Time Value of MoneyTime Value of MoneyMoney Value of Time???Money Value of Time???

Interest RatesInterest Rates

Why interest rates are positive?Why interest rates are positive?

– People have ‘positive time preference’People have ‘positive time preference’ Behavior of human beingsBehavior of human beings

– Current resources have productive usesCurrent resources have productive uses Technology and natural processTechnology and natural process

Simple vs. Compound InterestSimple vs. Compound Interest Simple InterestSimple Interest

– No interest is earned on interest money paid in the No interest is earned on interest money paid in the previous periodsprevious periods

– Money grows at a slower rateMoney grows at a slower rate

Compound InterestCompound Interest– Interest is earned on interest money paid in the Interest is earned on interest money paid in the

previous periodsprevious periods– Money grows at a faster rateMoney grows at a faster rate

Simple Interest ExampleSimple Interest Example $100 at 8% simple annual interest for 2 $100 at 8% simple annual interest for 2

yearsyears– First year interestFirst year interest

100 x (.08) = $8100 x (.08) = $8 Total = 100 + 8 = $___Total = 100 + 8 = $___

– Second year interestSecond year interest 100 x (.08) = $8100 x (.08) = $8 Total = 100 + 8 + 8 = $___Total = 100 + 8 + 8 = $___

– Total Interest after 2 years:Total Interest after 2 years:8 + 8 = $__8 + 8 = $__

Another exampleAnother example You deposit $5000 into a savings account You deposit $5000 into a savings account

that earns 13% simple annual interest. that earns 13% simple annual interest. What is the amount in the account after 6 What is the amount in the account after 6 years?years?

Answer:_________Answer:_________

What is the total amount of interest earned?What is the total amount of interest earned?

Answer:_________Answer:_________

Compound Interest ExampleCompound Interest Example

Invest $100 at 8% compounded annually Invest $100 at 8% compounded annually for 2 yearsfor 2 years

– Total after first year:Total after first year: 100 x (1 + .08) = $108100 x (1 + .08) = $108

– Total after second yearTotal after second year 108 x (1 + .08) = $_____108 x (1 + .08) = $_____

– Total Interest = 116.64 - 100 = $______Total Interest = 116.64 - 100 = $______

Compound Interest ExampleCompound Interest Example

YearYearBegin. AmountBegin. Amount Interest Earned Interest Earned Ending Amount Ending Amount

1 1 $100.00$100.00 $10.00$10.00 $110.00$110.00

2 2 110.00110.00 11.0011.00 121.00121.00

3 3 121.00121.00 12.1012.10 133.10133.10

4 4 133.10133.10 13.3113.31 146.41146.41

5 5 146.41146.41 14.6414.64 161.05161.05

Total interest $61.05Total interest $61.05

[What would be the total interest earned in simple [What would be the total interest earned in simple interest case? Ans: $_______ ]interest case? Ans: $_______ ]

Notice thatNotice that

– 1.1. $110 $110 = = $100 (1 + .10)$100 (1 + .10)

– 2.2. $121 $121 = = $110 (1 + .10) = $100 * 1.1 * 1.1 = $100 * $110 (1 + .10) = $100 * 1.1 * 1.1 = $100 * 1.11.122

– 3.3. $133.10 = $133.10 = $121 (1 + .10) = $100 * 1.1 * 1.1 * 1.1$121 (1 + .10) = $100 * 1.1 * 1.1 * 1.1

== $100 ________$100 ________

In general, the future value, FVIn general, the future value, FVtt, of $1 invested today at r, of $1 invested today at r% for t periods is% for t periods is

FVFVtt = $1 * (1 + r)= $1 * (1 + r)tt

The expression (1 + r)The expression (1 + r)t t is called the is called the future value factor.future value factor.

Future Value for a Lump SumFuture Value for a Lump Sum

FV on CalculatorFV on Calculator What is the FV of $5000 invested at 12% What is the FV of $5000 invested at 12%

per year for 4 years compounded annually?per year for 4 years compounded annually?

