Compound Interest What would you rather have: $1000 a year ago, - PDF document

Mt020.02 Slide 1 on4/10/00 Compound Interest What would you rather have: • $1000 a year ago, • $1000 today, or • $1000 next year? Probably you said $1000 a year ago, since you would have been able to invest it and so it would be worth more than $1000 today. We’ll be looking into how time affects the value of money, and measure it using exponential functions. When you deposit your money in a bank, you are, in effect, loaning your money to the bank. The bank uses your money and should pay you rent for the time during which it holds your raj

Mt020.02 Slide 2 on4/10/00 money. This “money rent”, the money paid to you by the bank for using your money, is called interest. The amount of interest paid to you depends on two things: how much money you loaned the bank in the first place (we call that amount P, the principal) and how long the bank has kept this money under its control (we call that time t, the term of the loan). Typically your interest is computed as a percent of the principal which will be paid to you per year, or nominal percentage rate (NPR). For example, a 6% NPR means that if your principal, P, is held for a year by the bank, the interest that the bank will pay for that privilege will be 6% of P. Obviously if that bank holds your money for 6 months, you should expect 3% of P in interest to be paid to you. raj

Mt020.02 Slide 3 on4/10/00 In this simple interest model, the “rent”, or interest is paid once, at the end of the loan (deposit). If the loan stretches on for t years, then the interest paid should be P(6%)(t), so a three year loan (deposit) should earn 18% in interest paid at the end. In general, the interest paid at the end of a simple interest loan extended at NPR = r over t years will be I = interest = Prt. The amount accumulated at the end of the term will be A, A = accumulation = principal + interest = P + Prt A = P(1+rt) <simple interest model> What characterizes simple interest is that one single interest payment is made at the end of the term. raj

Mt020.02 Slide 4 on4/10/00 Compound interest is what happens when the interest is paid periodically throughout the term of the loan (deposit), so that interest earns interest. Here’s an example: Example: Helen deposits $2000 into a bank paying 6% NPR but the bank compounds interest 4 times each year (quarterly). How much will Helen have after a year? Solution: Let A k represent how much Helen will have after k quarters (k = 0, 1, 2, 3, 4). The idea here is to view each of the three month quarters (called conversion periods for this loan) as smaller terms for simple interest loans. Each quarter will pay (6%)/4 = 1.5% interest on the balance at the start of the quarter. A 0 = 2000, A 1 = A 0 (1+1.5%) = 2000(1+.015) = 2000(1.015) raj

Mt020.02 Slide 5 on4/10/00 A 2 = A 1 (1+1.5%) = A 1 (1.015) = A 0 (1.015)(1.015) = A 0 (1.015) 2 = 2000(1.015) 2 and so on... A 3 = 2000(1.015) 3 and A 4 = 2000(1.015) 4 . After a year Helen will have A 4 = $ 2122.73 which you should compare to what she would have accumulated in a simple interest model A = $2000 (principal) + $120 (interest) = $ 2120. With compound interest, we have m conversion periods per year, and during each one we earn interest at a rate of r/m. Putting this all together we have • A = accumulated value, at the end • P = principal, the initial amount invested • r = the annual nominal interest rate (NPR) • t = the term (in years) for the loan (investment) • m = the number of conversion periods per year A = P( 1 + r/m ) (mt) raj

Mt020.02 Slide 6 on4/10/00 Example: Rosalind receives a gift of $10,000 on her 0-th birthday and her parents buy a twenty year bond which pays interest at 10% annually (nominal) but compounds interest monthly. How much will Rosalind have when she is 20? Solution: We have m =12 conversion periods per year, and with each period we expect to pick up r/m = (10%)/12 interest. The accumulated value after t=20 years will be $10,000( 1 + (0.10 12 ) ) (12)(20) = 10,000(1.00833) (12)(20) = $73,280.74 How much should Rosalind’s parents add to the original gift to make the accumulated value after 18 years be $120,000 for her college education? Solve this equation for A 0 : A 0 (1+(0.10 12 )) (12)(18) = 120,000 raj

Mt020.02 Slide 7 on4/10/00 And find that A 0 = $19,984.37, which says that Rosalind’s folks have to nearly double the gift, add $9,984 to achieve their goal. Having several options with various numbers of conversion periods per year proves to be confusing to the consumer who wants to compare situations. For example, what would be more advantageous, investing a) at 7.5% (NPR) compounded monthly (m=12) or b) at 7.75% (NPR) compounded quarterly (m=4)? Let’s check. Invest $1000 in each situation: 12   a) A = $1000 1 + 0.075 = $1,077.63 whereas   12 4   b) A=$1000 1 + 0.0775 = $1,079.78.   4 Clearly option b) is better. Note that option b) behaves as if the investment were a simple raj

Mt020.02 Slide 8 on4/10/00 interest investment over a year at 7.978%, where a) behaves as a simple interest problem over a year at 7.763%. We say these situations, a) and b) have effective interest rates of 7.978% and 7.763% respectively, since the result of the scenario is the same as if the same principal were invested at the effective rate over the same term. Effective rate (APR) is a useful way to compare loan deals or investment options.   m APR = r eff = effective rate = 1 + r − 1   m For this example, scenario b) computes to 4   r eff = 1 + 0.0775 − 1 = 1.07978 − 1 = 7.978%   4 raj