Investment Tools – Time Value of MoneyInvestment Tools – Time Value of Money

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Concepts Covered in This Section

– Future value

– Present value

– Perpetuities

– Annuities

– Uneven Cash Flows

– Rates of return

Time Value of Money

3

Time lines show timing of cash flows.

CF0 CF1 CF3CF2

0 1 2 3i%

Tick marks at ends of periods.• Time 0 is today; Time 1 is the end of Period 1; or

the beginning of Period 2.

90% of getting a Time Value problem correct is setting up the timeline correctly!!!

Interest Rate

Cash Flows

4

Finding FVs (moving to the right on a time line) is called compounding.

• Compounding involves earning interest on interest for investments of more than one period.

What’s the FV of an initial $100 after 3 What’s the FV of an initial $100 after 3 years if i = 10%?years if i = 10%?

FV = ?100

0 1 2 310%

Future Values

5

Single Sum Single Sum - Future & Present Value- Future & Present Value

• Assume that you can invest PV at interest rate i to receive future sum, FV• Similar reasoning leads to Present Value of a Future sum today.

1 2 30

FV1 = (1+i)PV

FV3 = (1+i)3PV

PV

FV2 = (1+i)2PV

1 2 30

PV = FV1/(1+i)

FV1

PV = FV2/(1+i)2

FV2

PV = FV3/(1+i)3

FV3

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PV =

FV

1+ i = FV

11+ i

nn n

n

FVn = PV(1 + i )n for given PV

$100 = 0.7513 = $75.13.1.10

PV = $1001

3

PV Calculation for $100 received in 3 years

if interest rate is 10%

Single Sum – FV & PV Formulas

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Question on PV of a given FVQuestion on PV of a given FV

Ex 1. An investor wants to have $1 million when she retires in 20 years. If she can earn a 10 percent annual return, compounded annually, on her investments, the lump-sum amount she would need to invest today to reach her goal is closest to:

A.$100,000.

B.$117,459.

C.$148,644.

D.$161,506.

• This is a single payment to be turned into a set future value FV=$1,000,000 in N=20 years time invested at r=10% interest rate.

PV =[ 1/(1+r) ]N FV

PV = [ 1/(1.10) ]20 $1,000,000

PV10 = [0.14864]($1,000,000)

PV10 = $148,644

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Perpetuities

Perpetuity is a series of constant payments, A, each period forever.

1 2 3 4 5 6 7

A

0

A A A A A A

Intuition: Present Value of a perpetuity is the amount that must invested today at the interest rate i to yield a payment of A each year without affecting the value of the initial investment.

PVperpetuity = [A/(1+i)t] = A [1/(1+i)t] = A/i

PV1 = A/(1+r)

PV2 = A/(1+r)2

PV3 = A/(1+r)3

PV4 = A/(1+r)4

etc. etc.

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• Regular or ordinary annuity is a finitefinite set of sequential cash flows, all with the same value A, which has a first cash flow that occurs one period from now.

• An annuity due is a finitefinite set of sequential cash flows, all with the same value A, which has a first cash flow that is paid immediatelyimmediately.

Annuities

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Time line for an ordinary annuity of $100 for 3 years.Time line for an ordinary annuity of $100 for 3 years.

$100 $100$100

0 1 2 3i%

Ordinary Annuity Timeline

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PMT PMT

0 1 2 3i%

PMT

Annuity Due

PV FV

Ordinary Annuity

PMT PMTPMT

0 1 2 3i%

Difference between an ordinary annuity and an annuity due?

Ordinary Annuity vs. Annuity Due

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Annuity Formula and Perpetuities

2 4 6 8 10 12 14

1. Perpetuity of A per period in Period 0 -- PV1 = A/i

A

0

A A A A A AA A AA AAA

2 4 6 8 10 12 14

2. Perpetuity of A per period in Period 8 -- PV8 = [1/(1+i)]8 x (A/i)

0

A A AAAA

2 4 6 8 10 12 14

3. Annuity of A for 8 periods -- PV =PV = PVPV11 – PV – PV88 = (A/i) = (A/i) x x { 1 – [1/(1+i)]{ 1 – [1/(1+i)]88 } }

A

0

A A AA A AA

Intuition: Formula for a N-period annuity of A is: PV of a Perpetuity of A today minus PV of a Perpetuity of A in period N

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Annuities & Perpetuities AgainAnnuities & Perpetuities Again

• Rather than memorize the annuity formula, it is easier to calculate it as the difference between two perpetuities with the same payment.

