Mathematics of Finance. Learning objectives. Compound Interest

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1 Mathematics of Finance Section 3.2 Learning objectives Compound interest Continuous compound interest Growth and time Annual percentage yield (APY) Compound Interest Compound interest arises when interest is added to the principal, so that from that moment on, the interest that has been added also itself earns interest. This addition of interest to the principal is called compounding (i.e. the interest is compounded). A loan, for example, may have its interest compounded every month: in this case, a loan with $100 initial principal and 1% interest per month would have a balance of $101 at the end of the first month, $ at the end of the second month, and so on.

2 Compound Interest This is the compound interest formula: Compound Interest The notion of compound interest is central to understanding the mathematics of finance. The term itself merely implies that interest paid on loan or an investment is added to the principle. As a result, interest is earned on interest. Compounding is the arithmetic process of determining the final value of a cash flow or series of cash flow or series of cash flows when compound interest is applied. Compound Interest Year Principal Interest earned amount Cumulative amount 1 200,000 40,000 (20% of 200,000) 240, ,000 48,000 (20% of 240,000) 280, ,000 57,600(20% of 288,000) 345,600 etc.

3 Example of compound interest An amount of $ is deposited in a bank paying an annual interest rate of 4.3%, compounded quarterly. Find the balance after 6 years. A. Using the formula above, with P = 1500, r = 4.3/100 = 0.043, n = 4, and t = 6: So, the balance after 6 years is approximately $1, Compounding Periods Annually: once a period Quarterly: Four times a period Semiannually: Twice a period Monthly: Twelve times a period Compounding Periods If $1000 is invested at 8% compounded at (a) semiannually, (b) annually, and (c) monthly what is the future value after 5 years?

4 Continuous Compound Interest The formula for continuous compound interest is: A = Pe rt where, P = principal amount (initial investment) r = annual interest rate (as a decimal) t = number of years A = amount after time t NOTE: is sometimes called the shampoo formula (Pert). Continuous Compound Interest An amount of $2, is deposited in a bank paying an annual interest rate of 3.1%, compounded continuously. Find the balance after 3 years. Use the continuous compound interest formula, A = Pe rt,with P = 2340, r = 3.1/100 = 0.031, 031 t = 3. Recall that e stands for the Napier's number (base of the natural logarithm) which is approximately However, one does not have to plug this value in the formula, as the calculator has a built-in key for e. Therefore, So, the balance after 3 years is approximately $2, Continuous Compound Interest If you invest $1,000 at an annual interest rate of 5% compounded continuously, calculate the final amount you will have in the account after five years.

5 Present Value The current worth of a future sum of money or stream of cash flows given a specified rate of return. The basis is that receiving $1,000 now is worth more than $1,000 five years from now, because if you got the money now, you could invest it and receive an additional return over the five years. Present Value What is the present value of the following future payments: a) $1,210 in 2 years at a 10% discount rate: b) $1, in 3 years at an 8% discount rate: c) $1,040 in 1 year at a 4% discount rate: a) present value = $1,210/(1.10) 2 = $1000 b) present value = $1,259.71/(1.08) 3 = $1000 c) present value = $1,040/(1.04) 1 = $1000 Computing Growth Rate A $20,000 investment in a European mutual fund over a 10-year period would have grown to $250,000. What annual nominal rate would produce the growth if (a) interest were compounded annually? And (b) interest were compounded continuously?

6 Annual Percentage Yield (APY) The effective interest rate from the standpoint of a person receiving interest. For instance, compare two interest-bearing accounts: one pays 1 percent interest monthly and the other pays 3 percent interest quarterly. The account paying monthly interest will have a higher annual percentage yield than the account paying quarterly interest, because interest is credited to the account more frequently, which increases the amounts of funds in the account that interest is paid upon. Annual Percentage Yield (APY) If a principal is invested at the annual (nominal) rate r compounded m time a year, the APY is: APY = (1 + r/m) m 1 If a principal is invested at the annual (nominal) rate r compounded continuously, then the APY is: APY = r -1 Using APY to Compare Investments Bank Rate Compounding Glen Falls Bank 3.92% Monthly Community Bank 3.96% Daily Wells Fargo 3.97% Quarterly Bank North 3.94% Continuously Comparing the CD s in the table above find the APY s and express your answer in a percentage. Which bank has the highest yield?

7 Finding the Effective Rate Find the future value for an investment of $6,300 at 3.8% compounded monthly for 4 years. Using the graphing calculator. Find the Present Value How much money must you invest now a 4.6% compounded semi-annually in order to have $10,000 in 5 years? Find the Future Value Find the future value of an investment of $6300 at 3.8% compounded monthly for 4 years. $

8 Computing Growth Time How long will it take $5000 to grow to $7000 at 6% compounded quarterly? Present Value Compounded Monthly I will give you $1000 in 5 years on a loan. How much money should you give me now to make it fair to me? You say a good interest rate is 6%. What are we trying to find?

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