Kenneth R. Szulczyk
We calculated a present value of $212.06 in Equation 15.ܸܲ=(ଵା.଼ଷଷഥ)భ+(ଵା.଼ଷଷഥ)ల+(ଵା.଼ଷଷഥ)భయ=$212. 06 ( 15 )
Compounding frequency has a special case. As m approaches infinity (݉→∞), the
compounding equation transforms into Equation 16, called continuous compounding.
Continuous compounding means for every fraction of a second; your balance earns interest.
Abbreviation, lim, refers to the limit and defines how the function behaves when m becomes
very large. Thus, the number e is a constant and equals approximately 2.1828. Number e is
similar to pi, and its digits do not repeat any patterns.
ܸܨ்=lim→ஶܸܲቀ 1 +ோ
ቁ
்∙
=ܸܲ∙݁ோ∙் ( 16 )
For example, you deposit $50 into your bank and leave it alone for 70 years. If the bank
uses continuous compounding, then your savings grow into $9,528.31, calculated in Equation
17 : Every fraction of a second over 70 years, you earn interest on your account. On the other
hand, if your bank uses monthly compounding, subsequently, your savings would grow into
$9,373.90, yielding $154.41 less than the standard compounding.
ܸܨ்=ܸܲ∙݁ோ∙்=$50∙݁.ହ∙=$9, 528. 31 ( 17 )Banks and financial institutions rarely use continuous compounding to calculate market
values of financial securities. Financial analysts and mathematicians use continuous
compounding to simplify complex calculations of financial formulas and mathematical models.
Annuities and Mortgages
Financial analysts use the present value formula to calculate an annuity. An annuity is an
investment for people who plan for retirement. An annuity has two parts. As people work, they
make periodic deposits into an annuity account. Subsequently, once they retire, they receive
periodic payments until death.
We define annuities as an ordinary annuity or an annuity due. If a person pays into an
annuity at the end of period, then it is an ordinary annuity. However, if a person pays into an
annuity at the beginning of the period, he or she receives one extra payment that earns interest
over the life of the annuity, called an annuity due. Consequently, they only differ when payment
is applied to the annuity account and when payments begin earning interest.
For this chapter, we stick to ordinary annuities. For example, you invest in an annuity that
earns 9% APR interest with annual compounding. Calculate the value of your annuity in five
years if you pay $20,000 into the annuity. We computed the annuity growth as $119,694.21 in
Equation 18.