Clear all memory: CLEAR ALLClear all memory: CLEAR ALL Ensure # compounding periods is 1: 1 Ensure # compounding periods is 1: 1 Enter amount invested today: -5000 Enter amount invested today: -5000 Enter # of years: 4Enter # of years: 4 Enter interest rate: 12 Enter interest rate: 12 Find Future Value:Find Future Value:

Answer: $___________Answer: $___________

PV

N

I/YR

FV

P/YR

Notice..Notice.. You entered $5000 as a You entered $5000 as a negativenegative amount amount You got FV answer as a You got FV answer as a positivepositive amount amount Why the negative sign?Why the negative sign?

It turns out that the calculator follows ‘cash It turns out that the calculator follows ‘cash flow convention’flow convention’– Cash Cash outflowoutflow is negative (i.e. money going out) is negative (i.e. money going out)– Cash Cash inflowinflow is positive (i.e. money coming in) is positive (i.e. money coming in)

Another exampleAnother example

Calculate the future value of $500 invested Calculate the future value of $500 invested today at 9% per year for 35 yearstoday at 9% per year for 35 years

Answer: ________Answer: ________

Present ValuesPresent Values Here you simply reverse the questionHere you simply reverse the question

You are givenYou are given– Future ValueFuture Value– Number of PeriodsNumber of Periods– Interest RateInterest Rate

and need to find the sum (PRESENT and need to find the sum (PRESENT VALUE) needed today to achieve that FVVALUE) needed today to achieve that FV

Q. Suppose you need $20,000 in three years to Q. Suppose you need $20,000 in three years to pay tuition at SU. If you can earn 8% on your pay tuition at SU. If you can earn 8% on your money, how much do you need today?money, how much do you need today?

A.A. Here we know the future value is $20,000, the rate Here we know the future value is $20,000, the rate (8%), and the number of periods (3). What is the (8%), and the number of periods (3). What is the unknown present amount (called the unknown present amount (called the present valuepresent value)? )?

From before:From before: FVFVtt = PV x (1 + r)= PV x (1 + r)tt

$20,000$20,000 = = PV __________PV __________

Rearranging:Rearranging:

PVPV = $20,000/(1.08)= $20,000/(1.08)33

= = $_____________$_____________

Present Value for a Lump SumPresent Value for a Lump Sum

In general, the present value, PV, of a $1 to In general, the present value, PV, of a $1 to be received in t periods when the rate is r isbe received in t periods when the rate is r is

PV = PV = FVFVtt

(1+r)(1+r)tt

Present Value FactorPresent Value Factor = = 11 (1+r)(1+r)tt

‘‘r’ is also called the r’ is also called the discount ratediscount rate

PV on CalculatorPV on Calculator Your friend promises to pay you $5,000 Your friend promises to pay you $5,000

after 3 years. How much are you willing to after 3 years. How much are you willing to pay her today? You can earn 8% pay her today? You can earn 8% compounded annually elsewhere.compounded annually elsewhere.

Clear all memory: CLEAR ALLClear all memory: CLEAR ALL Ensure # compounding periods is 1: 1 Ensure # compounding periods is 1: 1 Enter amount future value: 5000 Enter amount future value: 5000 Enter # of years: 3Enter # of years: 3 Enter interest rate: 8 Enter interest rate: 8 Find Present Value:Find Present Value:

Answer: $___________Answer: $___________

FVN

I/YR

PV

P/YR

Another PV exampleAnother PV example Vincent van Gogh painted Vincent van Gogh painted Portrait of Dr. Portrait of Dr.

GachetGachet in 1889. It sold in 1987 for $82.5 in 1889. It sold in 1987 for $82.5 million. How much should he have sold it million. How much should he have sold it in 1889 if annual interest rate over the in 1889 if annual interest rate over the period was 9%?period was 9%?

Answer: _____________Answer: _____________

Vincent Van Gogh

The Portrait of Dr Gachet

Present Value of $1 for Different Periods and Rates Present Value of $1 for Different Periods and Rates

PresentPresentvaluevalueof $1 ($)of $1 ($)

Time(years)

r = 0%

r = 15%

1 2 3 4 5 6 7 8 9 10

1.00

.90

.80

.70

.60

.50

.40

.30

.20

.10

r = 10%

r = 20%

r = 5%

Notice...Notice...

As time As time increasesincreases, present value , present value declinesdeclines

As interest rate As interest rate increasesincreases, present value , present value declinesdeclines

The rate of decline is The rate of decline is notnot a straight line! a straight line!

