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Time Value of Money Bond Valuation Risk and Return Stock ValuationPowerPoint Presentation

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.

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

0

1

2

3

10%

100

FV = ?

FV = ?

FV = ?

Finding FVs (moving to the right

on a time line) is called compounding.

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.

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

0

1

2

3

10%

100

100

100

110

121

FV = 331

N 10%?

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.

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

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

0

1

2

10%

100

PV = ?

Solve FV 10%?n = PV(1 + i )n for PV:

2

1

PV

=

$100

=

$100

PVIF

i,

n

1.10

=

$100

0.8264

=

$82.64.

N 10%?

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.

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%?

months before retirement

years after retirement

1

2

360

1

2

19

20

0

...

...

PMT

PMT

PMT

-145k

-145k

-145k

-145k

How much must you have in your account on the day you retire if i = 10%?

years after retirement

0

1

2

19

20

...

...

-145k

-145k

-145k

-145k

How much do you need on this date?

N if

I/YR

PV

FV

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

20 10 -145000 0

PMT

OUTPUT

1,234,467

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.

months before retirement

1

2

360

0

...

...

PMT

PMT

PMT

N order to have the $1,234,467 in your account?

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.

Key Features of a Bond order to have the $1,234,467 in your account?

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 order to have the $1,234,467 in your account?

must be repaid. Declines.

4. Issue date: Date when bond

was issued.

PV annuity order to have the $1,234,467 in your account?

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

10 10 100 1000

N I/YR PV PMT FV

-1,000

OUTPUT

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

INPUTS

10 13 100 1000

N I/YR PV PMT FV

-837.21

OUTPUT

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

What would happen if inflation fell, and r r =d declined to 7%?

INPUTS

10 7 100 1000

N I/YR PV PMT FV

-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 ($) maturity. What would happen to its value over time if the required rate of return remained at 10%, or at 13%,

rd = 7%.

1,372

1,211

rd = 10%.

M

1,000

837

rd = 13%.

775

30 25 20 15 10 5 0

Years remaining to Maturity

- At maturity, the value of any bond must equal its par value. maturity. What would happen to its value over time if the required rate of return remained at 10%, or at 13%,
- 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 maturity. What would happen to its value over time if the required rate of return remained at 10%, or at 13%,

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

Assume the FollowingInvestment AlternativesWhat is unique about maturity. What would happen to its value over time if the required rate of return remained at 10%, or at 13%,the T-bill return?

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

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.

Calculate the expected rate of return on each alternative. the economy?

^

r = expected rate of return.

^

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

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

+ 0.10(50%) = 17.4%.

^ the economy?

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?

. the economy?

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%.

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

^

/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.

- Standard deviation the economy?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 Return versus Risk the economy?

Expected

Risk,

CV

Security

return

HT 17.4% 20.0% 1.2

Market 15.0 15.3 1.0

USR 13.8 18.8 1.4

T-bills 8.0 0.0 0.0

Collections 1.74 13.4 7.7

- Which alternative is best?

Portfolio Risk and Return the economy?

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

^

Calculate rp and p.

Portfolio Return, r the economy?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.

Alternative Method the economy?

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...)

- the economy?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.

Portfolio standard deviation in general the economy?

p= Portfolio standard deviation.

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

Two-Stock Portfolios the economy?

- 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?

Portfolio beta the economy?

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.

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 rp would remain relatively constant.

^

as more randomly picked stocks were added?p (%)

Company-Specific (Diversifiable) Risk

35

Stand-Alone Risk, p

20

0

Market Risk

10 20 30 40 2,000+

# Stocks in Portfolio

Stand-alone Market Diversifiable as more randomly picked stocks were added?

= + .

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 randomly picked stocks were added?

- 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? with its risk?

- 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 are betas calculated? with its risk?

- 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.

_ with its risk?

ri

Beta Illustration

Illustration of beta calculation:

Regression line:

ri = -2.59 + 1.44 rM .

.

20

15

10

5

^

^

.

Year rM ri

1 15% 18%

2 -5 -10

3 12 16

_

-5 0 5 10 15 20

rM

-5

-10

.

How is beta calculated? with its risk?

- 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.

How is beta interpreted? with its risk?

- 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?

_ with its risk?

b = 1.30

ri

HT

40

20

b = 0

T-Bills

_

rM

-20 0 20 40

-20

b = -0.87

Collections

Regression Lines of Three Alternatives

Expected Return versus Market Risk with its risk?

Expected

Risk,b

Security

return

HT 17.4% 1.30

Market 15.0 1.00

USR 13.8 0.89

T-bills 8.0 0.00

Collections 1.74 -0.87

- Which of the alternatives is best?

Use the SML to calculate each with its risk?alternative’s required return.

- 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%.

^

Required Rates of Return with its risk?

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%.

Expected versus Required Returns with its risk?

^

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: with its risk?

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

What is the required rate of return with its risk?on the HT/Collections portfolio?

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 with its risk?

.

What is a constant growth stock?

One whose dividends are expected to

grow forever at a constant rate, g.

What happens if g > r with its risk?s?

- 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.

Assume beta = with its risk?1.2, rRF = 7%, and rM = 12%. What is the required rate of return on the firm’s stock?

Use the SML to calculate rs:

rs = rRF + (rM - rRF)bFirm

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

= 13%.

D with its risk?0 was $2.00 and g is a constant 6%. Find the expected dividends for the next 3 years, and their PVs. rs = 13%.

0

1

2

3

4

g = 6%

D0 = 2.00

2.12

2.2472

2.3820

13%

1.8761

1.7599

1.6508

D with its risk?0(1 + g)

rs - g

D1

rs - g

= =

$2.12

$2.12

= = = $30.29.

0.13 - 0.06

0.07

What’s the stock’s market value? D0 = 2.00, rs = 13%, g = 6%.Constant growth model:

Rearrange model to rate of return form: with its risk?

^

Then, rs = $2.12/$30.29 + 0.06

= 0.07 + 0.06 = 13%.

If we have supernormal growth of with its risk?30% for 3 yrs, then a long-run constant g = 6%, what is P0? rs is still 13%.

^

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

Nonconstant growth followed by constant with its risk?

growth:

0

1

2

3

4

rs=13%

g = 30%

g = 30%

g = 30%

g = 6%

D0 = 2.00 2.60 3.38 4.394 4.6576

2.3009

2.6470

3.0453

46.1140

^

54.1072 = P0

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