Kenneth R. Szulczyk
Present Value (PV) in dollars at Time 0. Interest rate (i) is the discount rate. Subscripts reflect the time with the final time period being T.We write the formula in Equation 4.ܸܨଵ=ܸܲ( 1 +݅)்=$100( 1 + 0. 05 )ଵ=$13, 150. 13 ( 4 )You want your money today because one hundred years is very far away. Present value of
$13,150.13 in one hundred years is worth $100 to today because you can take that $100 today,
invest it in a savings account at 5% interest, and let it grow into $13,150.13. If you receive a
payment in the future, then we compute the present value in Equation 5.
ܸܲ=
ி
(ଵା)=
,ଵହ.ଵଷ
(ଵା.ହ)భబబ=$100^ (^5 )^
We use algebra to solve for unknown variables. For example, you have $10,000 to invest
and want the final balance to grow into $15,000 in four years. We can calculate the minimum
interest you must earn to achieve this goal. We show all steps of algebra in Equation 6, and the
minimum interest rate equals 10.66% annually.
ܸܨ=்ܸܲ( 1 +݅)் ( 6 )
$15, 000 =$10, 000 ( 1 +݅)ସ
,
,=( 1 +݅)ସ
√^1.^5
ర =రඥ( 1 +݅)ସ
1. 1066 = 1 +݅
݅= 0. 1066
We have an easy formula to calculate how long something doubles in size, known as the
Rule of 72. Interest rate, i, as a percentage, and the time indicates the number of years.
Accordingly, the product of the interest rate and time equals 72 in Equation 7.
݅∙ݐ=ݐݓݎ݃ℎ∙݁݉݅ݐ= 72 ( 7 )For example, if your bank deposit earns 4% interest per year, how long does your deposit
double in size? Just divide 72 by 4, and your bank account doubles in 18 years. What would
happen if your interest rate climbed to 7% per year? Then your bank deposit would double in
10.3 years, or 72 ÷ 7.