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Wize University Calculus 1 Textbook > Applications of Differentiation for Business & Econ

Compound Interest

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Compound Interest

Interest is the cost of borrowing money. Typically interest is compounded (interest earned on top of interest). Interest payments on an investment or loan are usually paid out a certain number of times per year. We can use limits at infinity to compound interest perpetually.

Compound Interest

Given a principle (initial deposit) of PP dollars for a term of tt years compounded nn times per year at an annual interest rate of rr, the ending balance AA is
A(t)=P(1+rn)nt\boxed{A(t) = P \left (1 + \frac{r}{n} \right )^{nt} }

Continuously Compounded Interest

Instead of being compounded nn times a year, if interest is compounded continuously the ending balance AA is
A(t)=lim⁡n→∞P(1+rn)nt=Pert\boxed{A(t) = \lim_{n \rightarrow \infty} P \left (1 + \frac{r}{n} \right )^{nt} = Pe^{rt}}

Wize Concept
The above limit can be solved using L’Hˆopital’s Rule as an 1∞1^\infin indeterminate form.

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Example: Compound Interest

How many years would it take for your current savings of $11,000 to grow to $19,750, assuming your bank account pays an interest rate of 5% compounded semi-annually?

A=P(1+rn)ntA=P\left(1+\dfrac{r}{n}\right)^{nt}

P=11,000P=11,000

A=19,750A=19,750

r=0.05r=0.05

n=2n=2

Interest compounded semi-annually:

19,750=11000(1+0.052)2t19,750=11000\left(1+\frac{0.05}{2}\right)^{2t}

19.7511=(1.025)2t\dfrac{19.75}{11}=\left(1.025\right)^{2t}

Take the ln of both sides:

ln⁡(19.7511)=ln⁡(1.025)2t\ln\bigg(\dfrac{19.75}{11}\bigg)=\ln\left(1.025\right)^{2t}

ln⁡(19.7511)=2tln⁡(1.025)\ln\bigg(\dfrac{19.75}{11}\bigg)=2t\ln\left(1.025\right)

t=ln⁡(19.7511)2ln⁡(1.025)t=\dfrac{\ln\bigg(\dfrac{19.75}{11}\bigg)}{2\ln\left(1.025\right)}

≈11.85\approx 11.85 (years)

Given an initial investment of $7000 continuously compounded, what is the rate of annual interest required for this principle to double in 5 years?






Extra Practice
Which investment option provides the best return?
Option X: an interest rate of 8.75% compounded annually
Option Y: an interest rate of 8.50% compounded daily
The amount of money S(t) in an account with a 6% annual interest rate compounded continuously is given by:
S(t)=S0e0.06tS\left(t\right)=S_0e^{0.06t}
Where S0S_0 is the amount of money at time t=0 (the initial amount) and t is the time in years.
Q:\bf{Q:}You have $5,000 to invest and the investment option you're considering pays 6.5% compounded annually. How long will it take for your investment to double?
Q:\bf{}Q: Which investment option provides the best return?
Option X: an interest rate of 7.45% compounded annually
Option Y: an interest rate of 7.20% compounded daily
Aaron deposits $5,000 into his bank account that earns an annual effective interest rate of 3%. How much interest will Aaron earn over five years if he withdraws the interest as it's earned?
Q:\bf{Q:} You are offered the following two savings opportunities:
You can invest $750 today and receive $950 in 4 years.
You can invest $750 today in a GIC that guarantees 5.55% a year compounded annually.
Q:\bf{Q:} You have some money to invest and the investment option you're considering pays 3% compounded continuously. How long will it take for your investment to quadruple? (Round your answer to 2 decimal places)
Compound Interest
Find the annual interest rate such that a principle of 10,000 dollars will quadruple in 20 years given that it is continuously compounded.
Compound Interest
Given an initial investment of $7000 continuously compounded, what is the rate of annual interest required for this principle to double in 5 years?
Compound Interest
A very cautious individual stores his money both in the bank, and in a safe hidden underneath the floorboards below his bed. He keeps $5000 underneath his bed for a rainy day while investing the other $15000 into a savings account that grows 2% annually that is compounded continuously. Write an expression for the expected value of your total savings after 10 years.