1 The Power of Compounding

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Bernadette Morgan
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1 1 The Power of Compoudig 1.1 Simple vs Compoud Iterest You deposit $1,000 i a bak that pays 5% iterest each year. At the ed of the year you will have eared $50. The bak seds you a check for $50 dollars. For the ext two years you receive a check for $50. I total you have $1150. Lets say the bak istead deposited the $50 ito your accout. The ext year they are payig iterest o $1,050 istead of $1,000 this would give you 0.05 $1, 050 = $52.50 eared. Now beig deposited ito your accout, your ew balace is $1, For the ext year the iterest o $1, is 0.05 $1, = $ Addig this to your total from last year gives you $ Which is compoud ad which is simple iterest? Defiitio 1: The pricipal i fiacial formulas is the balace upo which iterest is paid. Defiitio 2: Simple iterest is iterest paid oly o the origial ivestmet ad ot o ay iterest added at latter dates. Defiitio 3: Compoud iterest is iterest paid both o the origial ivestmet ad o all iterest that has bee added to the origial ivestmet. 1. While baks almost always pay compoud iterest, bods usually pay simple iterest. Suppose you ivest $1,000 i a savigs bod that pays simple iterest of 10% per year. How much total iterest will you receive i 5 years? If the bod paid compoud iterest, would you receive more of less total iterest? Explai. Solutio. We have i simple iterest 0.1 = $100. Havig this doe over five years meas we will receive $500 of iterest. It is clear followig the logic of the example above that the compoud iterest would give you more moey, but how much more? We would have This is $ more tha with simple iterest ) 5 = Theorem 1.1 The Compoud Iterest Formula). 1. A = Accumulated Balace After Y Years. 3. AP R = Aual Percetage Rate as a decimal) 4. Y = Number of Years A = P 1 + AP R) Y Notes: The startig pricipal P, is ofter called the preset value P V ), because we usually begi a calculatio with the amout of moey i a accout at the preset. The accumulated balace A, is ofte called the future value F V ), because it is the amout that will be accumulated at some time i the future. Whe usig this formula, you must express the apr as a decimal rather tha as a percetage. 1. You ivest $100 i two accouts that each pays a iterest rate of 10% per year, but oe pays simple iterest ad the other pays compoud iterest. What is the differece over a three year period? How about 5? How about 100? August 10, korey@hawaii.edu

2 Solutio. Every year the accout that has simple iterest will accumulate $1000.1)=$10. Thus will have a total of $130 after three years, $150 after five years, ad $1100 after a hudred years. The compoud iterest formula said we will have A = $ ) 3 = $133.1 A = $ ) 5 = $ A = $ ) 3 $ = $1, 378, Compoud Iterest as Expoetial Growth The compoud iterest formula is of the form of expoetial growth. This is give by the geeral formula x t = x x 1 ) t Where x 1 > O July 18, 1461, Kig Edward IV of Eglad borrowed the moder equivalet of $384 form New College of Oxford. The kig soo paid back $160, but ever repaid the remaiig $224. The debt was forgotte for 535 years. Upo its rediscovery i 1996 a New College Admiistrator wrote to the Quee of Eglad Askig for repaymet, with iterest. If the iterest rate is 2%, calculate the amout due to New College usig a) Simple iterest. Solutio. At a 2% iterest rate we would have to pay 0.02$224)=$4.48. Now over 535 years, this would amout to 535$4.48)=$ Addig this to the origial debt of $224, we have a grad total of $ b) Compoud iterest. Solutio. Usig the compoud iterest formula we would have a grad total of A = $ ) 535 $ = $8, 940, Compoud Iterest Paid More Tha Oce a Year Lets say that we have a 8% iterest rate o a ivestmet of $1,000. How much moey would we have after 1 year. We would have $1,000 +$80=$1,080. What if we were give that the ivestmet is paid quarterly at a rate of 2%. This meas every three moths we would gai 2% of our ivestmet as such August 10, korey@hawaii.edu

