Time Value of Money Bond Valuation Risk and Return Stock Valuation

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Copyright © 2002 South-Western

Time Value of Money

Bond Valuation

Risk and Return

Stock Valuation

WEB CHAPTER 28Basic Financial Tools: A Review

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Time lines show timing of cash flows.

CF0 CF1 CF3CF2

0 1 2 3i%

Tick marks at ends of periods, so Time 0 is today; Time 1 is the end of Period 1; or the beginning of Period 2.

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Time line for a $100 lump sum due at the end of Year 2.

100

0 1 2 Yeari%

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

100 100100

0 1 2 3i%

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What’s the FV of an initial $100 after1, 2, and 3 years if i = 10%?

FV = ?

0 1 2 310%

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

100 FV = ? FV = ?

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After 1 year:

FV1 = PV + INT1 = PV + PV (i)= PV(1 + i)= $100(1.10)= $110.00.

After 2 years:

FV2 = PV(1 + i)2

= $100(1.10)2

= $121.00.

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After 3 years:

FV3 = PV(1 + i)3

= $100(1.10)3

= $133.10.

In general,

FVn = PV(1 + i)n.

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What’s the FV in 3 years of $100 received in Year 2 at 10%?

100

0 1 2 310%

110

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What’s the FV of a 3-year ordinary annuity of $100 at 10%?

100 100100

0 1 2 310%

110 121FV = 331

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3 10 0 -100

331.00N I/YR PV PMT FV

Financial Calculator Solution

Have payments but no lump sum PV, so enter 0 for present value.

INPUTS

OUTPUT

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10%

What’s the PV of $100 due in 2 years if i = 10%?

Finding PVs is discounting, and it’s the reverse of compounding.

100

0 1 2

PV = ?

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Solve FVn = PV(1 + i )n for PV:

PV =

FV

1+ i = FV

11+ i

nn n

n

PV = $1001

1.10 = $100 PVIF

= $100 0.8264 = $82.64.

i,n

2

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What’s the PV of this ordinary annuity?

100 100100

0 1 2 310%

90.91

82.64

75.13248.69 = PV

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Have payments but no lump sum FV, so enter 0 for future value.

3 10 100 0N I/YR PV PMT FV

-248.69

INPUTS

OUTPUT

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How much do you need to save each month for 30 years in order to retire on $145,000 a year for 20 years, i = 10%?

0 360 2 201 2

PMT PMT PMT

...

1 19

months before retirement years after retirement

-145k -145k -145k -145k

...

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How much must you have in your account on the day you retire if

i = 10%?

How much do you need on this date?

2 20...

1 19

years after retirement

-145k -145k -145k -145k

...

0

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You need the present value of a20- year 145k annuity--or $1,234,467.

20 10 -145000 0

N I/YR PV FVPMT

1,234,467

INPUTS

OUTPUT

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How much do you need to save each month for 30 years in order to have the

$1,234,467 in your account?

You need $1,234,467

on this date.0 3601 2

PMT PMT PMT

...

months before retirement

...

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You need a payment such that the future value of a 360-period annuity

earning 10%/12 per period is $1,234,467.

360 10/12 0 1234467

N I/YR PV FVPMT

546.11

INPUTS

OUTPUT

It will take an investment of $546.11 per month to fund your retirement.

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Key Features of a Bond

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

2. Coupon interest rate: Stated interest rate. Multiply by par value to get dollars of interest.

Generally fixed.

(More…)

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3. Maturity: Years until bondmust be repaid. Declines.

4. Issue date: Date when bondwas issued.

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10 10 100 1000N I/YR PV PMT FV

-1,000

The bond consists of a 10-year, 10% annuity of $100/year plus a $1,000 lump sum at t = 10:

$ 614.46 385.54

$1,000.00

PV annuity PV maturity value PV annuity

===

INPUTS

OUTPUT

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10 13 100 1000N I/YR PV PMT FV

-837.21

When kd rises, above the coupon rate, the bond’s value falls below par, so it sells at a discount.

What would happen if expected inflation rose by 3%, causing k = 13%?

INPUTS

OUTPUT

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What would happen if inflation fell, and kd declined to 7%?

