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THE TIME VALUE OF MONEY

Himanshu Puri

Faculty, ABS

Amity University, Haryana

What is Time Value?

We say that money has a time value because that money can

be invested with the expectation of earning a positive rate of

return

In other words, a dollar received today is worth more than a

dollar to be received tomorrow

That is because todays dollar can be invested so that we have

more than one dollar tomorrow

The Interest Rate

Obviously, $10,000 today.

You already recognize that there is TIME

VALUE TO MONEY!!

Which would you prefer -- $10,000 today

or $10,000 in 5 years?

Why TIME?

TIME allows you the opportunity to postponeconsumption and earn INTEREST.

Why is TIME such an important element in

your decision?

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The Terminology of Time Value

Present Value - An amount of money today, or the current

value of a future cash flow

Future Value - An amount of money at some future time

period

Period - A length of time (often a year, but can be a month,

week, day, hour, etc.)

Interest Rate - The compensation paid to a lender (or saver)for the use of funds expressed as a percen tage for a period

(normally expressed as an annual rate)

Timelines

A timeline is a graphical device used to clarify the timingof the cash flows for an investment

Each tick represents one time period

0 1 2 3 4 5

PV FV

Today

Types of Interest

Compound InterestInterest paid (earned) on any previous interest

earned, as well as on the principal borrowed

(lent).

Simple Interest

Interest paid (earned) on only the originalamount, or principal, borrowed (lent).

Simple Interest Formula

Formula SI = P0(i)(n)

SI: Simple Interest

P0: Deposit today (t=0)

i: Interest Rate per Periodn: Number of Time Periods

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Simple Interest Example

SI = P0(i)(n)

= $1,000(.07)(2)

= $140

Assume that you deposit $1,000 in an account

earning 7% simple interest for 2 years. What

is the accumulatedinterestat the end of the 2ndyear?

Simple Interest (FV)

FV = P0 + SI

= $1,000+ $140

=$1,140

Future Value is the value at some future time of a

present amount of money, or a series of payments,

evaluated at a given interest rate.

What is the Future Value (FV) of the deposit?

Simple Interest (PV)

The Present Value is simply the $1,000

you originally deposited. That is the

value today!

Present Value is the current value of a future

amount of money, or a series of payments,

evaluated at a given interest rate.

What is the Present Value (PV) of the previous

problem?

Calculating the Future Value

Suppose that you have an extra $100 today that you wish

to invest for one year. If you can earn 10% per year on

your investment, how much will you have in one year?

0 1 2 3 4 5

-100 ?

FV1 100 1 0 10 110 .

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Calculating the Future Value (cont.)

Suppose that at the end of year 1 you decide to extend the

investment for a second year. How much will you have

accumulated at the end of year 2?

0 1 2 3 4 5

-110 ?

FVor

FV

2

2

2

100 1 0 10 1 0 10 121

100 1 0 10 121

. .

.

Generalizing the Future Value

Recognizing the pattern that is developing, wecan generalize the future value calculations asfollows:

FV PV iNN

1

If you extended the investment for a thirdyear, you would have:

FV33

100 1 0 10 133 10 . .

Compound Interest

Note from the example that the future value is increasing atan increasing rate

In other words, the amount of interest earned each year isincreasing Year 1: $10

Year 2: $11

Year 3: $12.10

The reason for the increase is that each year you areearning interest on the interest that was earned in previousyears in addition to the interest on the original principleamount

General Future Value Formula

FV1 = P0(1+i)1

FV2 = P0(1+i)2

General Future Value Formula:

FVn = P0 (1+i)n

or FVn = P0 (FVIFi,n) -- See Table 1

etc.

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FV2 = $1,000 (FVIF7%,2)= $1,000 (1.145)= $1,145 [Due to Rounding]

Using Future Value Tables

Period 6% 7% 8%1 1.060 1.070 1.080

2 1.124 1.145 1.1663 1.191 1.225 1.260

4 1.262 1.311 1.360

5 1.338 1.403 1.469

Story Problem Example

Julie Miller wants to know how large her deposit of

$10,000 today will become at a compound annual

interest rate of 10% for 5 years.

0 1 2 3 4 5

$10,000

FV5

10%

Story Problem Solution

Calculation based on Table I:FV5 = $10,000(FVIF10%, 5)

= $10,000(1.611)

= $16,110 [Due to Rounding]

Calculation based on general formula: FVn = P0 (1+i)n

FV5 = $10,000 (1+ 0.10)5= $16,105.10

Calculating the Present Value

So far, we have seen how to calculate the futurevalue of an investment

But we can turn this around to find the amountthat needs to be invested to achieve somedesired future value:

PV

FV

i

N

N

1

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Present Value

Assume that you need $1,000in 2 years. Letsexamine the process to determine how much youneed to deposit today at a discount rate of 7%compounded annually.

