Chapter 5 The Time Value of Money 1 Pr. Zoubida SAMLAL.

Chapter 5 The Time Value of Money

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Pr. Zoubida SAMLAL

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Obviously, $10,000 today.

You already recognize that there is TIME VALUE TO MONEY!!

The Interest RateThe Interest Rate

Which would you prefer -- $10,000 today or $10,000 in 5 years?

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TIME allows you the opportunity to postpone consumption and earn

INTEREST.

Why TIME?Why TIME?

Why is TIME such an important element in your decision?

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Types of InterestTypes of Interest

• Compound InterestInterest paid (earned) on any previous interest earned, as well as on the principal borrowed (lent).

Simple InterestInterest paid (earned) on only the original

amount, or principal, borrowed (lent).

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Simple Interest FormulaSimple Interest Formula

Formula SI = P0(i)(n)

SI: Simple InterestP0: Deposit today (t=0)

i: Interest Rate per Periodn: Number of Time Periods

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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.

Simple Interest (FV)Simple Interest (FV)

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

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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.

Simple Interest (PV)Simple Interest (PV)

• What is the Present Value (PV) of the previous problem?

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Types of TVM Calculations

• There are many types of TVM calculations• The basic types will be covered in this review module

and include:– Present value of a lump sum– Future value of a lump sum– Present and future value of cash flow streams– Present and future value of annuities

• Keep in mind that these forms can, should, and will be used in combination to solve more complex TVM problems

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FV1 = P0 (1+i)1 = $1,000 (1.07)= $1,070

Compound InterestYou earned $70 interest on your $1,000

deposit over the first year.This is the same amount of interest you would

earn under simple interest.

Future Value Single DepositFuture Value Single Deposit

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FV1 = P0 (1+i)1 = $1,000 (1.07)

= $1,070FV2 = FV1 (1+i)1

= P0 (1+i)(1+i) = $1,000(1.07)(1.07)= P0 (1+i)2 =

$1,000(1.07)2

= $1,144.90

You earned an EXTRA $4.90 in Year 2 with compound over simple interest.

Future Value Single DepositFuture Value Single Deposit

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FV1 = P0(1+i)1

FV2 = P0(1+i)2

General Future Value Formula:FVn = P0 (1+i)n

or FVn = P0 (FVIFi,n)

General Future Value Formula

General Future Value Formula

etc.

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What factor do we use?

Single-Sum Future Value

Number Table A-1 of Discount Rate

Periods 2% 4% 6% 8% 10%

1 1,02000 1,04000 1,06000 1,08000 1,10000 2 1,04040 1,08160 1,12360 1,16640 1,21000 3 1,06121 1,12486 1,19102 1,25971 1,33100 4 1,08243 1,16986 1,26248 1,36049 1,46410 5 1,10408 1,21665 1,33823 1,46933 1,61051

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Number

of Discount RatePeriods 2% 4% 6% 8% 10%

1 1.02000 1.04000 1.06000 1.08000 1.10000 2 1.04040 1.08160 1.12360 1.16640 1.21000 3 1.06121 1.12486 1.19102 1.25971 1.33100 4 1.08243 1.16986 1.26248 1.36049 1.46410 5 1.10408 1.21665 1.33823 1.46933 1.61051

Table A-1- FV of 1$

$10,000 x 1.25971 = $12,597

Single-Sum Future Value

Present Value

Factor Future Value

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Example of FV of a Lump Sum

• How much money will you have in 5 years if you invest $100 today at a 10% rate of return?

1. Draw a timeline

2. Write out the formula using symbols:FVt = CF0 * (1+r)t

0 1 2 3

$100 ?i = 10%

4 5

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Example of FV of a Lump Sum

3. Substitute the numbers into the formula:FV = $100 * (1+.1)5

4. Solve for the future value:FV = $161.055. Check answer using a financial calculator:i = 10%n = 5PV = $100PMT = $0FV = ?

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PV0 = FV1 (1+i)-1

PV0 = FV2(1+i)-2

General Future Value Formula:PV0 = FVn (1+i)-n

or PV0 = FVn (PVIFi,n)

General Present Value Formula

General Present Value Formula

etc.

