1 - Finance Basics

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Interest Fundamentals

Time and Money

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Time and Money

Money today and money tomorrow are different.

$1000 today is better than $1000 next year.Not just because of inflation, but because of the value that can be created by using that money productively or the pleasure that can result from spending it now.

$2000 next year might be better than $1000 today It depends on my needs and opportunities.

What about $1500 or $1100 or $1010?

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Time and Money

If you have money: You can spend it.You can use it to help others.You can use it to increase your wealth.You can let someone else use it.

However you choose to use your money, it should produce enhanced value over time.

You need a method to evaluate the productivity of money – return on investment.

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Interest

Interest is the parameter we use to evaluate the productivity of money over time.Used to evaluate:

LoansFinancial investmentsBusiness investmentsEngineering design tradeoffsHealth, safety and environmental regulationsRoads and other public sector investments

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Interest

Characterized byThe principal – the initial sum of moneyThe interest period (typically one year)The duration of the transaction

Often expressed as the number of interest periods

Interest is the increase in value per interest period expressed as a percentage of the beginning principal.

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Simple Interest

Simple interest is calculated only on the amount of the principal.Consider a $1000 deposit:

One year at 10% simple interest:Interest = 0.1 x $1000 = $100Total = $1100

Four years at 10% simple interest:Interest = 4 x (0.1 x $1000) = $400Total = $1400

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Compound Interest

Interest is paid on the principal plus any accumulated, unpaid interest.

Consider a 4-year deposit at 10% interest compounded at the end of each year:

Year 0: Balance = $1000Year 1: Interest = $100 Balance = $1100Year 2: Interest = $110 Balance = $1210Year 3: Interest = $121 Balance = $1331Year 4: Interest = $133 Balance = $1464

The difference between $1464 and $1400 is the interest on the interest.

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Factors Affecting Compounding

The impact of compounding will be greater when:

The interest rate is higher.The compounding interval is more frequent.The duration is longer.

In real life all interest rates are compound, not simple.

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Note to Students

In this course all interest is compounded annually

unless otherwise indicated.

If simple interest is desired, it will be clearly stated.

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Equivalence: Definition

Two amounts of money or series of money flows at different points in time are equivalent if they are equal to each other at some point in time at a given interest rate.

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Equivalence

Equivalence: a logical individual would be indifferent between the two cases.

There is no economic reason to prefer one to the other.The interest rate used must properly reflect the value of money to that individual.

Different individuals have different rates.The interest rate is the required rate of return.

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Time and Money

Equivalence can be used to move sums of money to different points in time for comparison.If 10% interest correctly expresses our required rate of return, then

$1000 now is equivalent to$1464 in four years.

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Equivalence

If two cash flows are equivalent at any point in time, they are equivalent at all other times.

At 10% interest, the balances at the end of years 0 through 4 are equivalent.

Year 0: Balance = $1000Year 1: Interest = $100 Balance = $1100Year 2: Interest = $110 Balance = $1210Year 3: Interest = $121 Balance = $1331Year 4: Interest = $133 Balance = $1464

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Lotto Example

$100 million Texas LottoGet $4 million per year for 25 years, orGet $50 million now (approximately).

Which is better?$50 million now is more than $4 million now.$100 million total is more than $50 million total.

The Lotto is indifferent – they are precisely equivalent.

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Present Worth

We refer to the starting point in time for an analysis as the present. (Symbol: P)

Like t=0 in an engineering calculation, it may not actually be the current moment.

Present worth is the equivalent worth at the present of a series of future cash flows.

An appropriate interest rate must be specified.

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Future Worth

The future (F) is some specific, meaningful point of time after the present.Future Worth is the equivalent worth at that point in time of a series of cash flows.

An appropriate interest rate must be specified.The point in time must be specified.

At 10% interest, $1000 in the present is equivalent to $1464 four years in the future.

The bank is indifferent between these two cases.

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Principles of Equivalence

Financial comparisons must be made at a common point in time.

Comparisons may require the conversion of multiple cash flows to a single cash flow.

Equivalence depends on the interest rate.Equivalence is independent of point-of-view (borrower/lender).

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What interest rate?

The difficulty of identifying the correct interest rate for a specific analysis is the weakness of the method.

Different people or organizations can have different rates for the same transaction.Different transactions can involve different rates.

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What interest rate?

Opportunity cost:How much could I earn with this money on the next best opportunity?Generally the most logical choice.

Cost of capital:How much do I have to pay to obtain investment capital?A lower bound on the required rate of return.

