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Chapter 2

Compound Interest

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Repeat Repeat

In chapter 1, you simply

computed once for

simple interest and

simple discount.In this chapter, interest

earned per period is

automatically

reinvested sometimes

once, twice or several

times.

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Formula

By Junius W. Yu

Total Number of Conversion Periods

Time/Term

Frequency of Conversion

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Frequency

When interest rate is

compounded annually,

interest is computed once

hence one conversion periodis equivalent to one year.

Frequency of conversion (m)

number of times that theinterest is computed in a span

of one year.

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Frequency Conversion Table

Conversion

Period

Time

(t)

Frequency of

Conversion(m)

Total number

of Conversion(n = tm)

Annually 1 year 1 year 1 1

Semi- Annually 6 months 5 years 2 10

Quarterly 3 months 3 years 4 12

Monthly 1 month 2 years 12 24

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Interest Rate per period

By Junius W. Yu

Interest Period Nominal Rate

(j)

Frequency of

Distribution

(m)

Interest rate

per period

(i = j/m)

Annually 1 year 10% 1 10%

Semi- Annually 6 months 12% 2 6%

Quarterly 3 months 14% 4 3.5%

Monthly 1 month 16% 12 1 and 1/3%

Note: Dont Round Off!!!!!

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Compound Amount

Rollover !!!

The sum of the original

principal and

compound interest is

compound amount

- the accumulated value

of P at the end of theterm

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Amount of investment

at the end of n period

Period (n) Principal (P) Interest (I) Amount (F)

0 P 0 P + 0 = P

1 P P (i) P + Pi = P(1 + i)

2 P (1 + i) P (1 + i) (i) P (1 + i)2

3 P (1 + i)2 P (1 + i)2 (i) P (1 + i)3

4 P (1 + i)3 P (1 + i)3(i) P (1 + i)4

. . ..

n1 P (1 + i)n2 P (1 + i)n2(i) P (1 + i)n1

n P (1 + i)n1 P (1 + i)n1 (i) P (1 + i)n

By Junius W. Yu

n is 2, the exponent of the accumulation factor (1 + i) is 2.

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Compound Amount Formula

By Junius W. Yu

Present Value or

Original Principal

number of

conversion

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Exercise

1. If Php14,500 is invested for 5 years in abank that pays 6% compounded quarterly,what sum will the investor receive after 5

years? How much interest was earned?

2. Accumulate Php50,000 for 19 months at24% compounded monthly? How much isthe compounded interest?

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Given: P = Php14,500; t = 5 years; m = 4; j = 6%

Find: F and Ic

Solution: i = j/m = 0.06/4 = 0.015

n = tm = (5)(4) = 20

F = P (1 + i)n

Php14,500 (1 + 0.015)20

Php19,529.40

Ic = Php19,529.40Php14,500Php5,029.40

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By Junius W. Yu

Given: P = Php50,000; t = 19 months;

m = 12; j = 24%

Find: F and Ic

Solution: i = j/m = 0.24/12 = 0.02

n = tm = 19/12(12) = 19

F = P (1 + i)n

Php50,000 (1 + 0.02)19

Php72,840.56

Ic = Php72,840.56Php50,000

Php22,840.56

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By Junius W. Yu

Given: P = Php100,000; t = 28 months;

m = 12; j = 12%

Find: F and Ic

Solution: i = j/m = 0.12/12 = 0.01

n = tm = 28/12(12) = 28

F = P (1 + i)n

Php50,000 (1 + 0.01)28

Php132,129.10

Ic = Php132,129.10Php100,000

Php32,129.10

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Present Value at Compound Interest

By Junius W. Yu

Present Value = F(1 + i)-n

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Exercises

1. What is the present value of Php41,000 due

in 9 years and 3 months if the value is

compounded quarterly at 12%?

2. If money is invested at 7% compounded

semi-annually, find the present value of

Php100,000 which is due in 4 years.

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Given: F = Php41,000; t = 9.25; j = 0.12; m = 4

Find: P

Solution: Solve for i and ni = j/m = 0.12/4 = 0.03

n = tm = 9.25(4)= 37

Input the valuesP = F (1 + i)-n = Php41,000(1+0.03)-37

P = Php13,734.30

Answer: The present value of Php41K that is dueat the end of 9.25 years is Php13,734.30

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Given: F = Php100K; t = 4 years; j = 7%; m = 2

Find: P

Solution: i = j/m = 0.07/2 = 0.035

n = tm = (4)(2) = 8

input the values

P = F (1 + i)-n

=Php100,000(1+0.035)-8

Php 75,941.16

Answer: The present value of Php100K that isdue at the end of 4 years is Php58,200.91

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Nominal Rate Formula

By Junius W. Yu

1

1n

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Exercises

1. A Php85,000 investment earned an interest

of Php4,000 in 3 years. At what nominal rate

compounded annually was the money

invested?

2. If a certain principal triples itself in 5 years,

find the interest rate compounded quarterly

at which it is invested.

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Given: P = Php85,000, Ic = Php4000; m = 1,

Find: j

Solution: n = tm = 1(3) = 3F=85,000+4000 = Php89,000

=

1 =

8

8

1 1= 0.0155 or 1.55%

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Given: P, F = 3P, t = 5 years; m = 4; n = 4(5) = 20

Find: j = ?