Notice Four ComponentsNotice Four Components

Present Value (PV)Present Value (PV) Future Value at time t (FVFuture Value at time t (FVtt))

Interest rate Interest rate per periodper period (r) (r) Number of periods (t)Number of periods (t)

Given any three, the fourth can be foundGiven any three, the fourth can be found

Finding ‘r’Finding ‘r’

You need $8,000 after four years. You have You need $8,000 after four years. You have $7,000 today. What annual interest rate must $7,000 today. What annual interest rate must you earn to have that sum in the future?you earn to have that sum in the future?

Answer: __________Answer: __________

Finding ‘t’Finding ‘t’

How many years does it take to double your How many years does it take to double your $100,000 inheritance if you can invest the $100,000 inheritance if you can invest the money earning 11% compounded annually?money earning 11% compounded annually?

Answer: __________Answer: __________

Note:Note:

When calculating When calculating future valuefuture value what you are what you are doing is doing is compoundingcompounding a suma sum

When calculating When calculating present valuepresent value, what you are , what you are doing is doing is discountingdiscounting a suma sum

FV - Multiple Cash FlowsFV - Multiple Cash Flows You depositYou deposit

$100 in one year$100 in one year$200 in two years$200 in two years$300 in three years$300 in three years

How much will you have in three years? How much will you have in three years? r = 7% per year.r = 7% per year.

Answer: ____________Answer: ____________ Draw a time line!!!Draw a time line!!!

PV - Multiple Cash FlowsPV - Multiple Cash Flows An investment pays: An investment pays:

$200 in year 1$200 in year 1 $600 in year 3 $600 in year 3 $400 in year 2 $400 in year 2 $800 in year 4$800 in year 4You can earn 12% per year on similar You can earn 12% per year on similar investments. What is the most you are investments. What is the most you are willing to pay now for this investment?willing to pay now for this investment?

Answer: __________Answer: __________ Draw time line!!!Draw time line!!!

Important…Important… You can add cash flows ONLY if they are You can add cash flows ONLY if they are

brought back (or taken forward) to the brought back (or taken forward) to the SAME point in timeSAME point in time

Adding cash flows occurring at different Adding cash flows occurring at different points in time is like adding apples and points in time is like adding apples and oranges!oranges!

Level Multiple Cash FlowsLevel Multiple Cash Flows Examples of constant level cash flows for Examples of constant level cash flows for

more than one periodmore than one period– AnnuitiesAnnuities– PerpetuitiesPerpetuities

Most of the time we assume that the cash Most of the time we assume that the cash flow occurs at the END of the periodflow occurs at the END of the period

Examples of AnnuitiesExamples of Annuities

Car loan paymentsCar loan payments Mortgage on a houseMortgage on a house Most other consumer loansMost other consumer loans Contributions to a retirement planContributions to a retirement plan Retirement payments from a pension planRetirement payments from a pension plan

Saving a Fixed SumSaving a Fixed Sum You save $450 in a retirement fund every You save $450 in a retirement fund every

month for the next 30 years. The interest month for the next 30 years. The interest rate earned is 10%. What is the rate earned is 10%. What is the accumulated balance at the end of 30 years?accumulated balance at the end of 30 years?

This is This is Future Value of an AnnuityFuture Value of an Annuity

Future Value CalculatedFuture Value Calculated

Future value

calculated by

compounding each

cash flow separately

Time(years)

0 1 2 3 4 5

$2,000 $2,000 $2,000 $2,000 $2,000.0

2,200.0

2,420.0

2,662.0

2.928.2

$12,210.20

x 1.14

x 1.13

x 1.12

x 1.1

Total future value

Save $2,000 every year for 5 years into an account that pays 10%. What is the accumulated balance after 5 years?

FV of AnnuityFV of Annuity

FVofAnnuity C

r

r

t

1 1

Important to understand inputsImportant to understand inputs

‘ ‘r’ is the interest rate r’ is the interest rate per periodper period ‘‘t’ is the # of t’ is the # of periodsperiods. .

For example,For example,– if ‘t’ is # of if ‘t’ is # of yearsyears, ‘r’ is , ‘r’ is annualannual rate rate– if ‘t’ is # of if ‘t’ is # of monthsmonths, ‘r’ is the , ‘r’ is the monthlymonthly rate rate

FV of Annuity ExampleFV of Annuity Example

You will contribute $5,000 per year for the You will contribute $5,000 per year for the next 35 years into a retirement savings plan. next 35 years into a retirement savings plan. If your money earns 10% interest per year, If your money earns 10% interest per year, how much will you have accumulated at how much will you have accumulated at retirement?retirement?

