General Pricing Formulas for Finite Cash Flow Streams

Again, assume the bond pays a coupon C for a definite period, n, as in:

Solving this finite power series yields

See Appendix 2.1 for the complete derivation. Suppose C stands for any payment (like a payment on an installment loan). It has no terminal face value, so we set M to zero. Then if P represents the loan amount, C (the payment) can be solved for as:

This is an amortization formula that gives the pay down on a loan over time n at interest rate r for n periods with principal P. It is of interest to amortize a loan into interest and principal payments.

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Example 2.2

This example calculates the amortization schedule for each year in a 10-year loan with annual payments on a principal of $1,000 and interest rate equal to 10 percent.

This example clearly separates the principal and interest payments on the loan—the total payments equal $1,627.45, of which $1,000 is on the original principal, while $627.45 goes to interest. Extending this example, suppose that you are interested in buying a home with price P and mortgage rate r that will be financed over a 30-year period with monthly mortgage payments each equal to C. Then the price of the home must equal the sum of its discounted stream of monthly mortgage payments:

Table 2.2 Amortization of a 10-Year Loan

Solving this series (just like we did earlier) results in the following relationship between the price of the home and the monthly mortgage payment:

where m = 12 and therefore . More importantly, the amortization formula is equal to:

This amortization formula is useful for computing the monthly payment for any n period loan with interest rate r and financing period n.

Go to the companion website for more details.

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Example 2.3

This example calculates the amortization schedule for each month in a 30-year mortgage with monthly payments on a principal of $200,000 and interest rate equal to 5 percent. Note that almost half of the cost of this loan is in the form of interest payments. This is the power of compound interest.

Table 2.3 Monthly Amortization Schedule for a 30-Year Mortgage.

Go to the companion website for more details.

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