3.5 Exponential and Logarithmic Models Compound Interest
3.5 Exponential and Logarithmic Models – Compound Interest
- If a principal P dollars is borrowed for a period of t years at a per annum (yearly) interest rate of r, expressed as a decimal, the interest I charged is (Simple Interest Formula)
We typically use the term payment period as follows:Annually = once per year
Semiannually = twice per year
Quarterly = four times per year
Monthly = 12 times per year
Daily = 365 times per year
- The amount A after t years due to a principal P invested at an annual interest rate r compounded n times per year is .
Ex) A credit union pays interest of 8% per annum compounded quarterly on a certain savings plan. If $1000
is deposited in such a plan and the interest is left to accumulate, how much is in the account after 1 year?
We use the simple interest formula, . The principal P is $1000 and the rate of interest is 8% = 0.08. After the first quarter of a year, the time t is year, so the interest earned is
The new principal is . At the end of the second quarter, the interest on this new principal is
At the end of the third quarter, the interest on the new principal of is
Finally, after the fourth quarter, the interest is
After one year the account contains $1082.43
Using the compound interest formula we find we get the same answer (P=$1000, r=0.08, t=1, n=4)
- Continuous Compounding
The amount Aafter tyears due to a principal Pinvested at an annual interest rate of rcompounded continuously is
Ex) On January 2, 2002, $2000 is placed in and IRA that will pay interest of 10% per annum compounded
continuously. (a) What will the IRA be worth on January 1, 2022?
(b) What is the effective rate of interest?
(a)The amount A after 20 years is
(b)First, we compute the interest earned on $2000 at r = 10% compounded continuously for 1 year
The interest earned is . Use the simple interest formula,
, with I = $210.34, P = $2000, and t = 1, and solve for r , for the effective rate of interest.
The effective rate of interest is 10.517%
- Rate of interest required to double an investment
Ex) What annual rate of interest compounded annually should you seek if you want to double your
investmentin 5 years?
If P is the principal and we want P to double, the amount A will be 2P. We use the compound
interest formula with n = 1 and t = 5 to find r.
Substitute for known variables
Divide both sides by P
Take 5th root of both sides
The annual rate of interest needed to double the principal in 5 years is 14.87%
Ex) How long will it take for an investment to double in value if it earns 5% compounded continuously?
If P is the initial investment and we want P to double, the amount A will be 2P. We use the
continuous compounding formula with interest of r = 0.05.
continuous compounding formula
insert known variables
divide both sides by P
rewrite as a logarithm
ln(e) = 1 and power rule
solve for t
It will take about 14 years to double the investment.
(** Show how to solve using intersection of two graphs in calculator **)