0 1 2

$1,000

7%

PV1PV0

Present Value

PV0 = FV2 / (1+i)2 = $1,000/ (1.07)2 = FV2

/ (1+i)2 = $873.44

0 1 2

$1,000

7%

PV0

General Present Value Formula

PV0= FV1 / (1+i)1

PV0 = FV2 / (1+i)2

General Present Value Formula:PV0 = FVn / (1+i)

n

or PV0 = FVn (PVIFi,n) -- See Table II

etc.

PV2 = $1,000 (PVIF7%,2)

= $1,000 (.873)= $873 [Due to Rounding]

Using Present Value Tables

Period 6% 7% 8%1 .943 .935 .926

2 .890 .873 .857

3 .840 .816 .7944 .792 .763 .735

5 .747 .713 .681

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Story Problem Example

Julie Miller wants to know how large of a deposit

to make so that the money will grow to $10,000 in

5 years at a discount rate of 10%.

0 1 2 3 4 5

$10,000

PV0

10%

Story Problem Solution

Calculation based on general formula:

PV0 = FVn / (1+i)n

PV0 = $10,000 / (1+ 0.10)5

= $6,209.21

Calculation based on Table I:

PV0 = $10,000 (PVIF10%, 5)

= $10,000(.621)

= $6,210.00 [Due to Rounding]

Present Value: An Example

Suppose that your five-year old daughter has just

announced her desire to attend college. After some

research, you determine that you will need about $100,000on her 18th birthday to pay for four years of college. If

you can earn 8% per year on your investments, how much

do you need to invest today to achieve your goal?

PV

100 000

10876979

13

,

.$36, .

Annuities

An annuity is a series of nominally equal payments equally

spaced in time

Annuities are very common: Rent

Mortgage payments

Car payment

Pension income

The timeline shows an example of a 5-year, $100 annuity

0 1 2 3 4 5

100 100 100 100 100

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The Principle of Value Additivity

How do we find the value (PV or FV) of an annuity?

First, you must understand the principle of value

additivity:

The value of any stream of cash flows is equal to the

sum of the values of the components

In other words, if we can move the cash flows to thesame time period we can simply add them all

together to get the total value

Present Value of an Annuity

We can use the principle of value additivity to find the

present value of an annuity, by simply summing the present

values of each of the components:

PV

Pmt

i

Pmt

i

Pmt

i

Pmt

iA

t

t

t

N

N

N

1 1 1 11

1

1

2

2

Example PVA

PVA3 = $1,000/(1.07)1 +

$1,000/(1.07)2 +

$1,000/(1.07)3

= $934.58 + $873.44 + $816.30

= $2,624.32

$1,000 $1,000 $1,000

0 1 2 3 4

$2,624.32 = PVA3

7%

$934.58$873.44

$816.30

Cash flows occur at the end of the periodPVAn = R (PVIFAi%,n)

PVA3 = $1,000(PVIFA7%,3)

= $1,000(2.624) =$2,624

Using Table IV

Period 6% 7% 8%1 0.943 0.935 0.926

2 1.833 1.808 1.783

3 2.673 2.624 2.5774 3.465 3.387 3.312

5 4.212 4.100 3.993

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Present Value of an Annuity (cont.)

Using the example, and assuming a discount rate of 10%

per year, we find that the present value is:

PV

A

100

110

100

110

100

110

100

110

100

11037908

1 2 3 4 5. . . . .

.

0 1 2 3 4 5

100 100 100 100 100

62.0968.3075.1382.6490.91

379.08

The Future Value of an Annuity

We can also use the principle of value additivity to find the

future value of an annuity, by simply summing the future

values of each of the components:

FV Pmt i Pmt i Pmt i PmtA tN t

t

N

N N

N

1 1 11

1

1

2

2

Example FVA

FVA3 = $1,000(1.07)2 +

$1,000(1.07)1 + $1,000(1.07)0

= $1,145+$1,070+$1,000=$3,215

$1,000 $1,000 $1,000

0 1 2 3 4

$3,215 = FVA3

7%

$1,070

$1,145

Cash flows occur at the end of the periodFVAn = R (FVIFAi%,n)

FVA3 = $1,000 (FVIFA7%,3)

= $1,000 (3.215) =$3,215

Using Table III

Period 6% 7% 8%1 1.000 1.000 1.000

2 2.060 2.070 2.080

3 3.184 3.215 3.2464 4.375 4.440 4.506

5 5.637 5.751 5.867

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The Future Value of an Annuity (cont.)

Using the example, and assuming a discount rate of 10%

per year, we find that the future value is:

FVA 100 1 10 100 1 10 100 110 100 110 100 610 514 3 2 1

. . . . .