10 7 100 1000N I/YR PV PMT FV

-1,210.71

If coupon rate > kd, price rises above par, and bond sells at a premium.

INPUTS

OUTPUT

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The bond was issued 20 years ago and now has 10 years to maturity.

What would happen to its value over time if the required rate of return

remained at 10%, or at 13%,or at 7%?

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M

Bond Value ($)

Years remaining to Maturity

1,372

1,211

1,000

837

775

30 25 20 15 10 5 0

kd = 7%.

kd = 13%.

kd = 10%.

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At maturity, the value of any bond must equal its par value.

The value of a premium bond would decrease to $1,000.

The value of a discount bond would increase to $1,000.

A par bond stays at $1,000 if kd remains constant.

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Assume the FollowingInvestment Alternatives

Economy Prob. T-Bill HT Coll USR MP

Recession 0.10 8.0% -22.0% 28.0% 10.0% -13.0%

Below avg. 0.20 8.0 -2.0 14.7 -10.0 1.0

Average 0.40 8.0 20.0 0.0 7.0 15.0

Above avg. 0.20 8.0 35.0 -10.0 45.0 29.0

Boom 0.10 8.0 50.0 -20.0 30.0 43.0

1.00

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What is unique about the T-bill return?

The T-bill will return 8% regardless of the state of the economy.

Is the T-bill riskless? Explain.

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Do the returns of HT and Collections move with or counter to the economy?

HT moves with the economy, so it is positively correlated with the economy. This is the typical situation.

Collections moves counter to the economy. Such negative correlation is unusual.

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Calculate the expected rate of return on each alternative.

.k = k Pi i

i=1

n

k = expected rate of return.

kHT = 0.10(-22%) + 0.20(-2%) + 0.40(20%) + 0.20(35%) + 0.10(50%) = 17.4%.

^

^

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k

HT 17.40%

Market 15.00

USR 13.80

T-bill 8.00

Collections 1.74

^

HT has the highest rate of return.

Does that make it best?

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What is the standard deviationof returns for each alternative?

= Variance = 2

n

1=ii

2i P)k̂(k =

= Standard deviation.

.

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n

1=ii

2i P)k̂(k =

T-bills = 0.0%.HT = 20.0%.

Coll = 13.4%.USR = 18.8%. M = 15.3%.

.

= ((-22 - 17.4)2 0.10 + (-2 - 17.4)2 0.20 + (20 - 17.4)2 0.40 + (35 - 17.4)2 0.20 + (50 - 17.4)2 0.10)1/2 = 20.0%.

HT:

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The coefficient of variation (CV) is calculated as follows:

CVHT = 20.0%/17.4% = 1.15 1.2.

CVT-bills = 0.0%/8.0% = 0.

CVColl = 13.4%/1.74% = 7.7.

CVUSR = 18.8%/13.8% = 1.36 1.4.

CVM = 15.3%/15.0% = 1.0.

/k.^

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Prob.

Rate of Return (%)

T-bill

USR

HT

0 8 13.8 17.4

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Standard deviation measures the stand-alone risk of an investment.

The larger the standard deviation, the higher the probability that returns will be far below the expected return.

Coefficient of variation is an alternative measure of stand-alone risk.

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Expected Return versus Risk

SecurityExpected

return Risk,

HT 17.4% 20.0% 1.2Market 15.0 15.3 1.0USR 13.8 18.8 1.4T-bills 8.0 0.0 0.0Collections 1.74 13.4 7.7

Which alternative is best?

CV

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Portfolio Risk and Return

Assume a two-stock portfolio with $50,000 in HT and $50,000 in Collections.

Calculate kp and p.^

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Portfolio Return, kp

kp is a weighted average:

kp = 0.5(17.4%) + 0.5(1.74%) = 9.6%.

kp is between kHT and kColl.

^

^

^

^

^ ^

^ ^

kp = wikin

i = 1

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Alternative Method

kp = (3.0%)0.10 + (6.4%)0.20 + (10.0%)0.40 + (12.5%)0.20 + (15.0%)0.10 = 9.6%.

^

Estimated Return

Economy Prob. HT Coll. Port.

Recession 0.10 -22.0% 28.0% 3.0%Below avg. 0.20 -2.0 14.7 6.4Average 0.40 20.0 0.0 10.0Above avg. 0.20 35.0 -10.0 12.5Boom 0.10 50.0 -20.0 15.0

(More...)