• PV of an N-period annuity of $A per period is:

PVN =

(A/i) x { 1 – [1/(1+i)]N}

• Calculating the PV of an annuity has 3 steps:

1. Calculate (A/i)– PV of a Perpetuity with

payments of $A per period.

2. Calculate [1/(1+i)]N

– Discount factor associated with end of the annuity.

3. Calculate PVN = (A/i) x { 1 - [1/(1 + i)]N }– I think this is easier

under pressure than memorizing the formula.

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Question on FV of Annuity DueQuestion on FV of Annuity Due

Ex 2.An individual deposits $10,000 at the beginning of each of the next 10 years, starting today, into an account paying 9 percent interest compounded annually. The amount of money in the account at the end of 10 years will be closest to:

A.$109,000.

B.$143,200.

C.$151,900.

D.$165,600.

• This is an annuity due of A=$10,000 for N=10 years at i=9% interest rate.

• Annuity due must be adjusted by (1+i) to reflect payment is made at beginning rather than end of period.

• Also must adjust PV formula by (1+i)N for FV of annuity.

PVN = (1+i)N(1+i)[(A/i) { 1 – [1/(1+i)]N}]

PV10 = (1.09)11 ($10K/.09) {1 – [1/1.09]10}

PV10 = (2.58)($111,111){1 – [0.42]}

PV10 = $165,601

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Time line for uneven CFs: $100 at end Time line for uneven CFs: $100 at end of Year 1 (t = 1), $200 at t=2, and$300 at of Year 1 (t = 1), $200 at t=2, and$300 at

the end of Year 3.the end of Year 3.

$100 $300 $200

0 1 2 3i%

Uneven Cash Flows

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Question on Uneven Cash FlowsQuestion on Uneven Cash Flows

Ex 3.An investment promises to pay $100 one year from today, $200 two years from today, and $300 three years from today. If the required rate of return is 14 percent, compounded annually, the value of this investment today is closest to:

A. $404.

B.$444.

C. $462.

D. $516.

• This is a set of unequal cash flows. You could do it as a sum of annuities but it is easier to calculate it directly in this case.

• Interest rate is i =14%.

PV = [ 1/(1+i) ]t FVt

PV = $100/(1.14) + $200/(1.14)2 + $300/(1.14)3

PV = $87.72 + $153.89 + $202.49

PV = $444.10

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Uneven Cash Flows

Intuition: PV of uneven cash flows is equal to the sum of the PV’s of regular cash flows that sum to the uneven cash flows.

2 4 6 8 10 12 14

1. Uneven cash Flows over 10 periods – PV = PV10 + PV45

0

$100 $100 $100$100 $100 $500$500 $500 $100$500

2. Annuity of $100 per period for 10 periods -- PV10 = { 1 - [1/(1+i)]10 } x (A/i)

2 4 6 8 10 12 140

$100 $100 $100$100 $100 $100$100 $100 $100$100

3. Annuity of $400 per period for 4 periods from period 5 -- PV4

5 = [1/(1+i)]5 x [ (A/i) x { 1 – [1/(1+i)]4 } ]

2 4 6 8 10 12 140

$400$400 $400 $400

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Comparison of Compounding Periods

Annually: FV3 = $100(1.10)3 = $133.10.

Semiannually: FV6 = $100(1.05)6 = $134.01.

0 1 2 310%

100 133.10

0 1 2 3

5%

4 5 6

134.01

1 2 30

100

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Questions on Time ValueQuestions on Time Value

• Develop an approach to problems on Time Value.