Time preference:How much earned interest do I require to accept deferred gratification?

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What about inflation?

The time value of money includes an allowance for inflation, but is not solely due to inflation.

Even with no inflation, people would expect to be compensated for giving up the use of their money for a period of time.All normal interest rates include an allowance for anticipated future inflation.

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Cash Flow Diagrams

Two dimensional diagram of cash flows and time

Up arrows are positive cash flows (in).Down arrows are negative cash flows (out).Y-axis is time (in interest periods).

1 2 3 4 5 1 2 3 4 5 1 2 3 4 5

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End of Period Convention

All cash flows occur at year end unless otherwise specified.

It makes little sense to make a payment on a loan on the day the loan is initiated.Would just reduce the amount of money received.

By convention, the end of one year is identical to the beginning of the next.

The end of “year zero” is “the present”.It is actually the beginning of year one.

P

1 2 3 4 50

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Compound Interest Formulas

Future WorthPresent Worth

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Future Worth

The interest on a present sum (P) for one period is P x i.

The future worth of P after one period isF1 = P + P x i = (1+i) x P

The future worth after two periods isF2 = (1+i) x F1 = (1+i)2 x P

The future worth after n periods isFn = (1+i)n x P

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Present Worth

If Fn = (1+i)n x P, then

P = Fn x (1+i)-n

The process of moving a sum of money from the future to the present is called “discounting”.The interest rate is called the “discount rate”.

It is common to make comparisons of cash flow series based on their present worth.

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FW Example

If I deposit $10,000 in an account paying 12% interest compounded annually, what will it be worth at the end of the 10th year?

F10 = $10,000 (1.12)10

F10 = $10,000 (3.1058) = $31,058P

F

1 2 3 4 5

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FW Example

$0

$5,000

$10,000

$15,000

$20,000

$25,000

$30,000

$35,000

0 1 2 3 4 5 6 7 8 9 10

F10 = $10,000 (1.12)10 = $31,058

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PW Example

If I want to have $100,000 in 20 years, how much should I deposit today in an account paying 12% interest compounded annually?

P = $100k (1.12)-20 = $10,367P

F

1 2 3 4 5

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PW Example

$0

$10,000

$20,000

$30,000

$40,000

$50,000

$60,000

$70,000

$80,000

$90,000

$100,000

$110,000

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Years

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Irregular Flows Example

If I deposit $2000 in an account paying 12% now and again at the end of the second year, and withdraw $3000 after the 3rd year, what will be value of the account after 5 years?Let’s proceed through the sequence:

P = $2000F2 = $2000 (1.12)2 + $2000 = $2509 + $2000 = $4509F3 = $4509 (1.12)1 - $3000 = $5050 - $3000 = $2050F5 = $2050 (1.12)2 = $2571

1 2 3 4 5

F5

Cash flows must be moved to a common point in time before they can be combined or compared.

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If I deposit $2000 in an account paying 12% now and again at the end of the second year, and withdraw $3000 after the 3rd year, what will be value of the account after 5 years?Present worth approach:

P = $2000 + $2000 (1.12)-2 - $3000 (1.12)-3

P = $2000 + $1594 - $2135 = $1459F5 = $1459 (1.12)5 = $2,571

1 2 3 4 5

F5


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If I deposit $2000 in an account paying 12% now and again at the end of the second year, and withdraw $3000 after the 3rd year, what will be value of the account after 5 years?Future worth approach:

F5 = $2000 (1.12)5 + $2000 (1.12)3 - $3000 (1.12)2

F5 = $3,525 + 2,810 – 3,763 = $2,571

1 2 3 4 5

F5


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Summary: PW and FW

Present WorthP = Fn x (1+i)-n

Future WorthFn = (1+i)n x P


Compound Interest Formulas

Equal Payment Series

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Equal Payment Series

Financial transactions often involve a series of equal payments over a fixed period.

Examples: Auto loans, home mortgages, annuities

These periodic payments are denoted by the symbol “A”.We will derive formulas to relate A to the equivalent present and future amounts.

These will be covered in an order which facilitates the derivations, not the order of importance.

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End of Period Convention

A typical use of the equal payment series is for loan transactions. It makes no sense to receive the principal on a loan and then to simultaneously make the first payment. Therefore, the formulas for the equal payment factors all assume that the payments are received at the end of each period of time.

P

1 2 3 4 5

A

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Equal Payment Series: Future Worth

Find the future worth of a series of equal payments.