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p = ((3.0 - 9.6)2 0.10 + (6.4 - 9.6)2 0.20 + (10.0 - 9.6)2 0.40 + (12.5 - 9.6)2 0.20 + (15.0 - 9.6)2 0.10)1/2 = 3.3%.

p is much lower than:

either stock (20% and 13.4%).

average of HT and Coll (16.7%).

The portfolio provides average return but much lower risk. The key here is negative correlation.

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Portfolio standard deviation in general

rww2ww = 2,1212122

22

21

21p

p = Portfolio standard deviation.

Where w1 and w2 are portfolio weights and r1,2 is the correlation coefficient between stock 1 and 2.

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Two-Stock Portfolios

Two stocks can be combined to form a riskless portfolio if r = -1.0.

Risk is not reduced at all if the two stocks have r = +1.0.

In general, stocks have r 0.65, so risk is lowered but not eliminated.

Investors typically hold many stocks.

What happens when r = 0?

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Portfolio beta

bp = Portfolio beta

Where w1 and w2 are portfolio weights, and b1 and b2 are stock betas. For our portfolio of 50% HT and 50% Collections,

bp = 0.5(1.30) + 0.5(-0.87) = 0.215 0.22.

bp = w1b1 + w2b2

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What would happen to the riskiness of an average portfolio as more randomly

picked stocks were added?

p would decrease because the added stocks would not be perfectly correlated, but kp would remain relatively constant.

^

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Large

0 15

Prob.

2

1

1 35% ; Large 20%.Return

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# Stocks in Portfolio10 20 30 40 2,000+

Company-Specific (Diversifiable) Risk

Market Risk

20

0

Stand-Alone Risk, p

p (%)

35

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Stand-alone Market Diversifiable

Market risk is that part of a security’s stand-alone risk that cannot be eliminated by diversification.

Firm-specific, or diversifiable, risk is that part of a security’s stand-alone risk that can be eliminated by diversification.

risk risk risk

= + .

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Conclusions

As more stocks are added, each new stock has a smaller risk-reducing impact on the portfolio.

p falls very slowly after about 40 stocks are included. The lower limit for p is about 20% = M .

By forming well-diversified portfolios, investors can eliminate about half the riskiness of owning a single stock.

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No. Rational investors will minimize risk by holding portfolios.

They bear only market risk, so prices and returns reflect this lower risk.

The one-stock investor bears higher (stand-alone) risk, so the return is less than that required by the risk.

Can an investor holding one stock earn a return commensurate with its risk?

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Market risk, which is relevant for stocks held in well-diversified portfolios, is defined as the contribution of a security to the overall riskiness of the portfolio.

It is measured by a stock’s beta coefficient, which measures the stock’s volatility relative to the market.

What is the relevant risk for a stock held in isolation?

How is market risk measured for individual securities?

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How are betas calculated?

Run a regression with returns on the stock in question plotted on the Y- axis and returns on the market portfolio plotted on the X-axis.

The slope of the regression line, which measures relative volatility, is defined as the stock’s beta coefficient, or b.

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Year kM ki

1 15% 18%

2 -5 -10

3 12 16

.

.

.ki

_

kM

_-5 0 5 10 15 20

20

15

10

5

-5

-10

Illustration of beta calculation:

Regression line:ki = -2.59 + 1.44 kM .^ ^

Beta Illustration

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How is beta calculated?

The regression line, and hence beta, can be found using a calculator with a regression function or a spreadsheet program. In this example, b = 1.44.

Analysts typically use five years’ of monthly returns to establish the regression line.

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If b = 1.0, stock has average risk.

If b > 1.0, stock is riskier than average.

If b < 1.0, stock is less risky than average.

Most stocks have betas in the range of 0.5 to 1.5.

Can a stock have a negative beta?

How is beta interpreted?

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HT

T-Bills

b = 0

ki

_

kM

_-20 0 20 40

40

20

-20

b = 1.30

Collectionsb = -0.87

Regression Lines of Three Alternatives

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Expected Return versus Market Risk

SecurityExpected

return Risk,b

HT 17.4% 1.30Market 15.0 1.00USR 13.8 0.89T-bills 8.0 0.00Collections 1.74 -0.87

Which of the alternatives is best?

