Mathematics of Compound
Interest
Chapter 11
Compound Interest
• Compound interest formula: expresses value of
principal sum of money left on deposit (or
invested) for given number of years at give rate of
interest.
Vn = P(1 + i)n
• Where
–
–
–
–
Vn = value at end of n years
P = principal amount deposited or invested
i = interest rate per year
n = number of years
Compound Interest
• Exhibit 11.1: Investment Growth at 6%
Year
Beginning value Interest Ending Value Compound Interest Factor
1
$100.00
$6.00
$106.00
2
$106.00
$6.36
$112.36
3
$112.36
$6.74
$119.10
$119.10
$7.15
$126.25
($100)(1.06) = $106.00
($100)(1.06)(1.06) =
($100)(1.06)2 = $112.36
($100)(1.06)(1.06)(1.06) =
($100)(1.06)3 = $119.10
($100)(1.06)(1.06)(1.06)(1.06)
=
4
($100)(1.06)4 = $126.25
Compound Interest
• Generally, if interest compounds for n years, the
$100 investment will grow to ($100)(1.06)n.
– Ex. If investment is left to compound for 12 years, its
ending value will be:
$100(1.06)12 = ($100)(2.012) - $201.20, or
approximately double amount originally invested
• Rule of 72: by dividing rate of compound interest
into 72, one may estimate number of years
required to double original investment
– Ex. 6%/72 indicates that it takes approximately 12
years for investment to double in value if it earns
interest at 6% compounded annually.
Compound Interest
Compound Interest Tables
• See Exhibit 11.2: Sample of compound
interest factors.
• Compound interest factor: amount to which
$1 will grow at end of n years at an interest
rate of i percent
Compound Interest
Semiannual and Other Compounding Periods
• If interest is compounded more than once a
year (i.e. semiannually, quarterly, monthly,
daily), then adjust compound interest formula:
Vn = P[(1 + i/m)nm]
• Where m = number of times per year that
interest is compounded and V, P, i, and n are
as previously defined
Compound Interest
Semiannual and Other Compounding Periods
• If interest rate if 8% per year and
compounded semiannually, value of $100 left
to compound for two years:
V2 = ($100)[(1 + 0.08/4)(2)(4)]
V2 = ($100)(1 + 0.04)4
V2 = ($100)(1.04)4 = ($100.(1.170)
V2 = $117.00
Compound Interest
Semiannual and Other Compounding Periods
• If compounded quarterly:
V2 = ($100)[(1 + 0.08/4)(2)(4)]
V2 = ($100)(1 + 0.02)8
V2 = ($100)(1.02)8 = ($100)(1.172)
V2 = $117.20
Compound Interest
Semiannual and Other Compounding Periods
• If compounded monthly:
V2 = ($100)[(1 + 0.08/12)(2)(12)]
V2 = ($100)(1.00667)24 = ($100)(1.181)
V2 = $118.10
Compound Interest
Semiannual and Other Compounding Periods
• As interest is compounded more often, ending
value (terminal value) of investment becomes
larger.
• In all cases, ending value is higher than that
which would be obtained by earning interest at
8% compounded annually:
V2 = ($100)(1.08)2
V2 = ($100)(1.166)
V2 = $116.60
Compound Interest
Semiannual and Other Compounding Periods
• If compounding more often than annually,
use compound interest factor for relevant
total number of periods and interest rate
period.
– Ex. If $100 if to compound quarterly for 2 years at
8% per year, appropriate interest factor is for 8
periods (= 4 times per year times 2 years) at 2
percent (8% annually divided by 4 compounding
periods).
Compound Interest
Semiannual and Other Compounding Periods
• If interest is compounded more often than
annually, higher terminal value of investment
results  effective rate of interest earned is
higher.
– Original investment “earns interest on interest” more
often  effective rate of return is higher
• To calculate effective rate of interest, calculate
one-year compound interest factor for given
annual interest rate and number of compounding
periods per year.