Typical example is a series of equal deposits into an investment account.Each payment is made at the end of the period (including the first one).

No payment made at the present.The final value is simultaneous with and includes the final payment of the series.

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Equal Payment Series: Future Worth

Derivation:F = A + A(1 + i) + A(1 + i)2 + … + A(1 + i)N-1

F(1 + i) = A(1 + i) + A(1 + i)2 + … + A(1 + i)N

F(1 + i) – F = -A + A(1 + i)N = A[(1 + i)N –1]i F = A [(1 + i)N - 1]

F = (A/i) [(1 + i)N - 1] F

1 2 3 4 5

A

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Compound Interest Factors

The PW and FW formulas are simple.Easy to remember and easy to apply.

The series formulas are slightly more complex and it is handy to have a shorthand notation for them.These are called interest factors.

Factors are tabulated in Appendix A.

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Equal Payment SeriesCompound Amount Factor

(F/A, i%, N) = F/A =[(1 + i)N - 1] / iThe quantity (F/A, i, N) is called the Equal Payment Series Compound Amount Factor.This is read:

Find F given A at i% interest for N periods.

The process of moving a sum of money into the future is often called “compounding”.To find F multiply this factor by A.

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Example Using (F/A, i%, N)

Invest $5000 annually for 40 years into a mutual fund that provides a return of 5%.

All returns are reinvested at the same 5% rate.The first investment occurs at the end of the first year.

F = $5000 (F/A, 5%, 40) Use the tables to find (F/A, 5%, 40) = 120.7998

F = $5000 (120.7998) = $604,000

F

1 2 3 4 5

A

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Equal Payment SeriesPresent Worth Factor

What present amount is equivalent to a series of equal payments in the future?

An example is the Texas Lotto cash value.

To derive the factor, discount the future value back to the present:

P = [A (F/A, i%, N)] (1+i)-N

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(P/A, i%, N) Derivation

( ) ( )( )

( )( )( )

( )NiAPAii

iAP

iiiAP

iNiAFAP

N

N

N

N

N

,,/1

11

1111

1,,/

×=⎥⎦

⎤⎢⎣

⎡

+−+

=

⎥⎦

⎤⎢⎣

⎡+

×⎥⎦

⎤⎢⎣

⎡ −+=

+××= −

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Example Using (P/A, i%, N)

You have won the $100 million Texas Lotto. What is the cash value at 6% interest?

The first $4 million payment is made now.The remaining 24 payments are made at the end of each year.

P = $4 + $4(P/A, 6%, 24) P = $4 + $4 (12.55) = $4 + $50.2P = $54.2 million

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(A/P, i%, N) Derivation

( ) ( )( )

( )( )

( )( )NiFAF

iiFA

NiPAPi

iiPA

NiAPNiAPAP

N

N

N

,,/11

Similarly,

,,/11

1

,,/1P A then ,,,/ If

=⎥⎦

⎤⎢⎣

⎡

−+=

=⎥⎦

⎤⎢⎣

⎡

−++

=

⎥⎦

⎤⎢⎣

⎡=×=

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(A/P, i%, N) Example

You have $20,000 in the bank earning 6% annual interest. You are starting college and want to withdraw a uniform amount of money each month for 4 years to pay your living costs. How much can you withdraw monthly?

A = $20,000 (A/P,6%/12, 4x12) A = $20,000 (0.0235)A = $470 per month for 48 months

Total = $470 x 48 = $22,560

P

1 2 3 4 5

A

The account will be empty after the 48th monthly withdrawal.

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(A/F, i%, N) Example

You have had a baby and want to save $100,000 for its college education. How much should you deposit on each birthday (starting with the 1st) at 6% for 18 years?A = $100,000 (A/F, 6%, 18)A = $100,000 (0.0324) = $3,240 F

1 2 3 4 5

A

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Capitalized Equivalent Method

Some projects generate benefits over an extremely long time.

Bridge, dam, university endowment

Capitalized Equivalent is an NPW tool for evaluating these long-lived projects.CE is the sum of money in the present that is equivalent to an infinite stream of uniform annual benefits at a specified interest rate.

See section 5.4.2 of the text.

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University Endowment Example

Suppose you want to establish a perpetual scholarship of $5,000 per year at UT.

Assume zero inflationUT can earn 5% per year on its investments.How much do you need to donate?

)(%55000$ Endowment×=

000,100$05.0

5000$==Endowment

$100,000 is the capitalized equivalent of $5000 per year at 5% interest.

