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Econ 407: Applications of Exponential and Logarithmic Functions in Economics and Finance:

First Example: Discrete Compounding: you compute interest plus principal once,twice or several times a year. You invest $100 at the interest rate of 10% compounded once a year. What is the total principal plus interest at the end of one year, two years, 10 years? Let S be the fture Sum (total accumulation), p amount invested today, i: the interest year, and t the number of years.

> S:=100.0*(1+.1)^1;

> S:=100.0*(1+.1)^2;

> S:=100.0*(1+.1)^10;

Let us carry out the example with discrete compounding twice a year, three times a year, nine years, twenty times a year. Please note that t=1 year.

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> S:=100*(1+.1/2)^2;

> S:=100*(1+.1/3)^3;

> S:=100*(1+.1/9)^9;

> S:=100*(1+.1/360)^360;

> S:=100*(1+.1/12)^12;

Now let us investigate continous compounding, use the following formula.

> S:=100*exp(.1);

> S:=100*(1+0.1/(365*24))^(365*24);

> S:=p*exp(r*t);

> eval(S,{p=5, r=.05, t=100.0});

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An Application to inflation: What is the worth of a $1, in 100 years if the rate of inflation is five percent year compounded continously.

> eval(S,{p=1, r=-.05, t=100.0});

> 742*.0067;

An application to population growth: Given that the world population is 6 billion, and it is growing at the rate of 3 percent a year, what will be the population in 10 years, in 20 years?

> eval(S,{p=6000000000.0, r=.03, t=10.0});

> eval(S,{p=6000000000.0, r=.03, t=20.0});

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