100 100 100 100 100

0 1 2 3 4 5

146.41133.10121.00110.00 } = 610.51at year 5

Uneven Cash Flows

Very often an investment offers a stream of cash

flows which are not either a lump sum or an annuity

We can find the present or future value of such astream by using the principle of value additivity

Uneven Cash Flows: An Example (1)

Assume that an investment offers the following cash flows.

If your required return is 7%, what is the maximum price

that you would pay for this investment?

0 1 2 3 4 5

100 200 300

PV

100

107

200

107

300

10751304

1 2 3. . .

.

Uneven Cash Flows: An Example (2)

Suppose that you were to deposit the following amounts in

an account paying 5% per year. What would the balance

of the account be at the end of the third year?

0 1 2 3 4 5

300 500 700

FV 300 1 05 500 1 05 700 1 555 752 1

. . , .

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Mixed Flows Example

Julie Miller will receive the set of cash flows

below. What is the Present Value at a discount

rate of 10%.

0 1 2 3 4 5

$600 $600 $400 $400 $100

PV0

10%

Piece-At-A-Time

0 1 2 3 4 5

$600 $600 $400 $400 $100

10%

$545.45$495.87$300.53$273.21$ 62.09

$1677.15 = PV0of the Mixed Flow

Non-annual Compounding

So far we have assumed that the time period is equal to a

year

However, there is no reason that a time period cant be any

other length of time

We could assume that interest is earned semi-annually,

quarterly, monthly, daily, or any other length of time

The only change that must be made is to make sure that therate of interest is adjusted to the period length

Non-annual Compounding (cont.)

Suppose that you have $1,000 available for investment.

After investigating the local banks, you have compiled the

following table for comparison. In which bank should youdeposit your funds?

Bank Interest Rate CompoundingFirst National 10% AnnualSecond National 10% MonthlyThird National 10% Daily

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To solve this problem, you need to determine which bank

will pay you the most interest

In other words, at which bank will you have the highest

future value?

To find out, lets change our basic FV equation slightly:

FV PVi

m

Nm

1

In this version of the equation m is the number ofcompounding periods per year


We can find the FV for each bank as follows:

FV 1 0 00 110 1 1001

, . ,

FV

1000 1

0 10

12110471

12

,.

, .

FV

1000 1

0 10

365110516

365

,.

, .

First National Bank:

Second National Bank:

Third National Bank:

Obviously, you should choose the Third National Bank

Impact of Frequency

Julie Miller has $1,000 to invest for 2 Years at an annualinterest rate of 12%.

Annual FV2 = 1,000(1+ [.12/1])(1)(2)

= 1,254.40

Semi FV2 = 1,000(1+ [.12/2])(2)(2)

= 1,262.48

Impact of Frequency

Qrtly FV2 = 1,000(1+ [.12/4])(4)(2)

= 1,266.77

Monthly FV2 = 1,000(1+ [.12/12])(12)(2)

= 1,269.73

Daily FV2= 1,000(1+[.12/365])(365)(2)

= 1,271.20

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Effective AnnualInterest Rate

Effective Annual Interest Rate

The actual rate of interest earned (paid) after

adjusting the nominal ratefor factors such asthe number of compounding periods per

year.

(1 + [ i/ m ] )m- 1

BWs EffectiveAnnual Interest Rate

Basket Wonders (BW) has a $1,000 CD at the bank.The interest rate is 6%compounded quarterly for 1

year. What is the Effective Annual Interest Rate(EAR)?

EAR = ( 1 + 6% / 4 )4 - 1= 1.0614 - 1 = .0614 or 6.14%!

Steps to Amortizing a Loan

1. Calculate the payment per period.

2. Determine the interest in Period t.

(Loan Balance at t-1) x (i% / m)

3. Compute principal payment in Period t.

(Payment- Interestfrom Step 2)

4. Determine ending balance in Period t.

(Balance - principal paymentfrom Step 3)

5. Start again at Step 2 and repeat.

Amortizing a Loan Example

Julie Miller is borrowing $10,000 at a compound

annual interest rate of 12%. Amortize the loan if

annual payments are made for 5 years.

Step 1: Payment

PV0

= R (PVIFAi%,n

)

$10,000 = R (PVIFA 12%,5)

$10,000 = R (3.605)

R = $10,000 / 3.605 = $2,774

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Amortizing a Loan Example

End ofYear

Payment Interest Principal EndingBalance

0 --- --- --- $10,000

1 $2,774 $1,200 $1,574 8,426

2 2,774 1,011 1,763 6,663

3 2 774 800 1 974 4 689

4 2,774 563 2,211 2,478

5 2,775 297 2,478 0

$13,871 $3,871 $10,000

[Last Payment Slightly Higher Due to Rounding]

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