Application of Logarithmic Functions to Finance
MHF4UI
Tuesday October 23rd , 2012
Financial Problems use
Exponential Growth and Decay
We can apply what we have learned about Exponentials to
the World of Finance to model the growth of money.
The reason we can do this is because all loans and
investments charge/reward interest to investors/creditors.
Compounding refers to adding an amount of interest to
the original investment (principal).
Loans and investments often have interest compounded
over varying periods.
How Often do we Calculate Interest?
Compounding Period
Annually
Semi-annually
Quarterly
Monthly
Weekly
Daily (Continuous)
# of times interest is charged
per year
1
2
4
12
52
365
The General Formula for Problems of
Exponential Growth and Decay
The standard form of an exponential growth or decay is:
𝐴 𝑡 = 𝐴0 (𝑏)𝑡
Where,
𝐴0 is the initial quantity at t = 0
b is the factor of growth (1 + i) or decay (1 – i)
i is the percent rate of growth or decay (must be expressed as a decimal)
t is the number of growth or decay periods
A(t) is the quantity at time t
The Financial Formula for Growth
The formula for calculating the amount of an
investment or a loan after a given time-period is:
𝐴 𝑛 = 𝑃(1 + 𝑖)𝑛
Where,
P is the Principal (original value invested)
i is the interest rate divided by the number of compounding periods
per year
n is the number of years multiplied by the number of compounding
periods per year
A(n) is the value of the investment at time t
Investment Growth Example 1
Cathy invests $5,000 at 5% compounded quarterly. How much
will her investment be worth in 6 years?
Investment Growth Example 2
Cathy invests $5,000 at 5% compounded quarterly. How long
will it take her investment to Triple?
Investment Growth Example 3
We want to invest $50,000 today at an interest rate of 4%
compounded monthly. If we want to have $1,000,000 at
retirement, in how many years will be able to retire?
Finding Interest Rates Example 1
Steven’s parents invested $15,000 for him when he was born.
He just turned 18 and his investment has reached a value of
$115,000. What was the annual interest rate of the investment
if interest was compounded Semi-Annually?
The Time Value of Money
Have you ever wondered why stuff becomes more expensive over time?
If I give you $100 dollars and you put in under you mattress for 10 years,
will it still be worth $100 after ten years?
The answer is yes, a $100 bill is still $100, BUT the important idea is this:
If you spend $100 today you can buy 5 CDs.
If you spend $100 in 10 years you will only be able to buy 4 CDs.
This is referred to as the Time Value of Money
Time Value of Money Example 1
The annual rate of inflation is 2%. If you put $100 under your
mattress 10 years ago, how much would your money be worth
in todays dollars?
Time Value of Money Example 2
The annual rate of inflation is 3%. What would $1,000,000 today
be worth back in the year 1912?
Homework Questions:
• Complete Worksheet Handout