Pre-Session Review Part 2: Mathematics of Finance

[Pages:10]Pre-Session Review Part 2: Mathematics of Finance

For this section you will need a calculator with logarithmic and exponential function keys (such as log , ln , and x y )

D. Exponential and Logarithmic Functions

Exponential Functions are of the form f(x) = a x

where the variable x appears as a power (exponent) in the equation.

Typical application: Growth rate problems

Example:

Suppose the U.S. population is growing 3% per year. What will be the population in 10 years if the current population is P ?

Ans:

in one year: P + (.03) P = (1 + .03) P

in two years: (1 +.03) (1 + .03) P = P (1 + .03) 2

After 10 years:

P (1 + .03) 10

Properties of exponents: a0 = 1 aX @ aY = aX+Y aX / aY = aX!Y

( ) a X Y = a XY

(a b)X = aX bX

1

aX = X a

by definition, for all a ... 0 (0X = 0 for all x > 0) example: 23 @ 24 = (2 @ 2 @ 2) @ (2 @ 2 @ 2 @ 2) = 23+4 = 27

so a-Y = a0!Y = a0 / aY = 1/aY , also written as Y a

example: (23)2 = (2 @ 2 @ 2) @ (2 @ 2 @ 2) = 26

so

a b

X

=

aX bX

for b ... 0

Y

Y

1

so

a X = X aY ,

sin ce

aX

=

Y

aX

=

X

aY

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Graphically: 1. f(x) = aX

if a > 1

x

f(x) = a X

-2

a-2 = 1/a2

-1

a-1 = 1/a

0

1

1

a

2

a2

2. f(x) = aX

if a = 1

x

f(x) = a X

-2

1-2 = 1/12 = 1

-1

1-1 = 1/1 = 1

0

1

1

1

2

12 = 1

...so we have a series of similar graphs when a > 1 , approaching a horizontal line at f(x) = 1 when a decreases toward one.

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f(x) = aX when 0 < a < 1 :

x

f(x) = a X

-2

(?)-2 = 1/(?)2

= 4

-1

(?)-1 = 1/(?)

= 2

0

1

1

?

2

(?)2 = 1

...so we get a family of curves for different values of a when 0 < a < 1 :

What if a = 0 ? ? evaluate a X if x = -2 : (0) -2 = 1/(0) 2 = 1/0 = ??

What if a < 0? ? evaluate a X if x = ? and a = -2 : (-2) -? = - 2 = ?

(not defined) ("imaginary number")

Moral: We only work with exponential functions f(x) = a X when a > 0.

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A very useful constant in finance and calculus applications is e = 2.71828 . . . , the base of the natural log function (e is an irrational number).

This leads to a special exponential function

f(x) = e X

which is one of the family of exponential functions already graphed:

Logarithmic Functions Log functions provide an inverse function for exponential functions.

Rule:

If a X = b , then log a b = x

(Since a X = b , log a b = log a a X = x)

Examples:

10 2 = 100 23 = 8

so log 10 100 = log 10 (10) 2 = 2 so log 2 8 = log 2 (2) 3 = 3

Conventions: Log 10 is called the common log and is usually written without the subscript: log Log e is called the natural log and is written: ln

Examples: log 16 = x ? 10 x = 16 ? So x is greater than 1 but much less then 2 (x . 1.20412) ln 16 = y ? e y = 16 ? So y is greater than 2 but less than 3 (y . 2.7726)

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Properties of Logs

log (ab) = log a + log b

log (a/b) = log a ? log b

log a b = b (log a)

log c c a = a

(ie, the log function is the inverse of the exponential function.)

Graphically: (base 10)

x 1/100 1/10

1 10 100 1000

f(x) = log x -2 -1 0 1 2 3

Note: log (a) is not defined if a # 0 . Again, there is a family of curves for log functions with different bases:

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Inverse Function Notation Since the inverse of (log c x) is c X , shorthand notation has been developed to write the statement more compactly:

Notation:

antilog c x / c X antiln x / e X

or

log

?1 c

/

c

X

or

ln ?1 x / e X

Using anti-logs:

log c x = y

?

log

?1 c

log

c

x

x

x

= = =

log log c y

?1 c?1 c

y y

(apply anti-log to both sides to solve for x)

ln a = b

? ln ?1 ln a = ln ?1 b a = ln ?1 b a = eb

(apply anti-ln to both sides to solve for a)

Examples and Applications:

Solve for x: 1. log x = 2.5 ?

log ?1 log x = log ?1 2.5 x = 10 2.5 x = 316.2278

2. ln x = 8.2 ? ln ?1 ln x = ln ?1 8.2 x = e 8.2 x = 3640.95

3. e X = .5

? ln e X = ln (.5) x = - .6931

4. You have $A to invest and you want to double your money in 5 years. What interest rate must be earned?

(Use compound interest formula $A (1 + i) 5 = 2($A) and solve for i )

$A (1 + i) 5 = 2($A) ?

(1 + i) 5 = 2 (1 + i) = 2 .2

i = 2 .2 ? 1 = 1.1487 ? 1 = .1487

(Both sides to 1/5 power) (Subtract 1 from both sides)

So nearly a 15% return is needed.

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5. 5 X = 1.02

(Note: this is not a base 10 or base e problem, but base 5! You don't have a base 5 function key on your calculator, so how do you solve for x ?)

5 X = 1.02 ? ln 5 X = ln 1.02

take ln of both sides (or log would work) (we do have ln key on the calculator)

x (ln 5) = ln 1.02 use properties of exponential functions

x=

ln1.02 ln 5

=

.0198 1.6094

= .0123

For you to try ? solve for x:

6.

10 log 5 = x

7. log 2 4.3 = x 8. 7 X = 126

9. If log a x = 4.2 and if log a y = 1.4, show that loga xy = 2.8

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E. Mathematics of Finance

Idea: Apply the exponential function tools of the last section to finance problems:

Present value Future value Annuities

Overview:

1. People are willing to accept payment in the future only if they will receive a greater amount.

2. The further in the future that payment is due, the smaller is the current equivalent amount.

3. The greater the rate of interest, the smaller is the equivalent value of a fixed future amount.

Simple Interest

Let P = principal (the amount invested or borrowed, to which interest is applied) I = amount of interest (in $) i = interest rate (annual rate, in decimal form) n = number of years

Simple interest formula: I = P @ i @ n

Example:

How much interest is earned on an investment of $80 at 8.5% interest for 1? years?

I = P @ i @ n

= (80) (.085) (1.5)

= 10.2 ,

or $10.20

The simple interest formula allows us to find any of I, P, i , or n if the other three values are known.

Problems:

1. How long will it take a deposit of $1000 at 12% to earn $50 simple interest?

2. How much interest is due on a $15,000,000 loan at 4.5% for 1 day? (Think of a bank borrowing overnight from the Federal Funds market.)

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