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.
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.
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.