Draw a time line!!!Draw a time line!!!

Time LineTime Line

NoticeNotice: Payment begins at the : Payment begins at the endend of first year of first year

0 353421

-5000 -5000-5000-5000

FV of Annuity on CalculatorFV of Annuity on Calculator Clear all memory: CLEAR ALLClear all memory: CLEAR ALL Ensure # compounding periods is 1: 1 Ensure # compounding periods is 1: 1 Enter payments: -5000 Enter payments: -5000 Enter # of payments: 35Enter # of payments: 35 Enter interest rate: 10 Enter interest rate: 10 Find Future Value:Find Future Value:

Answer: $___________Answer: $___________

N

I/YR

FV

P/YR

PMT

FV Annuity - A Twist..FV Annuity - A Twist.. You estimate you will need $1 million to You estimate you will need $1 million to

live comfortably in retirement in 30 years. live comfortably in retirement in 30 years. How much must you save How much must you save monthlymonthly if your if your money earns 12% interest money earns 12% interest per yearper year??

Note: Payments are Note: Payments are monthlymonthly, interest , interest quoted is quoted is annualannual!!!!!!

Two ways to adjust for Two ways to adjust for compounding periodscompounding periods

Divide annual interest rate by 12 and enter Divide annual interest rate by 12 and enter interest rate per monthinterest rate per month into calculator as the into calculator as the interest rate and interest rate and leave “P/YR” as 1leave “P/YR” as 1

Set “P/YR” on calculator as 12: 12Set “P/YR” on calculator as 12: 12and enter the and enter the annualannual interest rate interest rate

ORP/YR

‘‘N’ on calculatorN’ on calculator

You can either:You can either: Enter # of periods directly (360 in the Enter # of periods directly (360 in the

example)example)

If you have set 12 as the P/YR then you can If you have set 12 as the P/YR then you can also enter it as 30 also enter it as 30 – (notice it appears as 360)(notice it appears as 360)

N

OR

FV Annuity on Calculator (2)FV Annuity on Calculator (2) Clear all memory: CLEAR ALLClear all memory: CLEAR ALL Monthly-> # compounding periods is 12: 12Monthly-> # compounding periods is 12: 12 Enter Future Value: 1,000,000 Enter Future Value: 1,000,000 Enter # of payments: 30Enter # of payments: 30 Enter interest rate: 12 Enter interest rate: 12 Find payments:Find payments:

Answer: $___________Answer: $___________

N

I/YR

PMT

P/YR

FV

Note thedifference!

Present Value of AnnuitiesPresent Value of Annuities

Here we bring multiple, level cash flows Here we bring multiple, level cash flows back to the present (year 0)back to the present (year 0)

Typical examples are consumer loans where Typical examples are consumer loans where the loan amount is the PV and the fixed the loan amount is the PV and the fixed payments are the cash flowspayments are the cash flows

PV of Annuity ExamplePV of Annuity Example

Cash flow per period (CFt) = $500Cash flow per period (CFt) = $500 Number of periods (t) = 4 yearsNumber of periods (t) = 4 years Interest Rate (r) = 9% per yearInterest Rate (r) = 9% per year What is the present value (PV) = ?What is the present value (PV) = ?

ALWAYS DRAW A TIME LINE!!!ALWAYS DRAW A TIME LINE!!!

PV of Annuity on CalculatorPV of Annuity on Calculator Clear all memory: CLEAR ALLClear all memory: CLEAR ALL Ensure # compounding periods is 1: 1 Ensure # compounding periods is 1: 1 Enter payments: 500 Enter payments: 500 Enter # of payments: 4Enter # of payments: 4 Enter interest rate: 9 Enter interest rate: 9 Find Present Value:Find Present Value:

Answer: $___________Answer: $___________

N

I/YR

PV

P/YR

PMT

PV of AnnuityPV of Annuity

PVofAnnuity Cr

r

t

11

1( )

AgainAgain: ‘r’ and ‘t’ must match – i.e. if t is # of : ‘r’ and ‘t’ must match – i.e. if t is # of months, r must be monthly ratemonths, r must be monthly rate