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Use the SML to calculate eachalternative’s required return.

The Security Market Line (SML) is part of the Capital Asset Pricing Model (CAPM).

SML: ki = kRF + (kM - kRF)bi .

Assume kRF = 8%; kM = kM = 15%.

RPM = kM - kRF = 15% - 8% = 7%.

^

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Required Rates of Return

kHT = 8.0% + (15.0% - 8.0%)(1.30)= 8.0% + (7%)(1.30)= 8.0% + 9.1% = 17.1%.

kM = 8.0% + (7%)(1.00) = 15.0%.

kUSR = 8.0% + (7%)(0.89) = 14.2%.

kT-bill = 8.0% + (7%)(0.00) = 8.0%.

kColl = 8.0% + (7%)(-0.87) = 1.9%.

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Expected versus Required Returns

^ k k HT 17.4% 17.1% Undervalued Market 15.0 15.0 Fairly valuedUSR 13.8 14.2 OvervaluedT-bills 8.0 8.0 Fairly valuedColl 1.74 1.9 Overvalued

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..Coll.

.HT

T-bills

.USR

kM = 15

kRF = 8

-1 0 1 2

.

SML: ki = 8% + (15% - 8%) bi.

ki (%)

Risk, bi

SML and Investment Alternatives

Market

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What is the required rate of returnon the HT/Collections portfolio?

kp = Weighted average k= 0.5(17%) + 0.5(2%) = 9.5%.

Or use SML:

bp = 0.22 (Slide 2-45)

kp = kRF + (kM - kRF) bp

= 8.0% + (15.0% - 8.0%)(0.22)= 8.0% + 7%(0.22) = 9.5%.

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P

D

k

D

k

D

k

D

ks s s s

01

12

23

31 1 1 1

. . .

One whose dividends are expected togrow forever at a constant rate, g.

Stock Value = PV of Dividends

What is a constant growth stock?

.

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For a constant growth stock,

t

tt

202

101

g1DD

g1DD

g1DD

gk

D

gk

g1DP̂

s

1

s

00

If g is constant, then:

.

.

.

.

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D D gtt 0 1

PVD

D

kt

tt

1

P PVDt0

$

0.25

Years (t)0

If g > k, P0 = negative

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What happens if g > ks?

If ks< g, get negative stock price, which is nonsense.

We can’t use model unless (1) g ks and (2) g is expected to be constant forever. Because g must be a long-term growth rate, it cannot be ks.

.PD

k gg

s0

1

requires ks

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Assume beta = 1.2, kRF = 7%, and kM = 12%. What is the required rate of

return on the firm’s stock?

ks = kRF + (kM - kRF)bFirm

= 7% + (12% - 7%) (1.2)= 13%.

Use the SML to calculate ks:

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D0 was $2.00 and g is a constant 6%. Find the expected dividends for the

next 3 years, and their PVs. ks = 13%.

0 1

2.2472

2

2.3820

3g = 6% 4

1.87611.75991.6508

D0 = 2.0013%

2.12

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What’s the stock’s market value? D0 = 2.00, ks = 13%, g = 6%.

Constant growth model:

= = = $30.29.0.13 - 0.06

$2.12 $2.12

0.07

0P̂ = =D0(1 + g)

ks - gD1

ks - g

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Rearrange model to rate of return form:

.PD

k g

D

Pg

s0

1 1

0

to ks

Then, ks = $2.12/$30.29 + 0.06= 0.07 + 0.06 = 13%.

^

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If we have supernormal growth of 30% for 3 yrs, then a long-run constant

g = 6%, what is P0? ks is still 13%.

Can no longer use constant growth model.

However, growth becomes constant after 3 years.

^

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Nonconstant growth followed by constantgrowth:

0

2.3009

2.6470

3.0453

46.1140

1 2 3 4ks=13%

54.1072 = P0

g = 30% g = 30% g = 30% g = 6%

D0 = 2.00 2.60 3.38 4.394 4.6576

^

$66.53770.060.13

$4.65763

P

ˆ

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Time Value of Money Bond Valuation Risk and Return Stock Valuation