Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc Mathematics of Finance 5 Compound Interest Annuities Amortization and Sinking Funds

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc

Mathematics of Finance5• Compound Interest

• Annuities

• Amortization and Sinking Funds

• Arithmetic and Geometric Progressions


Simple Interest

Interest: I = Prt

Accumulated amount: A = P(1 + rt)

Simple Interest - interest that is compounded on the original principal only.


Ex. $800 is invested for 9 years in an account that pays 12% annual simple interest. How much interest is earned? What is the accumulated amount in the account?

P = $800, r = 12%, and t = 9 years

Interest: I = Prt

= (800)(0.12)(9)

= 864 or $864

Accumulated amount = principal + interest

= 800 + 864 = 1664

or $1664


Compound Interest

(1 ) nA P i where and r

i n mtm

A = Accumulated amount after n periods

P = Principal

r = Nominal interest rate per year

m = Number of conversion periods per year

t = Term (number of years)

Compound Interest – interest is added to the original principal and then earns interest at the same rate.


Ex. Find the accumulated amount A, if $4000 is invested at 3% for 6 years, compounded monthly.

P = $4000, r = 3%, t = 6, and m = 12

(1 ) nA P i

So.03

.0025 and 12(6) 7212

ri n mt

m

724000(1 .0025)

4787.79

or $4787.79


Effective Rate of Interest

eff 1 1 m

rr

m where

reff = Effective rate of interest

r = Nominal interest rate per year

m = Number of conversion periods per year

Effective Rate – the simple interest rate that would produce the same accumulated amount in 1 year as the nominal rate compounded m times per year.


Ex. Find the effective rate that corresponds to a nominal rate of 6% compounded quarterly.

eff 1 1 m

rr

m

r = 6% and m = 4

4.06

1 1 4

.06136

So about 6.136% per year.


Present Value (Compound Interest)

(1 ) nP A i

Present Value (principal) – the amount required now to reach the desired future value.

where and r

i n mtm


Ex. Jackson invested a sum of money 10 years ago in an account that paid interest at a rate of 8% compounded monthly. His investment has grown to $5682.28. How much was his original investment?

A = $5682.28, r = 8%, t = 10, and m = 12

.08 and 12(10) 120

12

ri n mt

m

120.08

5682.28 1 12

P

2560.00

or $2560


AnnuityAnnuity – a sequence of payments made at regular time intervals.

Ordinary Annuity – payments made at the end of each payment period.

Simple Annuity – payment period coincides with the interest conversion period.


Future Value of an Annuity

(1 ) 1

niS R

i

The future value S of an annuity of n payments of R dollars each, paid at the end of each investment period into an account that earns interest at the rate of i per period is


Ex. Find the amount of an ordinary annuity of 36 monthly payments of $250 that earns interest at a rate of 9% per year compounded monthly.

(1 ) 1

niS R

i

R = 250, n = 36 and .09

12i

36.091 1 12250

.0912

S

10288.18 S or $10,288.18


Present Value of an Annuity

1 (1 )

niP R

i

The present value P of an annuity of n payments of R dollars each, paid at the end of each investment period into an account that earns interest at the rate of i per period is


Ex. Paige’s parents loaned her the money to buy a car. They required that she pay $150 per month, for 60 months, with interest charged at 2% per year compounded monthly on the unpaid balance. What was the original amount that Paige borrowed?

1 (1 )

niP R

i

60.021 1 12

150 .02

12

8557.85 or $8557.85


Amortization Formula

1 (1 ) n

PiR

i

The periodic payment R on a loan of P dollars to be amortized over n periods with interest charged at a rate of i per period is


Ex. The Kastners borrowed $83,000 from a credit union to finance the purchase of a house. The credit union charges interest at a rate of 7.75% per year on the unpaid balance, with interest computations made at the end of each month. The Kastners have agreed to repay the loan in equal monthly installments over 30 years. How much should each payment be if the loan is to be amortized at the end of the term?

P = 83000, n = (30)(12) = 360, and 0.0775

12i

Continued


360

.077583000 12 .07751 1 12

594.62

So a monthly installment of $594.62

1 (1 ) n

PiR

i


Ex. A bank has determined that the Radlers can afford monthly house payments of at most $750. The bank charges interest at a rate of 8% per year on the unpaid balance, with interest computations made at the end of each month. If the loan is to be amortized in equal monthly installments over 15 years, what is the maximum amount that the Radlers can borrow from the bank?

R = 750, n = (15)(12) = 180, and 0.08

12i

Continued


1 1n

iP R

i

180.081 1 12

750.08

12

78480.44

So they can borrow up to about $78480.44


Sinking Fund Payment

(1 ) 1n

iSR

i

The periodic payment R required to accumulate S dollars over n periods with interest charged at a rate of i per period is


Ex. Max has decided to set up a sinking fund for the purpose of purchasing a new car in 4 years. He estimates that he will need $25,000. If the fund earns 8.5% interest per year compounded semi-annually, determine the size of each (equal) semi-annual installment that Max should pay into the fund.

S = 25000, n = 4(2) = 8, and 0.085

4i

Continued


(1 ) 1n

iSR

i

8

.085 250004 .0851 14

2899.91

So semi-annual payments of about $2899.91


Arithmetic ProgressionsArithmetic progression – a sequence of numbers in which each term after the first is obtained by adding a constant d (common difference) to the preceding term.

Ex. 1, 8, 15, 22, 29, …

First term Common difference: d = 7


Arithmetic Progression

1 na a n d

The nth term with the first term a and common difference d is given by

2 ( 1) 2nn

S a n d

The sum of the first n terms with the first term a and common difference d is given by


Ex. Given the arithmetic progression1, 8, 15, 22, 29, …

find the 10th term and the sum of the first 10 terms.

10th term:

a = 1, d = 7, and n = 10.

1 na a n d

10 1 10 1 7 a = 64

Sum: 2 ( 1) 2nn

S a n d

1010

2(1) (10 1)(7) 2

S = 325


Geometric ProgressionsGeometric progression – a sequence of numbers in which each term after the first is obtained by multiplying the preceding term by constant r (common ratio).

Ex. 9, 3, 1, 1/3,…

Common ratio: r = 1/3


Geometric Progression

1 nna ar

The nth term with the first term a and common ratio r is given by

1 if 1 1

if 1

n

n

a rrS r

na r

The sum of the first n terms with the first term a and common ratio r is given by


Ex. Given the geometric progression4, 12, 36, 108, …

find the 8th term and the sum of the first 8 terms.

8th term:

a = 4, r = 3, and n = 8.

= 8748

Sum: 1

1

n

n

a rS

r

8

8

4 1 3

1 3S

= 13120

1 nna ar

8 18 4 3 a

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Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc Mathematics of Finance 5 Compound Interest Annuities Amortization and Sinking Funds