $10,832.87
Discrete vs.
continuous
compounding
A comparison of the investment's future values discretely
compounded for different periods, and continuously compounded,
shows how the value changes — depending on the frequency of
compounding.
Annually: $10,800.00
Monthly: $10,829.95
Weekly: $10,832.20
Continuously: $10,832.87
You can see that as the compounding period becomes smaller, the
future value of the investment increases. The increase occurs
because we are compounding the interest more frequently.
To further illustrate continuous compounding and compare it to
discrete compounding, refer to the next three graphs (Figures 3.1,
3.2, and 3.3). The first graph shows the growth of $100 of capital
invested at 25% for three years compounded annually. The second
graph shows $100 invested at 25% for three years compounded
semi-annually. The final graph represents the growth of $100
invested at 25% for three years compounded continuously.
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Figure 3.1: Annual Compounding: $100 at 25% for 3 Years
Figure 3.2: Semi-annual Compounding: $100 at 25% for 3 Years
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Figure 3.3: Continuous Compounding: $100 at 25% for 3 Years
In the first two graphs, the interest is calculated and added to the
account at discrete periods – at the end of each year (Figure 3.1) and
after each six-month period (Figure 3.2). Notice in Figure 3.3 that
when interest is calculated and added to the account continuously,
the graph is a smooth curve.
We can summarize the results as follows:
$100 at 25% for three years
Annual compounding: 195.31
Semi-annual: 202.73
Continuous: 211.70
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