Solution: =

1

=

3

1 4= 0.2259 or 22.59%

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Time?

We count the sands

of time and realize it

is too short to enjoyand yet too long to

endure?

Legend of the 9

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Finding Time

How long does it take for

an investment to reach

the desired results?

Investors want to know

the return on investment

on any given particular

time.

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John Napier

John Napierinventor of

logarithms.

- Famous mathematician

- Marvelous Merchiston

- Napier bones

- Occult Book of

Revelation- Apocalypse

- Travel box with spider

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Log time formula

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Exercise1. How long will it take Php65,000 to

accumulate to Php200,000 if invested at 5%

compounded quarterly?

2. How long will it take Php25,000 to become

Php55,000 if it is invested at 11% converted

monthly?

3. Find how long will it take for money to

double itself if invested at 9% compounded

annually, semi-annually, quarterly, monthly?

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Given: P = Php25,000; F = Php55,000; j=11%

Find: t

Solution: i = j/m = 0.11/12 = 0.009167

=log

55,00025,000

12 log(1 + .009167)

= 7.20 years

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Given: P, F= 2P, j = 9%, m = 1, 2, 4, 12

Find: t

Solution: log (2P/P) = log 2 = 0.30103

By Junius W. Yu

Annualy Semi

Annually

Quarterly Monthly

Interest

rate per

period

0.09 0.045 0.0225 0.0075

Log (1+i) 0.037427 0.01912 0.00966 0.00325

m log (1+i) 0.037426 0.038233 0.038653 0.038941

/log 2 8.043232 7.873651 7.787957 7.730481

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Compound Amount Formula (n)

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Nominal Rate equivalent to w

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Level I Exercise (+2,+1)

1. What nominal rate compounded monthly is

equivalent to 11% effective?

2. Find the effective rate equivalent to 7%

compounded quarterly. Apply the effective

and nominal rates in order to determine the

compound amount of Php10,000 for one

year.

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Problem 1

Given: w = 11%

Find: j (m = 12)

Solution: = 1 +

1

= 12 1 + 0.11

1

= 0.1048 or 10.48%

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Problem 2

Given: j = 0.07; m = 4

Find: w

Solution: = 1 +

1

= 1 +.

1

w = 0.071859 or 7.1859%

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Comparison

a. Comparison of Two

Nominal Rates

b. Comparison of

Nominal Rate andSimple Interest Rate

c. Comparison of

Nominal and Discount

Rate

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Comparison of Two Nominal RatesTwo rates are equivalent ifthey produce the samecompound amount for thesame span of time.

Equate F1 = F2

Then solve for j1 and j2

To compare two nominal

rates in terms of profit,compare their effectiverates. Higher rates, higherprofit.

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Nominal Rate equivalent to w

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Nominal Rate equivalent to w

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Exercises

1. What nominal rate compounded monthly is

equivalent to 15% compounded quarterly?

2. Mr. Wonderful was offered by Bank O, 6%

compounded quarterly for his investment.

Bank Z offers 6.5% compounded semi-

annually for the same amount of investment.

If you are Mr. Wonderful, what would youdo?

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Given: j2 = 15% (m2 = 4), m1 = 12

Find: j1 (m1 = 12)

Solution:= 1 +

1

= 12 1 +

0.15

4

1

= 0.148163 or 14.82%

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Formula r

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Formula j

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Exercises

1. What nominal rate compounded quarterly is

equivalent to 4.5% simple interest rate in a 4-

year transaction?

2. What simple interest rate is equivalent to

7.25% compounded monthly in a 4-year

transaction?

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Given: r = 4.5%; m = 4; t = 4

Find: j

Solution: n = tm = 4(4) = 16

= 1 +

1

= 12 1 + 0.045(4)

1

j = 0.04145 or 4.15%Answer: The nominal rate 4.15% compounded

monthly is equivalent to 4.5% simple interest

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Given: j = 7.25%; m = 12; t = 4Find: r

Solution: i = j/m = 0.0725/12 n = tm = 4(12) = 48

=:

;

=

:.

;

= 0.083815 or 8.38%

Answer: A simple interest rate of 8.38% isequivalent to a nominal rate of 7.25% (m = 12)

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Nominal Rate VS Discount Rate

Instead of simple interest

rate, simple discount rate

can be equivalent to a

given nominal rate

F simple discount rate =

F nominal rate

P/(1-dt) = P(1+i)n

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Formula d

=(1 +

)1

(1 + )

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Formula j

=

1

1

1

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Exercises

1. What nominal rate j, compounded quarterly

is equivalent to 12.5% discount rate in a 5-

year transaction?

2. Find the simple discount rate that isequivalent to a nominal rate of 8.6%

compounded semi-annually in an 8-year

transaction.

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Given: d = 12.5%, m = 4, t = 5, n = (5)(4) = 20

Find: j

Solution: = ;

1

= 41

1 (.125)(5)

1

= 0.201056 or 20.11%

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Given: j = 0.0860, m = 2, t = 8

Find: d

Solution: = (:

)

;

(:

)

=

(:.

);

8 (:.

) = 0.061267 or 6.13%