Module 8 -Time Value of Money

7/31/2019 Module 8 -Time Value of Money

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time value of money


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Have you

studied the

text for this

topic?


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Question?

Would a businessexpect to pay the

same for amachinepurchased andpaid for today, asthe same machineacquired todaybut not paid for fortwo years?


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Definition of Interest


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Simple Interest


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Compound Interest

Interest earned (simple interest formula) is left inbank and added to the original principal

The next simple interest calculation thus uses alarger P

P keeps growing each period

Result is that money grows exponentially insteadof linearly with compound interest


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Graphical View of Simple vs Compound

Compound Interest

Simple Interest

$$$

Time


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Application of

compoundinterest


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Power of Compounding

Example given in your text.

Put $2000/yr in RRSP starting at age 25 instead of35 ( an extra $20,000 of deposits)

You will have an extra $300,000* of accumulatedwealth at age 65

* Exact amount depends on the interest rate and frequency of compounding


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Interest on Interest Illustrated

Assume $1 invested at 10% for a year

Applying Simple interest formulaI = Prt

I = ($1) (.10) (1)I = 10 cents

Applying at end of year 2I = ($1.10) (.10) (1)I = 11 cents

This extra cent is HUGE! It represents interest on

interest and is the secret of compound interest


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Extend the calculations for 5

more years:

Look how interest increases each

year

$1.10

$1.21

$1.33

$1.46

$1.61

$1.77

$1.94

cc

c

c

c

c

c


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Results Compared


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Rule of 72Money doubles inapproximately

(72)( interest %)

years


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Question:

If $18,000 is invested to earn 8%compounded quarterly

How much will the investment be worth atthe end of 6 years?

Question type = FV of a single amount


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The Time Line

FV of $18,000 single amount in 6 years,

money worth 8%, quarterly compounding)

PV = $18,000

t = 0

FV = $28,951.20

t = 24


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If we use the simple interest

formula Lots of figuring.$18,000 x 2% = $360

$18,360 x 2% = $367.20

$18,727.20 x 2%

Etc

Etc

Etc

24 separate calculations required using I=Prt......... Bah humbug!!!!

Eventually get a result of $28,951.20

Most of you would not.high risk or clerical error


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The Compound Interest Formula


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The Compound Interest Formula

Pretty neat...........but requires scientific/business

calculator to do exponents

Note that 8% interest rate given = always bydefinition an annual (or effective) interest rate

We needed to express it in same time frame ascompounding period (a quarter) = 8/4 = 2

Note also n = number of compounding periods& not number of years


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(1+r)n..bah humbug


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Table for (1+r)n on Page 18


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Next StepMove from one amount

to multiple cash flows..an annuity

DEFINTIONAn identical stream of

cash flows (payment or receipt) made

each compounding period

Often called the periodic rent


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Future value of an annuity

Question:

If we invest $1,000 at the end of each of thenext 4 years at 10% can we afford to retire

to a beach?


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Time LineQuestion:

If we invest $1,000 at the end of each of the next 4 years at

10% can we afford to retire to a beach?

t = 0 t = 1t = 2 t = 3 t = 4

$1000 $1000 $1000 $1000


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$2.10

$3.31

$4.64*

$1 invested at

the end of each

of the next four

years at 10%

will be $4.64 at

the end of thefour years.

End Period 1

Period 2Period 3

Period 4

How do we get the $3.31?

There is $2.10 sitting all through period 3 earning

interest at 10% =21 cents + an additional deposit of

$1. So $2.10 + 0.21 + $1 = $3.31

c

c

c


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Choices


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FV of Annuity Tablepage 20 of Module 8


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Using the table

Question:If we invest $1,000 at the end of each of the next 4 years at 10%

can we afford to retire to a beach?

Rent = $1000

r = .10

n = 4

Factor from intersection of 10% column and n = 4 row = 4.6410

FV = rent x factor= $1000 x 4.6410

= $4,641


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Future value of an annuitythe power of

compounding kicks in as number ofcompounding periods increases


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Present ValueWe have now looked at FV of a singleamount and FV of an annuity

Next we look at PV of a single amount andPV of an annuity

PV is the amount at time zero


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Present Value of a single amount

New Question:

How much must be invested now so as tohave $30,000 five years from now, if theinterest rate is 8% compounded semi-annually?


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Time LineQuestion:How much must be invested now so as to have $30,000

five years from now, if the interest rate is 8%compounded semi-annually?

t = 0 t = 1t = 9

$30,000accumulation

at t10 = FV

Required deposit att0 = PV = ???

t = 10


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Present Value of single amount

formula

PV = FV

(1 + r )n

PV = $30,000 / (1.04)10

PV = $20,267

Notice this is the same formula asbefore butwith PV isolated instead of FV


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Present Value TablePage 19


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Present Value using factors

Question:How much must be invested now so as to have $30,000 five years from

now, if the interest rate is 8% compounded semi-annually?

FV = 30,000PV = ?

R = .04T = 10

Solution:$30,000 x .67556 (present value of $1 for 10 periods

@ 4%)= $20,267


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Accounting application

What would we capitalize an asset at (the

debit in the G/L) for if..

we agree to pay $30,000 in five years for an it

when interest rates are 8% compounded

semiannually


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Accounting application

Solution:$20,267 its present value

(not the future value of $30,000)


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Asset and Liability Valuation

When time periods are short (less thanone year), the time value of money isoften ignored.