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Numberof Discount Rate

Periods 4% 6% 8% 10% 12%

2 .92456 .89000 .85734 .82645 .79719

4 .85480 .79209 .73503 .68301 .63552

6 .79031 .70496 .63017 .56447 .50663

8 .73069 .62741 .54027 .46651 .40388

Table A-2- PV of 1$


Single-Sum Present Value

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Periods 4% 6% 8% 10% 12%

2 .92456 .89000 .85734 .82645 .79719

4 .85480 .79209 .73503 .68301 .63552

6 .79031 .70496 .63017 .56447 .50663

8 .73069 .62741 .54027 .46651 .40388

Table A-2

Single-Sum Problems

$20,000 x .63552 = $12,710

Future Value Factor Present Value

Table A-2- PV of 1$

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Example of PV of a Lump Sum• How much would $100 received five years from now be worth today

if the current interest rate is 10%?1. Draw a timeline

The arrow represents the flow of money and thenumbers under the timeline represent the time period.

Note that time period zero is today.

0 1 2 3 4 5

$100

?

i = 10%

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2. Write out the formula using symbols:PV = CFt / (1+r)t

3. Insert the appropriate numbers:PV = 100 / (1 + .1)5

4. Solve the formula:PV = $62.09

5. Check using a financial calculator:FV = $100n = 5PMT = 0 i = 10%PV = ?

Example of PV of a Lump Sum

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Annuities

(1) Periodic payments or receipts (called rents) of the same amount,

(2) The same-length interval between such rents, and

(3) Compounding of interest once each interval.

Annuity requires the following:

Ordinary annuity - rents occur at the end of each period.

Annuity Due - rents occur at the beginning of each period.

Two Types

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Future Value of an Ordinary Annuity

Rents occur at the end of each period.

No interest during 1st period.

Annuities

0 1

Present Value

2 3 4 5 6 7 8

$20,000

20,000 20,000 20,000 20,000 20,000 20,000 20,000

Future Value

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Bayou Inc. will deposit $20,000 in a 12% fund at the end of each year for 8 years beginning December 31, Year 1. What amount will be in the fund immediately after the last deposit?

0 1

Present Value

What table do we use?


2 3 4 5 6 7 8

$20,000

20,000 20,000 20,000 20,000 20,000 20,000 20,000

Future Value

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Periods 4% 6% 8% 10% 12%

2 2.04000 2.06000 2.08000 2.10000 2.12000 4 4.24646 4.37462 4.50611 4.64100 4.77933 6 6.63298 6.97532 7.33592 7.71561 8.11519 8 9.21423 9.89747 10.63663 11.43589 12.29969

10 12.00611 13.18079 14.48656 15.93743 17.54874

Table A-3- FV of an annuity payment of 1$ per year



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Number


2 2.04000 2.06000 2.08000 2.10000 2.12000 4 4.24646 4.37462 4.50611 4.64100 4.77933 6 6.63298 6.97532 7.33592 7.71561 8.11519 8 9.21423 9.89747 10.63663 11.43589 12.29969

10 12.00611 13.18079 14.48656 15.93743 17.54874

Table A-3


$20,000 x 12.29969 = $245,994

Deposit Factor Future Value

Table A-3- FV of an annuity payment of 1$ per year

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Example of FV of an Annuity

2. Write out the formula using symbols:FVAt = PMT * {[(1+r)t –1]/r}

3. Substitute the appropriate numbers:FVA20 = $100 * {[(1+.15)20 –1]/.15

4. Solve for the FV:FVA20 = $100 * 102.4436

FVA20 = $10,244.36

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Example of FV of an Annuity

5. Check using calculator:Make sure that the calculator is set to one period per year PMT = $100 n = 20 i = 15% FV = ?

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Jaime Yuen wins $2,000,000 in the state lottery. She will be paid $100,000 at the end of each year for the next 20 years. How much has she actually won? Assume an appropriate interest rate of 8%.

0 1

Present Value

What table do we use?

2 3 4 19 20

$100,000

100,000

100,000

100,000

100,000

Present Value of an Ordinary Annuity

. . . . .

100,000

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Number


1 0.96154 0.94340 0.92593 0.90900 0.89286 5 4.45183 4.21236 3.99271 3.79079 3.60478

10 8.11090 7.36009 6.71008 6.14457 5.65022 15 11.11839 9.71225 8.55948 7.60608 6.81086 20 13.59033 11.46992 9.81815 8.51356 7.46944

Table A-4



Table A-4- PV of an annuity payment of 1$ per year

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Number


1 0.96154 0.94340 0.92593 0.90900 0.89286 5 4.45183 4.21236 3.99271 3.79079 3.60478

10 8.11090 7.36009 6.71008 6.14457 5.65022 15 11.11839 9.71225 8.55948 7.60608 6.81086 20 13.59033 11.46992 9.81815 8.51356 7.46944

Table 6-4


$100,000 x 9.81815 = $981,815

Receipt Factor Present Value

Table A-4- PV of an annuity payment of 1$ per year

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Formulas of Annuities

Present value of an annuity:

PVA = PMT * {[1-(1+r)-t]/r}

Future value of an annuity:

FVA t = PMT * {[(1+r)t –1]/r}

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How about a stream of payments that are NOT equal??

