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

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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 yearsAmount in the account 0 1 2 n

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

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

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:

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.

Which investment is more attractive, one that pays 8.75% compounded quarterly or another that pays 8.7% compounded monthly? Comparing APYs = 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 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 years Now, confirm our answer graphically…

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 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…

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? 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 B Whiteboard Practice

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 Whiteboard Practice

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