When time periods are longer, assetsand liabilities are valued at the presentvalue of the future payments


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DiscountWhen the present value is LESS than the amount which

will ultimately be paid/received the difference is adiscount

Dr Asset $20,267Dr Discount on Note Payable $9,733

Cr Note Payable $30,000

Discount is a contra account the Note Payable wouldbe valued as a $20,267 liability initially


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Discount is reduced as time

passes .

Interest expense for the first six monthperiod

Dr Interest Expense $811

Cr Discount $811

($20,267 x 4% = 811)


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Discount amortization using the effective

interest method


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Another Accounting Question:

What is the journal entry to record theacquisition of a machine when thevendor agrees to wait 4 years for$150,000 payment ?

interest rates are 10% and moneycompounded annually.


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Question:

What is the journal entry to record the acquisition of amachine when the vendor agrees to wait 4 years for$150,000 payment interest rates are 10%

Solution:

Dr Machine $102,452

Dr Discount $ 47,548Cr Note Payable $150,000


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Discount amortization


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What if the payment terms are

different..

Business often negotiates installment paymentsor leases instead of agreeing to a single large

payment at the end of a long period of time

How much would four annual payments need tobe..

to make the deal equivalent to a cash paymentnow of $102,452 (or $150,000 in 4 years)?


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Time LineHow much would four annual payments need to be..

to make the deal equivalent to a cash payment now of$102,452 (or $150,000 in 4 years)?

t = 0 t = 1 t = 2 t = 3 t = 4

Pmt = ? Pmt = ? Pmt = ? Pmt = ?

FV = $150,000PV = $102, 452


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$2.10

$3.31

$4.64*

$1 invested at the end of each

of the next four years at 10%

will be $4.64 at the end of the

four years.

End Period 1

Period 2Period 3

Period 4

$4.64 is the future value of

an ordinary annuity

c

c

c

Recall this

slide fromearlier.the

FV of an

annuity is the

rent x 4.6410


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Relationship between FV and PV

if the FV = (periodic rent) x (annuity factor)

Then rearranging the equation gives:Rent = FV / annuity factor

Rent = $150,000 / 4.6410

Rent = $32,321


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Present value of an annuity


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$3.49

$3.84

$4.22$4.64*

Present Value of an

annuityis single amountto invest today which hassame future value * as anannuity

Period 4

$2.10

$3.31

$4.64*

End Period 1

Period 2Period 3

Invest $1 atend of four

years at 10%

c

c

c

c

c

c

c


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Formula for PV of Annuity

PV = (rent) x 1 - (1+r)-n

r

Question:

If assets and liabilities are valued at their present value what is thepresent value of an annuity of $32,321 for 4 years when rates are 10%?

PV = ($32,321) x 3.16986

PV = $102,453


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PV of Annuity TablePage 21


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PV of an annuityusing factor


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Amortization table for an annuity of payments

separating payments into repayments of

principal and interest component

($102,453 - $22,076 = $80,377)


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The Ordinary Annuity (in arrears)

vs the Annuity Due


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Look at our Time Line

t = 0 t = 1 t = 2 t = 3 t = 4

Pmt Pmt Pmt Pmt

If payments start one period earlier then

there is effectively one more full period

for compounding to occur


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Annuities Due

We will leave this topic until intermediate accounting

You should note that PV or NPV (net present value) asit is often called

Is taught in Finance, Math and numerous othercourses because it is a universally useful concept

Study it well as business studentsyou willencounter it time and time again


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Solving Time Value Problems

1.Asking for FV or PV?

2. Single amount or annuity?

3. Identify r and n in equivalent time frames

4. Find right table and right factor


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Complex Problem

Fred Flintstone is 62 years old and wishes todeposit equal amounts at the end of his 63rd, 64th

and 65th years so that starting at age 65 he can

withdraw $5,000 per year for ten years. Money

is worth 8%


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Complex Problem solved

The trick is to realize there are two differentcalculations involved

First translate the ten year annuity of $5,000pmts into the PV at age 65

Second find what annuity payments have a FVequivalent to that amount (the PV)


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Complex Problem solved

Factor for PV of Annuity (n = 10, r = 8) = 6.71008 PV = ($5,000) x 6.71008 = $33,550

FV = rent x (FV factor for annuity) $33,550 = rent x (3.2464) Rent = $33,550/3.2464 = $10,335

If Fred deposits $10,335 for next 3 years at 8%, he canwithdraw $5,000 a year for 10 years.


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Another Problem

Your mother put $1,000 a year in the bankstarting ten years ago to fund your college

education. Money is worth 12%. How much can

she give you today?

FV = rent x factor for FV of annuity

FV = 1,000 x (17.54874)

FV = $17,549

Better get a part time job


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Key Topics Module 8

Basic concept of the time value of money Simple Interest Compound Interest Future Value vs Present Value Time Lines to analyze problems Single amounts vs Annuities Formulas vs Tables Ordinary Annuities ($1 in Arrears) vs Annuities Due Accounting Applications of PV


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Time to go home!!!

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Module 8 -Time Value of Money

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