1. Draw the Time line for the cash flows.

2. Put in the cash flows from the problem.

3. Identify if single payment, annuity, annuity due, or perpetuity.

If uneven cash flows can you break it into sums of annuities?

4. Identify what is to be calculated – PV, FV, N or i ?

5. Write out the appropriate formula, put in values for the variables, and calculate.

• Best Study Tip: Do the problems, and then do some more and then do some more!! Practice using your calculator!!

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Possible Time Value QuestionsPossible Time Value Questions

• Present Value Formula1. Given FVN, i, N – solve for PVN

2. Given PVN , i, N – solve for FVN

3. Given PVN, FVN, N – solve for i

4. Given PVN, FVN, i – solve for N

• Perpetuity Formula1. Given A, i – solve for PVper

2. Given PVper, i – solve for A

3. Given PVper, A – solve for i

• Annuity Formula1. Given A, i, N – solve for PV

2. Given A, i, N – solve for FV

3. Given PV, i, N – solve for A

1

1

N

N NPV FVi

Perpetuity

APV

i

1

11

AnnuityN N

APV

i i

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Bonds and Their ValuationBonds and Their Valuation

• Key features of bonds• Bond valuation• Measuring yield• Assessing risk

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1.1. Par valuePar value:: Face amount; paid at maturity. Assume $1,000.

2.2. Coupon interest rateCoupon interest rate:: Stated interest rate. Multiply by par value to get dollars of interest. Often fixed but can float with market rate.

3.3. MaturityMaturity:: Years until bond must be repaid. Declines.

4.4. Issue dateIssue date:: Date when bond was issued.

5.5. Default riskDefault risk:: Risk that issuer will not make interest or principal payments.

Key Features of a BondKey Features of a Bond

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5

51 1 1

B t it i

i B B

CP FVP PV CF

k k

Valuing a 5-Period BondValuing a 5-Period Bond

Time = 0

1 2 3 4 5 6 7

Bond Price, PBt

Discounted Cash Flow ApproachDiscounted Cash Flow Approach• Current Bond Price = Present

value of all future Cash Flows (Interest & Principal) at required return, kB.

Coupon Interest, CPFace Value, FV

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The Right Discount FactorThe Right Discount Factor

• The discount rate (ki) is the opportunity cost of capital, i.e., the rate that could be earned on alternative investments of equal risk.

kkii = k* + IP + DRP + MRP + LP = k* + IP + DRP + MRP + LP

k* = Real rate of interest

IP = Inflation risk premium

DRP = Default risk premium

MRP = Maturity premium

LP = Liquidity risk premium

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What’s the value of a 10-year, 10% What’s the value of a 10-year, 10% coupon bond if kcoupon bond if kdd = 10%? = 10%?

$100 $100 $100 + $1,000VB = ?

0 1 2 1010% ...

= $90.91 + . . . + $38.55 + $385.54= $1,000.

Vk k

Bd d

$100 $1,

(1

000

(11 10 10 . . . +

$100

(1+ kd

++++ ) ) )

Bond Valuation Example

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Stocks and Their ValuationStocks and Their Valuation

• Features of common stock

• Determining common stock values

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• Represents ownership.

• Ownership implies control.

• Stockholders elect directors.

• Directors hire management.

• Management’s goal: Maximize stock price.

Features of Common StockFeatures of Common Stock

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Valuing Common StockValuing Common Stock

Time = 0

1 2 3 4 5 6 7

Stock Price, PSt

Uncertain Dividends, Dt+i

Dividend Discount ModelDividend Discount Model• Current Stock Price = Present value

of all future Expected Cash Flows (Dividends) at required return, kS.

1 1

es e t it i

i s

DP PV CF

k

29

P

D

k

D

k

D

k

D

ks s s s

01

12

23

31 1 1 1

. . .

Constant Growth stockConstant Growth stock • One whose dividends are expected to grow forever at a constant rate, g.• Can link this to earnings by assuming that firm pays out a fixed percentage of earnings as dividends

• i.e. Dt = k x Et where k equals payout ratio

Stock Value = PV of Dividends