Car Loan ExampleCar Loan Example

Car costs $ 20,000Car costs $ 20,000 Interest rate Interest rate per monthper month = 1% = 1% 5-year loan ---> number of months = t = 605-year loan ---> number of months = t = 60

What is the What is the monthlymonthly payment? payment? Answer: ___________Answer: ___________

Mortgage paymentsMortgage payments House cost $250,000House cost $250,000 Mortgage Rate = 7.5% annuallyMortgage Rate = 7.5% annually Term of loan = 30 yearsTerm of loan = 30 years Payments made Payments made monthlymonthly

What are your payments?What are your payments? Answer: _____________Answer: _____________

To Reiterate...To Reiterate... Be VERY careful about compounding Be VERY careful about compounding

periodsperiods Problem can state annual interest rate, but Problem can state annual interest rate, but

the cash flows can be monthly, quarterly…the cash flows can be monthly, quarterly… The convention is to state interest rate The convention is to state interest rate

annually (annually (AAnnual nnual PPercentage ercentage RRate)ate)

PerpetuityPerpetuity

Annuity foreverAnnuity forever

Examples: Preferred Stock, ConsolsExamples: Preferred Stock, Consols

PerpetuityPerpetuity

NoteNote: C and r measured over same interval: C and r measured over same interval

Perpetuity ExamplePerpetuity Example

Preferred stock pays $1.00 dividend per quarter. Preferred stock pays $1.00 dividend per quarter. The required return, r, is 2.5% per quarter.The required return, r, is 2.5% per quarter.

What is the stock value?What is the stock value?

Perpetuity ExamplePerpetuity Example Steve Forbes’s flat-tax proposal was Steve Forbes’s flat-tax proposal was

expected to save him $500,000 a year expected to save him $500,000 a year forever if passed. He spent $40,000,000 of forever if passed. He spent $40,000,000 of his own money for campaign his own money for campaign

ChargeCharge: He was running for presidency for : He was running for presidency for personal gainpersonal gain

Did the charge make senseDid the charge make sense

Forbes continued...Forbes continued...

What should be ‘r’ in the example?What should be ‘r’ in the example?

At what ‘r’ would Forbes have gained from At what ‘r’ would Forbes have gained from being a president and steamrolling flat-tax being a president and steamrolling flat-tax proposal?proposal?

Compounding PeriodsCompounding Periods

Interest can be compounded Interest can be compounded – Annually - SemiannuallyAnnually - Semiannually– Monthly - Daily - ContinuouslyMonthly - Daily - Continuously

SmallerSmaller the compounding period, the compounding period, fasterfaster is is the growth of moneythe growth of money

The same PV or FV formula can be used: The same PV or FV formula can be used: BUT UNDERSTAND THE INPUTS!!BUT UNDERSTAND THE INPUTS!!

Compounding exampleCompounding example Invest $5,000 in a 5-year CDInvest $5,000 in a 5-year CD Quoted Annual Percentage Rate (APR) = Quoted Annual Percentage Rate (APR) =

15%15% Calculate FVCalculate FV55 for annual, semi-annual, for annual, semi-annual,

monthly and daily compoundingmonthly and daily compounding

KeyKey: Adjust “P/YR” on calculator: Adjust “P/YR” on calculator

Answers:Answers: Annual:Annual: $10,056.78$10,056.78 Semi-annualSemi-annual $10,305.16$10,305.16 Monthly:Monthly: $10,535.91$10,535.91 Daily:Daily: $10,583.37$10,583.37

Continuous Compounding???Continuous Compounding???

Continuous compoudingContinuous compouding Compounded every instant “microsecond”Compounded every instant “microsecond”

r = interest rate per periodr = interest rate per period t = number of periods t = number of periods

Previous example answer: $ 10,585.00Previous example answer: $ 10,585.00

FV PV etr t

Continuous compounding Continuous compounding exampleexample

Invest $4,500 in an account paying 9.5% Invest $4,500 in an account paying 9.5% compounded continuouslycompounded continuously

What is the balance after 4 years?What is the balance after 4 years?Answer: _________Answer: _________

Quoted vs. Effective Interest Quoted vs. Effective Interest RatesRates

Quoted RateQuoted Rate: Usually stated annually along : Usually stated annually along with compounding period (APR)with compounding period (APR)– e.g. 10% compounded quarterlye.g. 10% compounded quarterly

