Mathematics of Finance It’s all about the $$$ in Sec. 3.6a

Interest Compounded Annually Suppose a principal of P dollars is invested in an account bearing an interest rate r expressed in decimal form and calculated at the end of each year. The value of the investment then follows the growth pattern shown below: Time in years Amount in the account 1 2 n

Interest Compounded Annually Interest computed in this way is called compound interest, because interest is eventually earned on the interest itself !!! If a principal P is invested at a fixed annual interest rate r, calculated at the end of each year, then the value of the investment after n years is where r is expressed as a decimal.

Interest Compounded k Times per Year What happens when the interest rate r is compounded multiple times per year??? (say, “k” times per year…) Then r/k is the interest rate per compounding period, and kt is the number of compounding periods. The amount A in the account after t years is Now, what happens when k gets really, really, really, really, really, really, really, enormously, gigantically, really big???

Interest Compounded Continuously Recall that When k approaches infinity, we say that the interest is being compounded continuously. The amount A after t years in such a situation is

Guided Practice Suppose you invest $500 at 7% interest compounded annually. Find the value of your investment 10 years later. with P = 500, r = 0.07, and n = 10

Guided Practice Suppose you now invest $500 at 9% annual interest which is compounded monthly (12 times a year). What is the value of your investment 5 years later? with P = 500, r = 0.09, k = 12, and t = 5

Guided Practice Now you’re investing $100 at 8% annual interest compounded continuously. Find the value of your investment at the end of each of the years 1, 2,…, 7. with P = 100, r = 0.08, and t = 1,2,…,7 After 1 year: To find the other values, let’s use a table!!! Let

It will take 20 years, 8 months Guided Practice Determine how much time is required for an investment to quadruple in value if interest is earned at the rate of 6.75%, compounded monthly. It will take 20 years, 8 months

Annual Percentage Yield With so many methods for compounding interest, how do we compare different investment plans? For example, would you prefer an investment earning 8.75% annual interest compounded quarterly or one earning 8.7% compounded monthly? We use… Annual Percentage Yield (APY) – the percentage rate that, compounded annually, would yield the same return as the given interest rate with the given compounding period.

Computing APY Uma invests $2000 with Crab Key Bank at 5.15% annual interest compounded quarterly. What is the equivalent APY? Let x = the equivalent APY Then the value of the investment at the end of 1 year using this rate is Thus, equating the two investment values:

Computing APY Uma invests $2000 with Crab Key Bank at 5.15% annual interest compounded quarterly. What is the equivalent APY?

Computing APY Uma invests $2000 with Crab Key Bank at 5.15% annual interest compounded quarterly. What is the equivalent APY? In other words, Uma’s $2000 invested at 5.15% compounded quarterly for 1 year earns the same interest and yields the same value as $2000 invested elsewhere paying 5.250% interest once at the end of the year.

Comparing APYs Which investment is more attractive, one that pays 8.75% compounded quarterly or another that pays 8.7% compounded monthly? = the APY for the 8.75% rate = the APY for the 8.7% rate The 8.7% rate compounded monthly is more attractive b/c its APY is higher than that for the 8.75% rate compounded quarterly…

Whiteboard Practice Judy has $500 to invest at 9% annual interest rate compounded monthly. How long will it take for her investment to grow to $3000? with P = 500, r = 0.09, k = 12, and A = 3000

Whiteboard Practice Now, confirm our answer graphically… years Judy has $500 to invest at 9% annual interest rate compounded monthly. How long will it take for her investment to grow to $3000? with P = 500, r = 0.09, k = 12, and A = 3000 Now, confirm our answer graphically… years

Whiteboard Practice Stephen has $500 to invest. What annual interest rate, compounded quarterly (4 times per year) is required to double his money in 10 years? with P = 500, k = 4, t = 10, and A = 1000

Whiteboard Practice Now, confirm our answer graphically… Stephen has $500 to invest. What annual interest rate, compounded quarterly (4 times per year) is required to double his money in 10 years? Now, confirm our answer graphically…

Whiteboard Practice Plan A Joey has to choose between three investment options. Plan A has a 5% APR compounded monthly, Plan B gives a 4.7% APR compounded continuously, and Plan C provides a 5.1% APR compounded annually. How long does it take for each of the investment options to double Joey’s money? Plan A

Whiteboard Practice Plan B Joey has to choose between three investment options. Plan A has a 5% APR compounded monthly, Plan B gives a 4.7% APR compounded continuously, and Plan C provides a 5.1% APR compounded annually. How long does it take for each of the investment options to double Joey’s money? Plan B

Whiteboard Practice Plan C Joey has to choose between three investment options. Plan A has a 5% APR compounded monthly, Plan B gives a 4.7% APR compounded continuously, and Plan C provides a 5.1% APR compounded annually. How long does it take for each of the investment options to double Joey’s money? Plan C

Returning to the “Joey” Which of Joey’s three plans offers the better APY? Does your answer agree with the results from our initial calculations for doubling times? Plan A Plan B Doubling Time = 13.892 yr. APY = 5.116% Doubling Time = 14.748 yr. APY = 4.812% Plan C Clearly, Plan A is the best value!!! Doubling Time = 13.935 yr. APY = 5.1%