Money, Banking, and International Financeܸܨହ=$20, 000 ( 1 + 0. 09 )ସ+$20, 000 ( 1 + 0. 09 )ଷ+⋯+$20, 000 ( 1 + 0. 09 ) ( 18 )
ܸܨହ=$20, 000 [ 1. 09 ସ+ 1. 09 ଷ+ 1. 09 ଶ+ 1. 09 ଵ+ 1. 09 ]
ܸܨହ=$119, 694. 21
Do you notice anything strange about the exponents? We raise the first term in Equation 18
to the fourth power because your initial payment occurred at the end of period 1, and has earned
four years of interest. Finally, the last term has a zero exponent, and the final $20,000 does not
earn interest. Moreover, mathematicians derived a formula to calculate an annuity without
calculating a long series of numbers. They derived a formula in Equation 19, and c is the
periodic payment into an annuity. Using the previous example, the value of the annuity still
equals $119,694.21.
ܸܨ்=ܥቂ
(ଵା)ିଵ
ቃ=$20,^000 ቂ
(ଵା.ଽ)ఱିଵ
.ଽ ቃ=$119,^694.^21 (^19 )^
We also have the other side of an ordinary annuity. For example, you saved a $60,000
annuity that earns 4% APR. You plan to withdraw equal annual payments over 10 years. How
much do you receive annually? Remember, you receive your first payment at the end of the first
period, which is the beginning of the second period. That $60,000 earns interest for the first
period. We compute an annual withdrawal payment of $7,397.46 in Equation 20.
ܸܨ=
∙
ଵି(ଵା)ష=
.ସ∙ ,
ଵି(ଵା.ସ)షభబ=$7,^397.^46 (^20 )^
Financial analysts use the present value formula to calculate mortgage payments, which is
vital to building an amortization table. An amortization table itemizes every payment for a
mortgage loan and decomposes every payment into interest and the amount that reduces the
principal. A mortgage is a bank loan for a property, and the property becomes the collateral. For
instance, if a person has a mortgage for a house and defaults on the loan, the bank can legally
take possession of the house. We use the present value formula to build an amortization table.
Mathematical notation for a mortgage is:
All future mortgage payments (FV) are equal and are usually monthly. Interest rate (i) is loan rate and becomes fixed throughout life of the loan. Bank loan is amount recorded for PV 0 because the bank loaned you money at time 0.We show a mortgage as a stream of cash flows to the bank in Equation 21.