3 After N quarters Iterest paid New balace 1st quarter 2% =$20 $1,000+$20=$1,020 2d quarter 2% $1, 020=$20.40 $1,020+$20.40=$1, rd quarter 2% $1, =$20.81 $1, $20.81=$1, th quarter 2% $1, =$21.22 $1, $21.22=$1, Notice this is larger tha the oe time paymet of iterest. What did we do? We first took the iterest ad Iterest rate broke it up ito parts, is the amout of times the iterest is payed i a year. That is. Similarly as the previous compoud iterest formula we did this how may times i oe year?. So we have a ew formula. Theorem 1.2 The Compoud Iterest Formula for Iterest Paid Times Per Year). 1. A = Accumulated Balace After Y Years. 3. AP R = Aual Percetage Rate as a decimal). 4. = Number of compoudig periods per year. 5. Y = Number of Years. A = P 1 + APR ) Y Note that Y is ot ecessarily a iteger, for example, a calculatios for six moths would have Y = You deposit $5,000 i a bak accout that pays a APR of 3% ad compouds iterests mothly. How much moey will you have after 5 years? Compare this amout to the amout you d have if iterest were paid oly oce a year. Solutio. P =$5,000, AP R=0.03, = for the moths i a year, ad Y =5. Pluggig ito the formula we get A = $5, ) 5) Where as i the compoud iterest formula, we would have This is about a $ differece. Exactly $11.71.) A = $5, ) Defiitio 4: The Aual percetage yield APY)- Also called the effective yield, relative icrease over oe year, or simply the yield- is the actual percetage by which a balace icreases i oe year. It is equal to the APR if iterest is compouded aually. It is greater tha the APR if the iterest is compouded more tha oce a year. 1. Say we have a $1,000 ivestmet at a 8% APR compouded quarterly. Fid the yield. August 10, korey@hawaii.edu

4 Solutio. We have already see that the total received will be $1, This is a icrease of $ The APY is give by APY = absolute icrease startig pricipal = $82.43 = = 8.243%. 2. Similarly as above, fid the APY usig a mothly compoud ad daily compoudig iterest. Solutio. First we calculate the balace of a mothly compoudig iterest. This is give by This gives us the formula A = ) 1) $ APY = $83 = = 8.3%. Now calculate the balace of a daily compoudig iterest. This is give by A = ) 3651) $ This gives us the formula APY = $83.28 = = 8.328%. 1.4 Cotiuous Compoudig Imagie we could compoud iterest ifiitely may times. This is give by the compoud iterest formula. Theorem 1.3 The Compoud Iterest Formula for Cotiuous Compoudig). 1. A = Accumulated Balace After Y Years. 3. AP R = Aual Percetage Rate as a decimal). 4. Y = Number of Years. A = P e APR Y The umber e is a special irratioal umber with the value of e You ca compute e to a power with the e x key o your calculator. We wo t prove this formula sice it requires calculus. 1. You deposit $100 i a accout with a APR of 8% ad cotiuous compoudig. How much will you have after 10 years? August 10, korey@hawaii.edu

5 Solutio. Pluggig ito the formula for Theorem 1.3 we get, A = P e APR = $100 e ) $ Suppose you put moey i a ivestmet with a iterest rate of APR=3%, ad leave it there for the ext 18 years. How much would you have to deposit ow to realize the $100,000 after 18 years, assumig that the iterest compouds aually, mothly, or cotiuously, Solutio. First lets calculate aually. This iterest is give by the formula, A = P 1 + AP R) Y = = P ) 18 = P = ) 18 = P $58, Next lets calculate mothly. This iterest is give by the formula, A = P 1 + APR = = P = P = ) ) = P $ ) Y ) 18) Fially lets calculate cotiuous. This iterest is give by the formula, A = P e APR Y = = P e = P = e = P $58, August 10, korey@hawaii.edu

1 Savings Plans and Investments

1 Savings Plans and Investments 4C Lesso Usig ad Uderstadig Mathematics 6 1 Savigs las ad Ivestmets 1.1 The Savigs la Formula Lets put a $100 ito a accout at the ed of the moth. At the ed of the moth for 5 more moths, you deposit $100