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• Future value of a cash flow stream: n

FV = S [CFt * (1+r)n-t] t=0

• Present value of a cash flow stream:–

n

– PV = S [CFt / (1+r)n-t] t=0

Formulas of Cash Flow stream

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Example of PV of a Cash Flow Stream

• Joe made an investment that will pay $100 the first year, $300 the second year, $500 the third year and $1000 the fourth year. If the interest rate is ten percent, what is the present value of this cash flow stream?

1. Draw a timeline:

0 1 2 3 4A-2 r=10% , n= 1

$100 $300 $500 $1000

i = 10%A-2 r=10% , n= 2

A-2 r=10% , n= 3

A-2 r=10% , n= 4

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Periods 4% 6% 8% 10% 12%

1 .92456 .89000 .85734 .82645 .79719

2 .85480 .79209 .73503 .68301 .63552

3 .79031 .70496 .63017 .56447 .50663

4 .73069 .62741 .54027 .46651 .40388

Table A-2- PV of 1$

CF1 * Factor 1 +

CF2 * Factor 2 +CF3 * Factor 3 +CF4* Factor 4 =

Present value of a cash flow stream: n

PV = S [CFt / (1+r)n-t] t=0


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2. Write out the formula using symbols:

n

PV = S [CFt / (1+r)t] t=0

ORPV = [CF1/(1+r)1]+[CF2/(1+r)2]+[CF3/(1+r)3]+[CF4/(1+r)4]

3. Substitute the appropriate numbers:

PV = [100/(1+.1)1]+[$300/(1+.1)2]+[500/(1+.1)3]+[1000/(1.1)4]

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4. Solve for the present value:PV = $90.91 + $247.93 + $375.66 + $683.01PV = $1397.51

5. Check using a calculator:– Make sure to use the appropriate rate of return, number of periods, and

future value for each of the calculations. To illustrate, for the first cash flow, you should enter FV=100, n=1, i=10, PMT=0, PV=?. Note that you will have to do four separate calculations.

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Example of FV of a Cash Flow Stream

• Joe made a decision to start saving money. He will pay $100 now year, $300 the first year, $500 the second year and $1000 the third year. If the interest rate is ten percent, what is the future value of this cash flow stream?

1. Draw a timeline:

0 1 2 3 4

A-1 r=10% , n= 1

$100 $300 $500 $1000

i = 10%

A-1 r=10% , n= 2

+

A-1 r=10% , n= 3

+

A-1 r=10% , n= 4

+

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Periods 2% 4% 6% 8% 10%

1 1,02000 1,04000 1,06000 1,08000 1,10000 2 1,04040 1,08160 1,12360 1,16640 1,21000 3 1,06121 1,12486 1,19102 1,25971 1,33100 4 1,08243 1,16986 1,26248 1,36049 1,46410

Table A-1- FV of 1$

CF3* Factor 1 +

CF2 * Factor 2 +CF1 * Factor 3 +CF0* Factor 4 =

Future value of a cash flow stream: nFV = S [CFt * (1+r)n-t]

Example of FV of a Cash Flow Stream

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Rule of Thumb

• The following are simple rules that you should always use no matter what type of TVM problem you are trying to solve:

1. Stop and think: Make sure you understand what the problem is asking. You will get the wrong answer if you are answering the wrong question.

2. Draw a representative timeline and label the cash flows and time periods appropriately.

3. Write out the complete formula using symbols first and then substitute the actual numbers to solve.

4. Check your answers using a calculator.• While these may seem like trivial and time consuming tasks, they will

significantly increase your understanding of the material and your accuracy rate.

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We will use the “Rule-of-72”.

Double Your Money!!!Double Your Money!!!

Quick! How long does it take to double $5,000 at a compound rate of 12% per year

(approx.)?

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Approx. Years to Double = 72 / i%

72 / 12% = 6 Years[Actual Time is 6.12 Years]

The “Rule-of-72”The “Rule-of-72”

Quick! How long does it take to double $5,000 at a compound rate of 12% per year

(approx.)?

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1. Read problem thoroughly2. Create a time line3. Put cash flows and arrows on time line4. Determine if it is a PV or FV problem5. Determine if solution involves a single CF, annuity stream(s), or mixed flow6. Solve the problem7. Check with financial calculator (optional)

Steps to Solve Time Value of Money Problems

Steps to Solve Time Value of Money Problems

Chapter 5 The Time Value of Money 1 Pr. Zoubida SAMLAL.

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