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Time Value of Money Bond Valuation Risk and Return Stock Valuation

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WEB CHAPTER 28Basic Financial Tools: A Review

- Time Value of Money
- Bond Valuation
- Risk and Return
- Stock Valuation

Time lines show timing of cash flows.

0

1

2

3

i%

CF0

CF1

CF2

CF3

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

0

1

2 Year

i%

100

0

1

2

3

i%

100

100

100

0

1

2

3

10%

100

FV = ?

FV = ?

FV = ?

Finding FVs (moving to the right

on a time line) is called compounding.

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.

After 3 years:

FV3= PV(1 + i)3

= $100(1.10)3

= $133.10.

In general,

FVn= PV(1 + i)n.

0

1

2

3

10%

100

110

0

1

2

3

10%

100

100

100

110

121

FV= 331

N

I/YR

PV

PMT

FV

Financial Calculator Solution

INPUTS

3 10 0 -100

331.00

OUTPUT

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

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

0

1

2

10%

100

PV = ?

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

2

1

PV

=

$100

=

$100

PVIF

i,

n

1.10

=

$100

0.8264

=

$82.64.

0

1

2

3

10%

100

100

100

90.91

82.64

75.13

248.69 = PV

N

I/YR

PV

PMT

FV

INPUTS

3 10 100 0

OUTPUT

-248.69

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

months before retirement

years after retirement

1

2

360

1

2

19

20

0

...

...

PMT

PMT

PMT

-145k

-145k

-145k

-145k

years after retirement

0

1

2

19

20

...

...

-145k

-145k

-145k

-145k

How much do you need on this date?

N

I/YR

PV

FV

INPUTS

2010-145000 0

PMT

OUTPUT

1,234,467

You need $1,234,467

on this date.

months before retirement

1

2

360

0

...

...

PMT

PMT

PMT

N

I/YR

PV

FV

You need a payment such that the future value of a 360-period annuity earning 10%/12 per period is $1,234,467.

INPUTS

360 10/12 0 1234467

PMT

OUTPUT

546.11

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

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…)

3.Maturity: Years until bond

must be repaid. Declines.

4.Issue date: Date when bond

was issued.

PV annuity

PV maturity value

PV annuity

$ 614.46

385.54

$1,000.00

=

=

=

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

INPUTS

1010 100 1000

NI/YR PV PMTFV

-1,000

OUTPUT

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

INPUTS

1013 100 1000

NI/YR PV PMTFV

-837.21

OUTPUT

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

INPUTS

10 7 100 1000

NI/YR PV PMTFV

-1,210.71

OUTPUT

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

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%?

Bond Value ($)

rd = 7%.

1,372

1,211

rd = 10%.

M

1,000

837

rd = 13%.

775

3025 20 15 10 5 0

Years remaining to Maturity

- 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 rd remains constant.

Economy

Prob.

T-Bill

HT

Coll

USR

MP

Recession0.10 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 0.108.0 50.0 -20.0 30.0 43.0

1.00

- The T-bill will return 8% regardless of the state of the economy.
- Is the T-bill riskless? Explain.

- 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.

^

r = expected rate of return.

^

rHT = 0.10(-22%) + 0.20(-2%)

+ 0.40(20%) + 0.20(35%)

+ 0.10(50%) = 17.4%.

^

r

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?

= Standard deviation.

.

.

HT:

= ((-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%.

T-bills = 0.0%.

Coll=13.4%.

USR=18.8%.

M=15.3%.

HT = 20.0%.

^

/r.

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.

Prob.

T-bill

USR

HT

0

8

13.8

17.4

Rate of Return (%)

- 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.

Expected

Risk,

CV

Security

return

HT 17.4% 20.0%1.2

Market15.015.3 1.0

USR13.8 18.81.4

T-bills8.00.00.0

Collections1.74 13.4 7.7

- Which alternative is best?

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

^

Calculate rp and p.

^

^

rp is a weighted average:

n

^

^

rp = wiri

i = 1

^

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

^

^

^

rp is between rHT and rColl.

Estimated Return

Economy

Prob.

HT

Coll.

Port.