Compound Interest
Semiannual and Other Compounding Periods
• Effective annual interest rate:
iE = (1 + i/m)nm - 1
Compound Interest
Semiannual and Other Compounding Periods
• Ex. continued: To calculate effective annual rate of interest,
find one-year compound interest factor for 2% per period
over 4 periods and then subtracting 1 (for return of
principal) from it:
iE = (1 + 0.02)4 -1
iE = (1.082) – 1
iE = 0.0824 = 8.24%
Where iE = effective annual interest rate
• Effective annual interest rate when 8% per year
compounded quarterly is 8.24%  8% per year
compounded quarterly provides same return as 8.24% per
year compounded annually
Present Value
• Present value of a dollar: represents “today’s
value” of sum of money to be received in
future, if money in hand today can be invested
at given interest rate
– Dollar received in future is less valuable than
dollar in hand today because dollar in hand today
can be invested to grow to more than a dollar in
future.
Present Value
• Derive present-value formula from compound
interest formula
– Let r = rate at which money currently in hand may be
invested (directly comparable to i in compound interest
formula)
– Present value of dollar found as follows
Vn = PV(1 + r)n
PV = Vn/(1 + r)n
PV = Vn [1/(1 + r)n]
Where PV = present value of sum Vn to be received n period
in future
r = discount rate per period
Present Value
• Rate at which money currently in hand may be
invested (r) is referred to as discount rate
rather than interest rate. Why?
– Present-value formula uses rate of return available
to “discount” future dollars to current (and lower)
present values
– Discounted present value or discounted cash flow:
sum of cash to be received in future
Present Value
• Ex. What is the present value of $1,500 to be
received eight years from now if money in
hand can be invested at 10%?
PV = ($1,500)[1/(1.10)8]
PV = ($1,500)(0.467)
PV = $700.50
• Present value of $1,500 to be received 8 years from now
given a 10% discount rate is $700.50 (or $700.50
invested today at interest rate of 10% will grow to be
$1,500 at end of 8 years)
Present Value
• Ex. (continued): Answer confirmed by
compound interest table in Exhibit 11.2
Vn = P(1 + i)n
Vn = ($700.50)(1.10)8
Vn = ($700.50)(2.144)
Vn = $1501.87
– Present-value factors are reciprocal of compound
interest factors.
– See Exhibit 11.3: Sample of present-value factors
Present Value of an Annuity
• Annuity: series of constant receipts (or
payments) that are received (or paid) at end
of each year for some number of years into
future
• Present value of annuity (An): present value
of stream of future cash receipts of fixed
amount received at end of each year for some
number of years into future, given discount
rate (r)
Present Value of Annuity
• Ex. Present value of future stream of receipts of $100 per year
to be received at end of each year for next 3 years given
discount count rate r = 6% (using Exhibit 11.3 to find
appropriate discount factors):
An = ($100)(0.943) + ($100)(0.890) + ($100)(0.840)
An = ($100)(0.943 + 0.890 + 0.840)
An = ($100)(2.673)
An = $267.30
• Present value of annuity of $100 per year for 3 years is equal
to present value of $100 received 1 year from now plus
present value of $100 received 2 years from now plus
present value of $100 received 3 years from now.
Present Value of Annuity
• Ex. (continued)
– Present-value factors for each of 3 years are
added together and then multiplied by constant
annual receipt.
– Tables of present-value annuity factors add
together individual year’s present-value factors
for number of years annuity is to run.
• See Exhibit 11.4 for sample present-value
annuity factors.
Present Value of Annuity
• To calculate present-value annuity factors directly:
An = R[1/(1 + r)] + R[1/(1 + r)2] +…+ R[1/(1 + r)n]
An = R[1/(1 +r) + 1/(1 + r)2 +…+ 1/(1+r)n]
An = R[(1-(1/(1 + r)n))/r]
Where A = present value of annuity
R = amount of future receipts
r = discount rate
n = number of years
• Expression within brackets gives present-value annuity
factors presented in Exhibit 11.4.
Present Value of Annuity
• Ex. continued: Sum of 3-year annuity was
found to be $267.30 given 6% discount rate.
– $267.30 represents amount of money that would
have to be invested today at 6% so that one would
withdraw $100 at end of each year for next 3
years before exhausting investment.
– Exhibit 11.5 illustrates this process.
