1 Savings Plans and Investments

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Trevor Jackson
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1 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 ito the accout. That accout accumulates % AR ad compouds mothly. How much moey would we have after 6 moths. Ed of... rior Balace Iterest o rior Balace Ed of Moth Deposit New Balace Moth 1 $0 $0 $100 $100 Moth 2 $100 1% $100 = $1 $100 $201 Moth 3 $201 1% $201 = $2.01 $100 $ Moth 4 $ % $ = $3.03 $100 $ Moth 5 $ % $ = $4.06 $100 $ Moth 6 $ % $ = $5.10 $100 $ Theorem 1.1 (Savigs la Formula, Regular aymets. This versio of the savigs pla formula assumes the same paymet, ad compoudig periods. For example, if paymets are made mothly, iterest also is calculated ad paid mothly. A R (y where 1. A=Accumulated savigs pla balace A = MT 2. MT = Regular paymet (deposit amouts 3. AR=Aual percetage rate (as a decimal 4. = Number of paymet periods per year 5. Y= Number of years ( A R As with compoud iterest, the accumulated balace (A is ofte called the future value (FV; the preset value is the startig pricipal (, which is 0 because we assume the accout has o balace before the paymets begi. 1. Use the savigs pla formula to calculate the balace after 6 moths for a AR of % ad mothly paymets of $100. Solutio. Usig the formula above,we have Same as above. A = MT = 100 = $ A R (y 0. ( A R ( At age 30, Michelle starts a IRA to save for retiremet by depositig $100 at the ed of each moth. If she ca cout o a AR of 6%, how much will she have whe she retires 35 years later at age 65? Compare the IRA s value to her total deposits over this time period. August 18, korey@hawaii.edu

2 4C Lesso Usig ad Uderstadig Mathematics 6 Solutio. Usig the same formula, we have A = MT = 100 A R (y 0.06 = $142, Over a 35 year period she would have deposited 35 year moth year ( A R ( $100 = $42, 000. moth The IRA has a balace of about $142,471-$42,000=$100,471 more tha she deposited. 1.2 laig Ahead with Savigs las 1. You wat to build a $100,000 college fud i 18 years by makig regular, ed of the moth deposits. Assumig a AR of 3%, calculate how much you should deposit mothly. How much of the fial value comes from actual deposits ad how much from iterest? Solutio. Similarly, we have A = MT = $100, 000 = MT = MT = $100, 000 [ (1 A R (y ( A R (18 1 (18 = MT = $100, 000 (18 = MT = $ Over a 18 year period you would have deposited 18 year moth year The iterest is about $100,000-$75,540=$24,460. $100 $75, 540. moth 2. You would like to retire 25 years from ow ad have a retiremet fud from which you ca draw a icome of $50,000 per year forever! How ca you do it? Assume a costat AR of 7%. Solutio. We wat to be able to withdraw $50,000 o iterest. That meas $50,000=0.07(total balace, August 18, korey@hawaii.edu

3 4C Lesso Usig ad Uderstadig Mathematics 6 hece total balace=$50, = $714, 286. Usig the formula we get A R (y A = MT = 714, 286 = MT = MT = 714, 286 [ 0.07 (1 (25 1 = MT = 714, 286 = $ ( A R 0.07 ( ( Total ad Aual Retur Cosider a ivestmet that grows from a origial pricipal to a later accumulated balace A. Defiitio 1.1: The total retur is the percetage chage i the ivestmet value: total retur = (A 100%. Defiitio 1.2: The aual retur is the aual percetage yield (AY that would give the same overall growth over Y years. The formula is aual retur = ( 1 A Y. This formula gives the aual retur as be a decimal; multiply by 100% to express it a as a percetage. 1. You ivest $3000 i the Clearwater mutual fud. Over 4 years, your ivestmet grows i value to $8400. What are your total ad aual returs for the 4 year period. Solutio. We have total retur = (A 100% $8400 $3000 = 100% $3, 000 = 180%, August 18, korey@hawaii.edu

4 4C Lesso Usig ad Uderstadig Mathematics 6 while ( A aual retur == = ( $8400 $3000 = = 29.4%. 1 Y 1 4 This says that over 4 years you will have icreased your ivestmet by 180%, or every year your ivestmet has grow about 29.4%. 2. You purchased shares i NewWeb.com for $2000;. Three years later, you sold them for $1100. What were your total retur ad aual retur o this ivestmet. Solutio. We have while total retur = (A 100% $1100 $2000 = 100% $2, 000 = 45%, ( A aual retur = = 1 Y ( $1100 $2000 = = 18%. 1 3 This says that over 3 years you will have lost moey by 45% of your iitial ivestmet, or every year your ivestmet has decreased by about 18%. 1.4 Types of Ivestmet Three Basic Types of Ivestmets Defiitio 1.3: stock gives you a share of owership i a compay. You ivest by purchasig shares of the stock, ad the oly way to get your moey out is to sell the stock. Because stock prices chage with time, the sale may give you wither a gai or a loss o your total ivestmet. Defiitio 1.4: A bod represets a promise of future cash. You usually buy bods issued by either govermet or corporatios. The issuer pays you simple iterest ad promises to pay back your iitial ivestmet plus iterest at some later date. Defiitio 1.5: Cash ivestmets iclude moey you deposit ito bak accouts, certificates of deposit (CD, ad U.S. Treasury Bills. Cash ivestmets geerally ear iterest. August 18, korey@hawaii.edu

5 4C Lesso Usig ad Uderstadig Mathematics 6 Ivestmet Cosideratios: Liquidity, Risk ad Retur Defiitio 1.6: Liquidity: How difficult is it to take out your moey? A ivestmet from which you ca withdraw moey easily, such as a ordiary bak accout, is said to be liquid. The liquidity of a ivestmet like real estate is much lower because real estate ca be difficult to sell. Defiitio 1.7: Risk: Is your ivestmet priciple at risk? The safest ivestmets are federally isured bak accouts ad U.S. Treasury bills. There is virtually o risk of losig the pricipal you ve ivested. Stocks ad bods are much riskier because they ca drop i value, i which case you may lose part or all of your pricipal. Defiitio 1.8: Retur: How much retur ca you expect o your ivestmet? A higher retur meas you ear more moey. I geeral, low risk ivestmets offer relatively low returs, while high risk ivestmets offer the prospects of higher returs, alog with the possibility of losig your pricipal. 1. Suppose your great great gradmother ivested $100 at the ed of 1900 i each of three fuds that tracked the averages of stocks, bods, ad cash, respectively. Assumig that her ivestmet grew at the rated give below, approximately how much would each ivestmet have bee worth at the ed of 20? Category Average Aual Retur Stocks 6.3% Bods 2.0% Cash 0.9% Solutio. Average aual retur is a compouded every year iterest, so usig that formula we have Stocks: A = (1 A R Y = $100 ( = $93, , Bods: A = (1 A R Y = $100 ( = $$918.80, Cash: A = (1 A R Y = $100 ( = $ Do t eed to do problems August 18, korey@hawaii.edu

1 The Power of Compounding

1 The Power of Compounding 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.