Recession 0.10-22.0% 28.0% 3.0%

Below avg. 0.20 -2.0 14.7 6.4

Average 0.40 20.0 0.0 10.0

Above avg. 0.20 35.0 -10.0 12.5

Boom 0.10 50.0 -20.0 15.0

^

rp = (3.0%)0.10 + (6.4%)0.20 + (10.0%)0.40

+ (12.5%)0.20 + (15.0%)0.10 = 9.6%.

(More...)

- 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.

p= Portfolio standard deviation.

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

- 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?

bp = Portfolio beta

bp = w1b1 + w2b2

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.

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

^

Prob.

Large

2

1

0

15

Return

135% ; Large20%.

p (%)

Company-Specific (Diversifiable) Risk

35

Stand-Alone Risk, p

20

0

Market Risk

102030 40 2,000+

# Stocks in Portfolio

= + .

risk risk risk

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.

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.

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

- 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.

How is market risk measured for individual securities?

- 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?

- 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.

_

ri

Beta Illustration

Illustration of beta calculation:

Regression line:

ri = -2.59 + 1.44 rM .

.

20

15

10

5

^

^

.

YearrMri

115% 18%

2 -5-10

312 16

_

-505101520

rM

-5

-10

.

- 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.

- 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?

_

b = 1.30

ri

HT

40

20

b = 0

T-Bills

_

rM

-2002040

-20

b = -0.87

Collections

Regression Lines of Three Alternatives

Expected

Risk,b

Security

return

HT17.4%1.30

Market15.01.00

USR13.80.89

T-bills 8.00.00

Collections1.74 -0.87

- Which of the alternatives is best?

- The Security Market Line (SML) is part of the Capital Asset Pricing Model (CAPM).
- SML: ri = rRF + (rM - rRF)bi .
- Assume kRF = 8%; rM = rM = 15%.
- RPM = rM - rRF = 15% - 8% = 7%.

^

rHT = 8.0% + (15.0% - 8.0%)(1.30)

= 8.0% + (7%)(1.30)

= 8.0% + 9.1%= 17.1%.

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

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

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

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

^

r

r

HT 17.4% 17.1% Undervalued

Market 15.0 15.0 Fairly valued

USR 13.8 14.2 Overvalued

T-bills 8.0 8.0 Fairly valued

Coll 1.74 1.9 Overvalued

SML:

ri = 8% + (15% - 8%) bi.

ri (%)

.

HT

.

.

rM = 15

rRF = 8

Market

.

USR

T-bills

.

Coll.

Risk, bi

-1 0 1 2

SML and Investment Alternatives

rp= Weighted average r

= 0.5(17%) + 0.5(2%) = 9.5%.

Or use SML:

bp= 0.22 (Slide 28-45)

rp= rRF + (rM - rRF) bp

= 8.0% + (15.0% - 8.0%)(0.22)

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

Stock Value = PV of Dividends

.

What is a constant growth stock?

One whose dividends are expected to

grow forever at a constant rate, g.

.

.

.

If g is constant, then:

.

$

0.25

If g > r, P0 = negative

0

Years (t)

- If rs< g, get negative stock price, which is nonsense.
- We can’t use model unless (1) g rs and (2) g is expected to be constant forever. Because g must be a long-term growth rate, it cannot be rs.

Use the SML to calculate rs:

rs= rRF + (rM - rRF)bFirm

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

= 13%.

0

1

2

3

4

g = 6%

D0 = 2.00

2.12

2.2472

2.3820

13%

1.8761

1.7599

1.6508

D0(1 + g)

rs - g

D1

rs - g

= =

$2.12

$2.12

= = = $30.29.

0.13 - 0.06

0.07

Constant growth model:

Rearrange model to rate of return form:

^

Then, rs= $2.12/$30.29 + 0.06

= 0.07 + 0.06 = 13%.

^

- Can no longer use constant growth model.
- However, growth becomes constant after 3 years.

Nonconstant growth followed by constant

growth:

0

1

2

3

4

rs=13%

g = 30%

g = 30%

g = 30%

g = 6%

D0 = 2.00 2.603.38 4.394 4.6576

2.3009

2.6470

3.0453

46.1140

^

54.1072 = P0