Present Value of Annuity
• Exhibit 11.5: Sample Annuity Schedule
YEAR
1
Beginning balance
$267.30
Annual interest @ 6%
16.04
Subtotal
$283.34
Annual withdrawal
(100.00)
Ending balance
$183.34
2
$183.34
11.00
$194.34
(100.00)
$94.34
3
$94.34
5.66
$100.00
(100.00)
$ -0-
Compound Value of an Annuity
• Compound value (future value) of an annuity
(Sn): ending value of series of constant
payments made at end of each year for
specified number of years that earn given rate
of interest per year
– “Flip side” of present value of annuity
Compound Value of Annuity
• Compound value of annuity
Sn = P(1 + i)(n-1) + P(1 + i)(n-2) +…+ P(1 + i) + P(1)
Sn = P[(1 + i)(n-1) + (1 + i)(n-2) +…+ (1 +i) +1]
Sn = P[((1 + i)(n -1))/i]
Where Sn = compound sum
P = principal amount deposited each year
i = interest rate
n = number of years
– Each deposit compounds for 1 year less than total number of years
annuity runs.
• First deposit earns interest for n-1 years, second deposit for n-2
years, and so forth.
• Last deposit earns no interest at all because annuity formula set
up so that deposits are made at end of each year.
Compound Value of Annuity
• If I invest a constant amount of money per year at end
of each year at given interest rate, what will be total
sum accumulated at end of given number of years?
• Ex. If $100 is invested at end of each year for 4 years,
ending value of investment will be:
S4 = ($100)(1 + i)(n-1) + ($100)(1 + i)(n-2) + ($100)(1 + i)(n-3) +
($100)(1 + i)(n-4)
S4 = ($100)(1.06)3 + ($100)(1.06)2 + ($100)(1.06)1 +
($100)(1.06)0
S4 = $437.50
• See Exhibit 11.6 for compound value annuity factors.
Compound Value of Annuity
Calculators and Personal Computers
• Few people use commonly available compound
interest and present-value tables to solve time value
of money problems.
• Hand-held calculators can solve most of these
problems, while business/scientific calculators can
solve more complex ones.
• Spreadsheet packages on PCs can solve any of these
problems easily.
Application to Personal Decision Making
• Ex. Child’s college expenses.
– Parent wants to provide $50,000 per year for 4 years
for infant’s college tuition 18 years from now.
– Parent wants to accumulate lump sum sufficient to
pay out $50,000 per year beginning 18 years from
now by making annual installment payments at end of
each of next 17 years and then make first withdrawal
at end of 18th year.
– Annual payment is invested to earn average effective
annual rate of return equal to 8%.
Application to Personal Decision Making
• Ex. (continued)
– Parent faces two problems that can be solved
using formulations for sum of annuity and present
value of annuity.
1.Determine how much of lump sum is needed to
pay out $50,000 per year for 4 years
2.Determine how much money must be deposited
each year over next 17 years in order to
accumulate lump sum
Application to Personal Decision Making
• Ex. (continued)
1. Present-value annuity formula (using presentvalue-of-annuity factor from Exhibit 11.4):
An = R[(1-(1/(1 + r)n))/r]
An = $50,000[(1-(1/1.08)4))/0.08]
An = $165,600
– Lump sum of $165,000 is required in order to
pay out $50,000 per year for 4 years.
Application to Personal Decision Making
• Ex. (continued)
2. Compound-value-of-an-annuity formula (using sum of
annuity factor from table):
Sn = P[(((1 + i)n) -1)/i]
$165,600 = P[(((1.08)17)-1)/0.08]
P = $4,906
–
–
Parent must invest $4,906 at end of each year for next 17
years in order to accumulate lump sum of $165,500 from
which $50,000 may be withdrawn each year for 4
consecutive years.
Different interest rate will result in different answer.
•
Lower interest rate will require larger lump sum to be
accumulated and larger annual deposits.
•
Higher interest rate will allow smaller lump sum and lower
annual deposits.
Application to Personal Decision
Making
• Ex. Corporation sets up sinking fund of some type.
– If company issues bond worth $100,000,000 that must
be redeemed at par (paid off at face value) 20 years
from date of issue, company may set aside fixed
annual contribution to sinking fund to redeem issue.
– How much money must be contributed to sinking
fund each year in order to retire issue?
– If annual deposit could be invested at 8%, this
annual payment would be required:
$100,000,000/(45.762) = $2,182,213