The Amortization Model

Written for the Envision It! Workshop

Steven McKelvey
Mathematics and Computer Science
Saint Olaf College

The Model

This model of loan amortization is an example of a useful STELLA model which does not have an equilibrium. The situation we model is this: assume we have a bank account which pays interest every unit of time (typically a month, quarter or year). Depending on the initial balance and the relative sizes of deposits and withdrawls the balance on this account may be increasing or decreasing. Developing a STELLA model of this process allows us to easily compute future account balances.

The difference equation we use to perform this computation is nicely suited for implementation in STELLA. The equation is:

P(i+1)=P(i) + r*P(i) + deposits - withdrawls

where r is the interest rate per unit time and deposits and withdrawls are the transactions during each time interval. The quantities P(i) are the principal balances at the various times during the model's run. These values will be represented by reservoirs in the STELLA model.

It is important to remember that the interest rate used in not necessarily the annual interest rate, but rather the interest rate per time period. For instance, if interest is paid monthly, then the interest rate r which should be used in the model is r = 0.08/12 = 0.00667.

The STELLA diagram for this model is:

This model can be used to answer a wide variety of questions concerning a savings account. The exercises below demonstrate some of the uses of this model.

Exercises

1. Build the STELLA model for amortization. Convince yourself that with reasonable interest rates and deposits/withdrawls this model does not have any equilibria.
2. Suppose you have $10,000 in a saving account paying 8% per annum interest, compounded monthly.
   1. If no further deposits or withdrawls are made, how long will it be before your account holds $15,000?
   2. What monthly withdrawl can you make if you want the balance to remain positive for exactly 48 months?
   3. How many months will the account last if you withdraw $300 from the account every month?
3. Starting with a copy of the STELLA model you created above, form a model where the principal is no longer a savings account balance but is, instead, the amount outstanding on an automobile or home loan. Begin by finding a relevant difference equation upon which to base the STELLA model.
   1. At 8% annual interest, how long will it take to pay off a $100,000 mortgage using $500 monthly payments?
   2. At 8% annual interest, how long will it take to pay off a $100,000 mortgage using $1000 monthly payments?
   3. At 8% annual interest, how long will it take to pay off a $100,000 mortgage using $1500 monthly payments?
   4. What monthly payment is required to pay off a $100,000 mortgage at 8% annual interest over 15 years? (30 years?)
4. (WARNING: This exercise can lead to serious depression!) Using the first model determine what the final savings balance would be if you were able to deposit the monthly mortgage payment calculated above into a saving account (where you received interest) instead of a mortgage where you pay the interest. (If you want to be REALLY depressed, do this for the 30 year mortgage.)

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