Effective Annual RateEffective Annual Rate (EAR): Interest rate (EAR): Interest rate actually earned IF the compounding period were actually earned IF the compounding period were one yearone year

EAREAR

m = number of compounding periods in a year

EAR on CalculatorEAR on Calculator What is the EAR for quoted rate of 15% per What is the EAR for quoted rate of 15% per

year compounded quarterly?year compounded quarterly? Set number of periods per year: 4Set number of periods per year: 4 Enter quoted annual rate: 15Enter quoted annual rate: 15 Compute EAR:Compute EAR: Answer: _______ Answer: _______

P/YR

I/YR

EFF%

EAR ExampleEAR Example Compute EAR for 12% compoundedCompute EAR for 12% compounded

– AnnuallyAnnually– QuarterlyQuarterly– MonthlyMonthly– DailyDaily

Answers: ____ , ____ , ____ , ____Answers: ____ , ____ , ____ , ____

EAR for Continuous EAR for Continuous compoundingcompounding

Example: Quoted rate is 10% compounded Example: Quoted rate is 10% compounded continuouslycontinuously

EAR = _____%EAR = _____%

EAR eq 1

Complicatons to TVMComplicatons to TVM

When payments begin beyond year 1When payments begin beyond year 1

PV and FV combinedPV and FV combined

When payments begin in year 0 (Annuities Due)When payments begin in year 0 (Annuities Due)

Payments beyond year 1Payments beyond year 1 A car dealer offers ‘no payments for next A car dealer offers ‘no payments for next

12 months’ deal on a $15,000 car. After 12 months’ deal on a $15,000 car. After that, you will pay monthly payments for the that, you will pay monthly payments for the next 4 years. r = 10% APR. What are your next 4 years. r = 10% APR. What are your monthly payments?monthly payments?

Answer: ___________Answer: ___________

PV and FV combinedPV and FV combined

How much must you invest per year to have an How much must you invest per year to have an amount in 20 years that will provide an annual amount in 20 years that will provide an annual income of $12,000 per year for 5 years? r = 8% income of $12,000 per year for 5 years? r = 8% annually.annually.

Answer: ___________Answer: ___________

PV and FV combined 2PV and FV combined 2 You have 2 optionsYou have 2 options::

– Receive $100 for next 10 years onlyReceive $100 for next 10 years only– Receive $100 forever beginning in year 11Receive $100 forever beginning in year 11

If r = 10% which one would you prefer?If r = 10% which one would you prefer? At what interest rate are you indifferent At what interest rate are you indifferent

between the two options?between the two options?

Annuities DueAnnuities Due

Payments begin in year 0Payments begin in year 0– Ex. Rent/Lease PaymentsEx. Rent/Lease Payments

TrickTrick:: Adjust BEG/END on calculator to BEGAdjust BEG/END on calculator to BEG

Leave to END, but multiply (1+r) for both Leave to END, but multiply (1+r) for both PV and FVPV and FV

OR

Annuity Due ExampleAnnuity Due Example

Find PV of a 4-year (5 payment), $400 annuity Find PV of a 4-year (5 payment), $400 annuity duedue. r . r = 10%= 10%

Find FV in year 5 of the above annuity dueFind FV in year 5 of the above annuity due

Answers:Answers:– PV = $1,667.95PV = $1,667.95

– FVFV5 5 = $2,686.24= $2,686.24

Time(years)

0 1 2 3 4 5

$400 $400 $400 $400 $400

FV

Another ExampleAnother Example You start to contribute $500 every month to You start to contribute $500 every month to

your IRA account beginning immediately. your IRA account beginning immediately. How much will you accumulate at the end How much will you accumulate at the end of first year? The return on your of first year? The return on your investment is 20% per year.investment is 20% per year.

NoteNote: ‘Return’ here is just another term for : ‘Return’ here is just another term for the interest ratethe interest rate

Answer: $_______Answer: $_______

Tricky but Legal...Tricky but Legal...