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Capitalized Equivalent

CE is the present value of the infinite stream of uniform annual payments at the specified interest rate.Even though the total amount of money is infinite, the PW is finite.

iAiCE =)(

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Equal Payment Series:Present Worth Using CE

The PW of an Equal Payment Series can be expressed using the CE:

( )( )

( )

[ ] NN

N

N

iCECEiiAP

NiAPAii

iAP

−− +−=+−=

×=⎥⎦

⎤⎢⎣

⎡

+−+

=

)1()1(1

,,/1

11

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Equal Payment Series:Present Worth Using CE

P = CE [1 - (1+i)-N] = (A/i) [1 - (1+i)-N]

(1+i)-N CE A

∞…

CE

CE

A

…1 2 3 N

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Present Worth and CE

Present Worth vs. Payments

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49

Number Payments

Pres

ent W

orth

/CE

4 Percent Interest6 Percent Interest8 Percent Interest

Compound Interest

A Few More ThingsYou Need to Know.

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Beginning of Period Payments

What if payments are made at the beginning of periods instead of at the end?

This looks like a series that begins at t = -1.F = A (F/A, i%,5) will give the value at the beginning of year 5 (end of year 4).Apply the factor (F/P,i%,1) = (1 + i) to move the sum to the end of year 5. F

1 2 3 4 5

A0

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Present Worth Factor

P/F = (1+i)-n = (P/F, i, N)The quantity (P/F, i, N) is called the Present Worth or Discounting Factor.This is read:

Find P given F at i% interest for N periods.The process of moving a sum of money from the future to the present is often called “discounting”.i% is sometimes called the “discount rate”.

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Compound Amount Factor

Fn/P = (1+i)n = (F/P, i, N)The quantity (F/P, i, N) is called the Compound Amount Factor.This is read:

Find F given P at i% interest for N periods.

The process of moving a sum of money from the present into the future is often called “compounding”.

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Excel Functions − PW

PV: The present value of a future sum or a series of uniform periodic payments.

P = PV(i, N, 0, -F) for a future sumP = PV(i, N, -A) for a uniform series

i is expressed as a fraction or using “%” notationi is the interest rate per periodThere is an option to use the beginning of period convention.

NPV: The present value of a series of equally spaced, non-uniform future cash flows.

P = NPV(i, value1, value2, …)Value1 is the cash flow at the end of year 1, not year 0.

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Excel Functions − FW

FV: The future value of a present sum or a series of uniform periodic payments.

F = FV(i, N, 0, -P) for a future sumF = FV(i, N, -A) for a uniform series

i is expressed as a fraction or using “%” notationi is the interest rate per periodThere is an option to use the beginning of period convention.

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Solving for N and i

Each compound interest factor involves 4 parameters. If you know any three you can solve for the fourth:

F = (1+i)N x Pi = (F/P)1/N – 1N = log(F/P) / log(1+i)

Factors involving A are more complex.Best to use Excel or a financial calculator.

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Other Excel Functions

Use the PMT function to find A:A = PMT (i, N, -P)A = PMT(i, N, 0, -F)

Use the RATE function to find i:i = RATE (N,0,P,-F)i = RATE (N,A,-P)i = RATE (N,A,0,-F)

Use the NPER function to find N:N = NPER (i,0,P,-F)

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Chain Rule

Compound interest factors are just numbers. They can be manipulated algebraically or chained together.For example:

P = F(P/F, i%, N) = F(A/F,i%,N)(P/A,i%,N)

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Cash Flows with Subpatterns

Some non-uniform cash flow sequences contain patterns that can be exploited to simplify solution.Example:

At 15%Find P

$100

1 2 3 4 5

$5096 7 8

$150$200

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Brute Force Solution:$ 50(P/F,15%,1) = $43.48$100(P/F,15%,2) = $75.61$100(P/F,15%,3) = $65.75$100(P/F,15%,4) = $57.18$150(P/F,15%,5) = $74.58$150(P/F,15%,6) = $64.85$150(P/F,15%,7) = $56.39$150(P/F,15%,8) = $49.04$200(P/F,15%,9) = $57.78Total = $543.72

$100

1 2 3 4 5

$5096 7 8

$150$200

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Vertical Groups:P = $50(P/A,15%,9) + $50(P/A,15,8)(P/F,15,1)+ $50(P/A,15,5)(P/F,15,4) + $50(P/F,15,9)P = 50(4.7716) + 50(4.4873)(0.8696)+ 50(3.3522)(0.5718) + 50(0.2843) = $543.72

$100

1 2 3 4 5

$5096 7 8

$150$200

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Horizontal Groups:P = $50(P/F,15%,1) + 100(P/A,15,3)(P/F,15,1)+$150(P/A,15,4)(P/F,15,4) + 200(P/F,15,9)

$100

1 2 3 4 5

$5096 7 8

$150$200

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Scaling

Interest factors are linear in the amount.

