Mathematical Modeling     Chapter 1.5    Financial Models

Example 1

You deposit $10,000 in a bank account. The account earns 4% annual interest (compounded annually), and you withdraw $1,200 at the end of each year. Let u(n) denote the value of the account after n years.

Here we model a 10 year period:

u[0]=10000;
Do[u[n]=1.04u[n-1]-1200,{n,1,10}];
data=Table[{n,u[n]},{n,0,10}];
ListPlot[data]

By changing the last line of code from ListPlot to Grid, we get a table of values instead:

u[0]=10000;
Do[u[n]=1.04u[n-1]-1200,{n,1,10}];
data=Table[{n,u[n]},{n,0,10}];
Grid[data]

The plot & table make clear that we can only keep this up for 10 years. In the 11th year, we are sure to be in the red.

Example 2

You deposit $10,000 in a bank account. The account earns 4% annual interest, compounded monthly, and you withdraw $100 at the end of each month. Just as in the first example, you are withdrawing $1,200 each year. Let u(n) denote the value of the account after n months.

The only difference between this example and the last is that we need to compound at the monthly rate of 4%/12=(1/3)%=1/300.

We again model a 10 year period (120 months):

u[0]=10000;
Do[u[n]=(1+1/300)u[n-1]-100.,{n,1,120}];
data=Table[{n,u[n]},{n,0,120}];
ListPlot[data]

And since we don't want to look at 120 rows in a full grid, let's just look at the last few months by adjusting the iterator on the third input line:

u[0]=10000;
Do[u[n]=(1+1/300)u[n-1]-100.,{n,1,120}];
data=Table[{n,u[n]},{n,115,120}];
Grid[data]

Note that we used 100. (with a decimal point) so that we get decimal values. Note also that under this scheme, we have less in our account after ten years than we did in example 1.

Exercise 1

You deposit $10,000 in a bank account. The account earns 4% annual interest, compounded monthly, and you withdraw $100 at the end of each month. Just as in the first example, you are withdrawing $1,200 each year. Let u(n) denote the value of the account after n months. (This is exactly the scenario of example 2.)

How many months can you do this before your account runs out?

Exercise 2

You owe $10,000 on your credit card. The card charges 24% interest, compounded monthly. You make a monthly payment of $225.

(a) How many months will it take to pay off the card, assuming you make no other charges on the account, and continue paying monthly payments of $225?

(b) Multiply your answer to (a) by 225. This is how much you paid in total for your $10,000 of credit card purchases.

(c) What is the smallest monthly payment that you could make and still (eventually) pay off your card?