Add-on InterestAdd-on InterestCalled ‘add-on’ interest because interest is Called ‘add-on’ interest because interest is added on to the principal added on to the principal beforebefore the the payments are calculatedpayments are calculated

Points on a LoanPoints on a Loan: : Percentage of loan amount reduced up frontPercentage of loan amount reduced up front– Used in home mortgagesUsed in home mortgages

Example: Add-on InterestExample: Add-on Interest You are offered the opportunity to borrow $1,000 for 3 You are offered the opportunity to borrow $1,000 for 3

years at 12% ‘add-on’ interest. The lender calculates years at 12% ‘add-on’ interest. The lender calculates the payment as follows:the payment as follows:Amt. owed in 3 years: $1000 x (1+.12)Amt. owed in 3 years: $1000 x (1+.12)33 = 1,405 = 1,405Monthly Payment = $1,405 / 36 = $39Monthly Payment = $1,405 / 36 = $39

What is the effective annual rate (EAR)?What is the effective annual rate (EAR)? StepsSteps::

– Calculate the APR interest (I/YR)Calculate the APR interest (I/YR)– Use answer to calculate the EARUse answer to calculate the EAR

Add-on Example (2)Add-on Example (2)

Calcuate the EAR on a 6-year, $7,000 loan at Calcuate the EAR on a 6-year, $7,000 loan at 13% ‘add-on’ interest. The payments are 13% ‘add-on’ interest. The payments are monthly.monthly.

Answer: ________Answer: ________

Example: Points on a LoanExample: Points on a Loan

1-year loan of $100. r = 10% + 2 points 1-year loan of $100. r = 10% + 2 points [Note: 1 point = 1% of loan amount. Hence [Note: 1 point = 1% of loan amount. Hence you pay upfront $2 to lender. Hence you you pay upfront $2 to lender. Hence you are actually getting only $98, not $100]are actually getting only $98, not $100]

What is the EAR?What is the EAR?

$110 = $98 (1+r)$110 = $98 (1+r)r = 12.24%r = 12.24%

Points on a loan (2)Points on a loan (2)

Calculate the EAR on a 10-year, $110,000 Calculate the EAR on a 10-year, $110,000 mortgage when interest rate quoted is 7.75% + 1 mortgage when interest rate quoted is 7.75% + 1 point. The payments are monthlypoint. The payments are monthly

Answer: _________Answer: _________

Balloon PaymentsBalloon Payments

Amount on the loan outstanding after a Amount on the loan outstanding after a certain number of payments have been certain number of payments have been mademade

– Sometimes called ‘residual’ on a loanSometimes called ‘residual’ on a loan e.g. when you want to pay off a loan earlye.g. when you want to pay off a loan early

Balloon ExampleBalloon Example You borrowed $90,000 on a house for 30 You borrowed $90,000 on a house for 30

years 10 years ago. The annual interest rate years 10 years ago. The annual interest rate then was 17%. The payments are monthly. then was 17%. The payments are monthly. Since interest rate has fallen, you want to Since interest rate has fallen, you want to payoff the remaining amount on the loan payoff the remaining amount on the loan and refinance it. What is the outstanding and refinance it. What is the outstanding amount to be paid off?amount to be paid off? (Note: Payments are $1,283.11)(Note: Payments are $1,283.11)

Answer: $__________Answer: $__________

Two ways to calculate BalloonsTwo ways to calculate Balloons First calculate paymentsFirst calculate payments Take the present value of the Take the present value of the remainingremaining

(unpaid) payments(unpaid) payments

Use amortization function on calculatorUse amortization function on calculator Enter the period : period Enter the period : period Enter , and then Enter , and then

INPUT

AMORT == =

OR

Another Example..Another Example..

What is the outstanding balance on a 5 year What is the outstanding balance on a 5 year $19,000 car loan at 11% interest after 2-1/2 $19,000 car loan at 11% interest after 2-1/2 years have passed? The payments are monthly.years have passed? The payments are monthly.

Answer: $____________Answer: $____________

TVM TIPSTVM TIPS Draw time line!Draw time line!

Check & set BEG/END on calculatorCheck & set BEG/END on calculator

Check & set P/YR on calculatorCheck & set P/YR on calculator

Check & set # of decimal places to 4Check & set # of decimal places to 4

TVM Tips Continued...TVM Tips Continued... Clear all previously stored #’s in memoryClear all previously stored #’s in memory

– Especially true when same problem requires Especially true when same problem requires multiple TVM calculationsmultiple TVM calculations

Make sure that for FV and PV calculation, Make sure that for FV and PV calculation, you have correctly signed (+/-) the cash you have correctly signed (+/-) the cash flowsflows