Time Value of Money

1
WEB CHAPTER 28Basic Financial Tools A Review

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

2

• Time lines show timing of cash flows.

0
1
2
3
i
CF0
CF1
CF3
CF2
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.
3
Time line for a 100 lump sum due at the end of
Year 2.
0
1
2 Year
i
100
4
Time line for an ordinary annuity of 100 for 3
years.
0
1
2
3
i
100
100
100
5
Whats the FV of an initial 100 after1, 2, and
3 years if i 10?
0
1
2
3
10
FV ?
100
FV ?
FV ?
Finding FVs (moving to the right on a time line)
is called compounding.
6
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.
7
After 3 years
FV3 PV(1 i)3 100(1.10)3 133.10.
In general,
FVn PV(1 i)n.
8
Whats the FV in 3 years of 100 received in Year
2 at 10?
0
1
2
3
10
100
110
9
Whats the FV of a 3-year ordinary annuity of
100 at 10?
0
1
2
3
10
100
100
100
110 121 FV 331
10
Financial Calculator Solution
INPUTS
3 10 0 -100 331.00
OUTPUT
Have payments but no lump sum PV, so enter 0 for
present value.
11
Whats the PV of 100 due in 2 years if i 10?
Finding PVs is discounting, and its the reverse
of compounding.
0
1
2
10
100
PV ?
12
Solve FVn PV(1 i )n for PV
2
1
?
?
?
?
?
PV

100


100
PVIF
?
?
?
i,
n
1.10
?
?


100
0.8264


82.64.
13
Whats the PV of this ordinary annuity?
0
1
2
3
10
100
100
100
90.91
82.64
75.13
248.69 PV
14
INPUTS
3 10 100 0
OUTPUT
-248.69
Have payments but no lump sum FV, so enter 0 for
future value.
15
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
0
360
2
20
1
2
1
19
...
...
PMT
PMT
PMT
-145k
-145k
-145k
-145k
16
How much must you have in your account on the day
you retire if i 10?
years after retirement
2
20
1
19
0
...
...
-145k
-145k
-145k
-145k
How much do you need on this date?
17
You need the present value of a20- year 145k
annuity--or 1,234,467.
INPUTS
20 10 -145000 0
PMT
OUTPUT
1,234,467
18
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
0
360
1
2
...
...
PMT
PMT
PMT
19
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.
20
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)
21
3. Maturity Years until bond must be repaid.
Declines. 4. Issue date Date when bond was
issued.
22
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
23
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 bonds
value falls below par, so it sells at a discount.
24
What would happen if inflation fell, and rd
declined to 7?
INPUTS
10 7 100 1000 N I/YR PV
PMT FV -1,210.71
OUTPUT
If coupon rate gt rd, price rises above par, and
bond sells at a premium.
25
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?
26
Bond Value ()
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
27

• 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.

28
Assume the FollowingInvestment Alternatives
29
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.

30
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.

31
Calculate the expected rate of return on each
alternative.

r expected rate of return.

rHT 0.10(-22) 0.20(-2) 0.40(20)
0.20(35) 0.10(50) 17.4.
32

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?

33
What is the standard deviationof returns for
each alternative?
Standard deviation.
.
34
.
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.
35
The coefficient of variation (CV) 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.

36
Prob.
T-bill
USR
HT
0
8
13.8
17.4
Rate of Return ()
37

• 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.

38
Expected Return versus Risk
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 Collec
tions 1.74 13.4 7.7

• Which alternative is best?

39
Portfolio Risk and Return
Assume a two-stock portfolio with 50,000 in HT
and 50,000 in Collections.

Calculate rp and ?p.
40
Portfolio Return, rp


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.
41
Alternative Method
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...)
42

• ?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.

43
Portfolio standard deviation in general
?p Portfolio standard deviation.
Where w1 and w2 are portfolio weights and r1,2 is
the correlation coefficient between stock 1 and 2.
44
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?

45
Portfolio beta
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.
46
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.

47
Prob.
Large
2
1
0
15
Return
?1 ??35 ?Large ??20.
48
?p ()
Company-Specific (Diversifiable) Risk
35
Stand-Alone Risk, ?p
20 0
Market Risk
10 20 30 40 2,000
Stocks in Portfolio
49
Stand-alone Market Diversifiable
.
risk risk risk

Market risk is that part of a securitys
stand-alone risk that cannot be eliminated by
diversification. Firm-specific, or diversifiable,
risk is that part of a securitys stand-alone
risk that can be eliminated by diversification.
50
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.

51
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.

52
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 stocks beta coefficient,
  which measures the stocks volatility relative to
  the market.
• What is the relevant risk for a stock held in
  isolation?

53
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 stocks
  beta coefficient, or b.

54
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
.
55
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.

56
How is beta interpreted?

• If b 1.0, stock has average risk.
• If b gt 1.0, stock is riskier than average.
• If b lt 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?

57
_
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
58
Expected Return versus Market Risk
Expected
Risk,?b
Security
return
HT 17.4 1.30 Market 15.0 1.00 USR 13.8 0.89 T-bil
ls 8.0 0.00 Collections 1.74 -0.87

• Which of the alternatives is best?

59
Use the SML to calculate eachalternatives
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.

60
Required Rates of Return
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.
61
Expected versus Required Returns

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
62
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
63
What is the required rate of returnon 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.
64
Stock Value PV of Dividends
.
What is a constant growth stock?
One whose dividends are expected to grow forever
at a constant rate, g.
65
For a constant growth stock,
.
.
.
If g is constant, then
.
66

0.25
If g gt r, P0 negative
0
Years (t)
67
What happens if g gt rs?

• If rslt g, get negative stock price, which is
  nonsense.
• We cant 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.

68
Assume beta 1.2, rRF 7, and rM 12. What
is the required rate of return on the firms
stock?
Use the SML to calculate rs
rs rRF (rM - rRF)bFirm 7 (12 - 7)
(1.2) 13.
69
D0 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
2.2472
2.3820
D0 2.00
2.12
13
1.8761
1.7599
1.6508
70
Whats the stocks market value? D0 2.00, rs
13, g 6.
Constant growth model
71
Rearrange model to rate of return form

Then, rs 2.12/30.29 0.06 0.07 0.06
13.
72
If we have supernormal growth of 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.

73
Nonconstant growth followed by constant growth
0
1
2
3
4
rs13
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