A = $ 1,000 (A/P, 10%, 5) = $263.80A = $10,000 (A/P, 10%, 5) = $2638

Interest factors are geometric in time.(F/P,9%,5) = 1.5386(F/P,9%,10) = 2.3674 = (1.5386)2

(F/P,9%,15) = 3.6425 = (1.5386)3

(F/P,9%,20) = 5.6044 = (1.5386)4

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Rule of 72

The time required for an investment to doubleis approximately (72 / i%).

72/2 = 36 (F/P,2%,36) = 2.039972/4 = 18 (F/P,4%,18) = 2.025872/6 = 12 (F/P,6%,12) = 2.012272/8 = 9 (F/P,8%, 9) = 1.999072/10 = 7.2 (F/P,10%,7.2) = 1.986272/12 = 6 (F/P,12%,6) = 1.973872/15 = 4.8 (F/P,15%,4.8) = 1.955972/18 = 4 (F/P,18%,4) = 1.938872/24 = 3 (F/P,24%,3) = 1.9066

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Gradient Series

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Gradient Series

Two series have been defined for cash flows which increase or decrease over time:

Linear Gradient SeriesGeometric Gradient Series

The corresponding interest factors can sometimes be useful.In general, I recommend using a spreadsheet for non-uniform series.

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Linear Gradient Series

Cash flows which increase (or decrease) by a fixed dollar amount (G) each period.

G

GG

G

0 1 2 3 4 5

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Separate into two series – one a uniform series and the other containing only the increases.

G

GG

G

0 1 2 3 4 5

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The remaining gradient is called a strict gradient series.

Note that there is no cash flow at the end of the first period.

G

GG

G

0 1 2 3 4 5

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The remaining gradient is called a strict gradient series.

There is no cash flow at the end of the first period.

The uniform series must be added to this.

G

0 1 2 3 4 5

2G3G

4G

( )( ) ⎥

⎦

⎤⎢⎣

⎡

+−−+

= N

N

iiiNiGP

111

2

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Geometric Gradient Series

Cash flows which increase (or decrease) by constant percentage (g) each period.

A1(1+g)N-1

A1

0 1 2 3 4 N

A1(1+g)A1(1+g)2

A1(1+g)3

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Geometric Gradient Series

A1(1+g)N-1

A1

0 1 2 3 4 N

A1(1+g)A1(1+g)2

A1(1+g)3

( ) ( )

( ) gii

NAP

gigi

igAPNN

=⎥⎦

⎤⎢⎣

⎡+

=

≠⎥⎦

⎤⎢⎣

⎡

−++−

=−

if1

if111

1

1

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Summary

Interest is a measure of our time preference for money or the “price” of money.To compare sums of money at different points in time, we must consider the time value of money.

Equivalence allows us to do this.

Compound interest factors allow conversion between sums in the present and future and between sums and series of payments.

Questions?

Interest Fundamentals

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In Class Exercise:Compound Interest

You have deposited $10,000 at 12% interest for 5 years, what is the withdrawal you can make if

a. Interest compounds annually and you make a single withdrawal at the end of 5 years?

b. Interest compounds annually and you make uniform withdrawals at the end of each year?

c. Interest compounds monthly and you make withdrawals at the end of each month?

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In Class Exercise:Compound Interest #1a

a. Interest compounds annually and you make a single withdrawal at the end of 5 years?

623,17$)7623.1(000,10$)5%,12,/(000,10$

=== PFF

P

F

1 2 3 4 5

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In Class Exercise:Compound Interest #1b

b. Interest compounds annually and you make withdrawals at the end of each year?

P

1 2 3 4 5

2774$)2774.0(000,10$)5%,12,/(000,10$

=== PAA

© R B McCann - 82

In Class Exercise:Compound Interest #1c

c. Interest compounds monthly and you make withdrawals at the end of each month?

P

1 2 3 4 5

222$)0222.0(000,10$)125,12/%12,/(000,10$

==×= PAA

R B McCann 42 1/5/2008

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