Lecture Notes on Simple Interest,Compound Interest, and Future

ValuesBy Dr. Mohammad Al-Khaleel

Yarmouk University - Jordan

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Outcomes• Understanding what is meant by "the time value

of money".• Understanding the relation between present and

future values.• Calculating the simple and compound interests

and the corresponding future and present valuesof an amount of money invested today.

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Outline• Time value of money and interest• The simple interest; Present and Future values• The compound interest; Present and Future values• Compounding more than once a year

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Time Value of Money and Interest• Which would you prefer$10000 Today or$10000

in 5 years?Obviously,$10000 today; you already recognizethat there is time value of money.

• If you put some money on a bank account for ayear, then the bank can do whatever it wants withthat money for a year; RIGHT? To reward you forthat, the bank pays you some interest.The more years the bank works with your money,the more rewards you would expect to get as anaccumulated interest!

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Why TimeWhy is time such an important element in yourdecision?

Time allows you the opportunity to postpone consump-

tion and earn compensation for lending your money as

an interest.

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Types of Interest• Simple Interest(SI); which is interest earned on

only the original amount, called Principal, lentover a period of time at a certain rate.

• Compound Interest(CI); which is interest earnedon any previous interests earned as well as on thePrincipal lent.

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Simple Interest FormulaThe Simple interest is given by

SI = P0 · i · n,

whereP0 : Present value today (depositedt = 0),i : interest rate per period of time,

n : number of time periods

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Simple Interest ExampleAssume that you deposit$1000 in an account paying7% annual simple interest for 2 years. What is theaccumulated interest at the end of the second year?

Solution:SI = P0 · i · n = 1000 ·7

100 · 2 = $140.

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Simple Interest and Future ValueFuture Value (FV) is the value at some future time ofa present amount of money evaluated at a giveninterest rate.What is the future value of the deposit?FVn = P0 + SI = P0(1 + n · i).For our example above,FVn = 1000 + 140 = $1140.

Note that there are 4 variables in the formula above.

Therefore, having any three of them, could be used to

find the fourth one.

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Simple Interest and Present ValuePresent Value (PV) is the current value of a futureamount of money evaluated at a given interest rate.What is the present value of the previous problem?

The present value is simply the$1000 you originally

deposited. That is the value tody.

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Why Compound Interest

0 5 10 15 20 25 300

2000

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year

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ture

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10% Compound interest7% Compound interest10% Simple interest7% Simple interest

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Why Compound Interest

2 4 6 8 10 120

500

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10% Compound interest7% Compound interest10% Simple interest7% Simple interest

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Future Value FormulaAssume you deposit$1000 at an annual compoundinterest rate of7% for 2 years.First Year: FV1 = P0(1 + 1 · i) = P0(1 + i)1 =1000(1.07) = $1070You earned$70 interest on your$1000 deposit overthe first year.This is the same amount of interest you would earnunder the simple interest.Second Year: FV2 = P0(1 + i)(1 + i) = P0(1 + i)2 =1000(1.07)2 = $1144.90

You earned extra$4.90 in year two with compound in-

terest over simple interest.

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General Future Value FormulaFV1 = P0(1 + i)1

FV2 = P0(1 + i)2

...FVn = P0(1 + i)n

where again,P0 : Present value today (depositedt = 0),i : interest rate per period of time,n : number of compounding periods.As we have seen in the Figure, the Future value hereis growing exponentially.

Note also that we have again 4 variables as forSI.

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ExampleA person wants to know how large his deposit of$10000 today will become at a compound annualinterest rate of10% for 5 years.Solution: Using the formula:

FV5 = 10000(1 + 0.1)5 = $16105.10.

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Double Your MoneyHow long does it take to double$5000 at a compoundrate of12% per year?Solution: Using the formula,FVn = P0(1 + i)n:10000 = 5000(1 + 0.12)n, i.e.,2P0 = P0(1 + 0.12)n.Hence,

2 = (1.12)n⇔ ln(n) = n ln(1.12)

⇔ n = ln(2)ln(1.12)

⇔ n = 6.1163 ≈ 6.12,

where the natural logarithmic function is used to solvethe equation;

ln ≡ log .

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It does not matter how much money you have at thestart.

Therefore,n = ln(2)ln(i+1) ≈

ln(2)i

, for small i, where we

use the first term of the Taylor expansion ofln(i + 1)

abouti = 0 as an approximation forln(i+1) for small

i.

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Frequency of CompoundingGeneral Formula:FVn = P0(1 + i)n

≡ P0(1 + r

m)mt,

whereFVn : Future Value,P0 : Principal, Present valuetoday (depositedt = 0),r : annual interest rate,m : number of compounding periods per year.t : time; in years,i : interest rate per period of time,n : total number of compounding periods.

Note thati := r

m, andn = mt, so, if interest is com-

pounded annually, theni = r andn = t.

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Impact of FrequencyA person has$1000 to invest for 2 years at an annualcompound interest rate of12%:annual: FV2 = 1000(1 + 0.12

1 )(1)(2) = $1254.40

semi-annual: FV2 = 1000(1 + 0.122 )(2)(2) = $1262.48

quarterly: FV2 = 1000(1 + 0.124 )(4)(2) = $1266.77

monthly: FV2 = 1000(1 + 0.1212 )(12)(2) = $1269.73

daily: FV2 = 1000(1 + 0.12365 )(365)(2) = $1271.20

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Present ValueAssume that you need$1000 in 2 years.Let us find how much you need to deposit today at arate of7% compounded annually.FVn = P0(1 + i)n, which implies,PV = P0 = FVn

(1+i)n.

Thus,PV = 1000(1.07)2 = 873.4387 · · · ≈ $873.44.

As a general Present Value Formula:

PV = FVn

(1+i)n.

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ExampleA person wants to know how large of a deposit tomake so that the money will grow to$10000 in 5years at a rate of10% compounded annually.solution:

PV = 10000(1+0.1)5 = 6209.21323 · · · ≈ $6209.21.

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Comparing SI and CI

• Suppose that you put your moneym years in oneaccount and thenn years in another account, andthat both accounts pay

(a) Compound interest at a ratei.(b) Simple Interest at ratei.

ForCI in (a), we haveFVm = P0(1 + i)m, andthen you withdraw the money and put it inanother account forn years and getFVn = P0(1 + i)m(1 + i)n = P0(1 + i)m+n.This is the same as what you would get if you hadkept the Principal in the same account form + nyears.

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Now, for SI in (b), we haveFVm = P0 + ISm = P0 + P0 · i · m.If you withdraw the money and put it in anotheraccount forn years you get

FVn = (P0 + P0 · i · m) + SIn

= (P0 + P0 · i · m) + (P0 + P0 · i · m) · i · n

= P0 + P0 · i · m + P0 · i · n + P0 · i2· m · n.

Whereas, you would get if you kept the Principal inthe same account form + n yearsFVm+n = P0+P0 ·i·(m+n) = P0+P0 ·i·m+P0 ·i·n.

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You could increase the interest you earn bywithdrawing your money halfway and open a newaccount with the same simple interest ratei.

This inconsistency means that simple interest is not

that often used in practice.

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• Simple interest: Principal aftern years growsLINEARLY; P0(1 + i · n).Compound interest: Principal aftern years growsEXPONENTIALLY; P0(1 + i)n.

• There is no difference between Simple interestand Compound interest in1 year; both lead toP0(1 + i).

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