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### 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%

YearBeginning valueInterestEnding ValueCompound Interest Factor

1 $100.00 $6.00 $106.00 ($100)(1.06) = $106.00

($100)(1.06)(1.06) =

2 $106.00 $6.36 $112.36 ($100)(1.06)2 = $112.36

($100)(1.06)(1.06)(1.06) =

3 $112.36 $6.74 $119.10 ($100)(1.06)3 = $119.10

($100)(1.06)(1.06)(1.06)(1.06) =

4 $119.10 $7.15 $126.25 ($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 raterather 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 2 3

Beginning balance $267.30 $183.34 $94.34

Annual interest @ 6% 16.04 11.00 5.66

Subtotal $283.34 $194.34 $100.00

Annual withdrawal (100.00)(100.00)(100.00)

Ending balance $183.34 $94.34 $ -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.
- Determine how much of lump sum is needed to pay out $50,000 per year for 4 years
- 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)
- Present-value annuity formula (using present-value-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)
- 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

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