Unformatted text preview: Business Calculus Vera Klimkovsky MATHEMATICS OF FINANCE:
SIMPLE AND COMPOUND INTEREST
1. SIMPLE INTEREST
Simple interest is interest that is computed on the
original principal only.
Let P represent the principal amount, r represent an
interest rate per year and t is time, in years, then we
can define an interest earned (paid on principal) as I = Prt . The accumulated amount , A , is the sum of the principal
and the interest paid on the principal, A = P + I = P + Prt = P (1 + rt ) Example:
A bank pays simple interest at a rate of 8% per year for
a deposit. If a customer deposits $1000 and makes no
withdrawals for 3 years, what is the total amount on
deposit at the end of 3 years? What is the interest earned
in that period of time?
Solution:
P = $1000 (principal), r = 0.08 (interest rate), t = 3 (years)
The total amount on deposit at the end of 3 years is given by A = P (1 + rt ) = 1000 (1 + 0.08 × 3) = $1, 240.00
The interest earned over the 3year period is given by I = Prt = 1000 × 0.08 × 3 = $240.00 . Business Calculus Vera Klimkovsky 2. COMPOUND INTEREST
Compound Interest is the interest computed on the
principle amount to which interest earned todate has
been added.
If amount of P dollars is invested over a term of t years,
earning interest at the rate of r per year compounded
annually, then the accumulated amount is A = P (1 + r ) t If interest at a nominal rate of r per year is compounded
m times a year on a principal of P dollars, then the
simple interest rate per conversion period is
i= r Annual interest rate
=
m
Periods per year Since there are n = mt (number of conversion periods in
a year times the term, number of years), we can write
the final general formula: A = P (1 + i ) n r⎞
⎛
= P ⎜1 + ⎟
⎝ m⎠ mt Example:
Suppose $1000 is deposited in a bank for a term of 3
years, earning interest at the nominal rate of 8% per
year compounded annually. Find the accumulated
amount at the end of 3 years.
Solution:
The accumulated amount at the end of the third year is A = P(1 + r )3 = 1000 (1 + 0.08 ) ≈ $1259.71
3 Business Calculus Vera Klimkovsky Example
Find the accumulated amount after 3 years if $1000 is
invested at 8% per year compounded
(a) annually (already done above)
(b) semiannually
(c) quarterly
(d) monthly
(e) daily.
Solution: Here P = 1000, r = 0.08, t = 3
(a) Annually, m = 1 A = P (1 + i )
(b) mt n r⎞
3
⎛
= P ⎜1 + ⎟ = 1000 (1 + 0.08 ) = 1259.71
⎝ m⎠ Semiannually, m = 2
mt 6 r⎞
⎛
⎛ 0.08 ⎞
A = P (1 + i ) = P ⎜1 + ⎟ = 1000 ⎜1 +
⎟ = 1265.32
2⎠
⎝ m⎠
⎝
(c)
Quarterly, m = 4
n A = P (1 + i )
(d) mt n 36 r⎞
⎛
⎛ 0.08 ⎞
= P ⎜1 + ⎟ = 1000 ⎜1 +
⎟ = 1270.24
m⎠
12 ⎠
⎝
⎝ Daily, m = 365 A = P (1 + i ) 12 r⎞
⎛
⎛ 0.08 ⎞
= P ⎜1 + ⎟ = 1000 ⎜1 +
⎟ = 1268.24
4⎠
⎝ m⎠
⎝ Monthly, m = 12 A = P (1 + i )
(e) mt n mt n 1095 r⎞
⎛
⎛ 0.08 ⎞
= P ⎜1 + ⎟ = 1000 ⎜1 +
⎟
365 ⎠
⎝ m⎠
⎝ = 1271.22 Business Calculus Vera Klimkovsky 3. CONTINUOUS COMPOUNDING
Continuous compounding occurs when the interest is
computed and added to the principal at every instance,
thus making the compounding period infinitely small and
therefore, making m infinitely large. m 1⎞
⎛
1 + ⎟ as m becomes very
If we examine the expression ⎜
⎝ m⎠
large, we see that it approaches some fixed value. This value is known as constant e (Euler’s number) e ≈ 2.71828182845904523536... Business Calculus Vera Klimkovsky m mt 1⎞
r⎞
⎛
⎛
1 + ⎟ and ⎜ 1 + ⎟ ,
Notice, that the two expressions, ⎜
⎝ m⎠
⎝ m⎠
are very similar but not exactly the same.
Now, it can also be shown (by using mathematical
manipulations) that
mt r⎞
⎛
A = P ⎜1 + ⎟ = Pe rt
⎝ m⎠ as m → ∞ (become infinitely large) Example
Find the accumulated amount after 3 years if $1000 is
invested at 8% per year compounded continuously.
Solution: A = Pe rt = 1000 ⋅ e0.08×3 = $1, 271.25 ...
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This note was uploaded on 10/19/2011 for the course MATH 110 taught by Professor Staff during the Spring '11 term at S.F. State.
 Spring '11
 Staff
 Calculus

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