chapter 5 notes on the significance of the logistal measures

1. Ch 5
notes

2. • A nominal interest rate is the growth rate of your
money
• A real interest rate is the growth rate of your
purchasing power
Nominal Interest Rate
Real Interest Rate
Inflation Rate
1
nom
real
nom
real
r
r
i
r i
r
i





 : real nom
Note r r i
 

4. • We expect higher nominal interest rates when
inflation is higher
• If E(i) denotes current expectations of inflation, the
Fisher hypothesis is
 
nom real
r r E i
 

7. • The breakeven inflation rate represents a measure
of expected inflation derived from 10-Year Treasury
Constant Maturity Securities (BC_10YEAR) and 10-
Year Treasury Inflation-Indexed Constant Maturity
Securities (TC_10YEAR). The latest value implies
what market participants expect inflation to be in
the next 10 years, on average.
Starting with the update on June 21, 2019, the
Treasury bond data used in calculating interest rate
spreads is obtained directly from the U.S. Treasury
Department

9. Returns
• Sources of investment risk
• Macroeconomic fluctuations
• Changing fortunes of various industries
• Firm-specific unexpected developments
• Holding period return (HPR), or realized rate of
return, is based on the price per share at year’s end
and any cash dividends collected

10. 5-10
Rates of Return: Single Period
Example
Ending Price = $110
Beginning Price = $100
Dividend = $4
HPR =
P1 − P0 + 𝐷1
P0
=
$110 − $100 + $4
$100
=
$110 − $100
$100
+
$4
$100
= 10% Capital Gains yield + 4% Dividend Yield
= 14% Holding Period Return

11. • Expected returns
• p(s) = probability of each scenario
• r(s) = HPR in each scenario
• s = scenario
( ) ( ) ( )
s
E r p s r s
 


12. Variance and standard deviation
     
2
2
s
p s r s E r
   
 
 

2
STD 


13. • Risk premium is the difference between the
expected HPR and the risk-free rate
• Provides compensation for the risk of an investment
• Risk-free rate is the rate of interest that can be
earned with certainty
• Commonly taken to be the rate on short-term T-bills
• Difference between actual rate of return and
risk-free rate is called excess return
• Risk aversion dictates the degree to which
investors are willing to commit funds to stocks

14. • When using historical data, each observation is
treated as an equally likely “scenario”
• Expected return, E(r), is estimated by arithmetic
average of sample rates of return

15. Careful here …notice the starting
value was 1 dollar!!
• Geometric rate of return
• measure of performance over the sample period is the
(fixed) annual HPR that would compound over the
period to the same terminal value obtained from the
sequence of actual returns in the time series.

16. Arithmetic Average: An Example
Year Beg. Value Ending Value HPR HPY
1 $100 $115 1.15 .15
2 115 138 1.20 .20
3 138 110.40 .80 -.20
AM=[(0.15)+(0.20)+(-0.20)] / 3 = 0.15/3=5%
1-16

17. Geometric Average: An Example
Year Beg. Value Ending Value HPR HPY
1 $100 $115 1.15 .15
2 115 138 1.20 .20
3 138 110.40 .80 -.20
GM=[(1.15) x (1.20) x (0.80)]1/3 – 1 =(1.104)1/3 -1=1.03353 -1
=3.353%
1-17

18. • If distribution is normal
• Geometric = arithmetic – ½ the variance
• Must use decimal values.

19.  
 




n
s
r
s
r
n 1
2
2 1
ˆ

Estimated variance
Expected value of squared deviations
 
 
2
1
1
1
ˆ 




n
j
r
s
r
n

Unbiased estimated standard deviation

20. • Sharpe ratio
• Evaluates performance of investment managers

21. n.b.

22. Deviations from Normality and Tail Risk.
These are normality measures!!- Moments
of the distribution.
3
3
( )
ˆ
R R
Skew Average

 

  
 
4
4
( )
3
ˆ
R R
Kurtosis Average

 

 
 
 

25. • Measures of downside risk
• Value at risk (VaR)
• Loss that will be incurred in the event of an extreme adverse
price change with some given, usually low, probability
• Expected shortfall (ES)
• Expected loss on a security conditional on returns being in the
left tail of the probability distribution
• Lower partial standard deviation (LPSD)
• SD computed using only the portion of the return distribution
below a threshold such as the risk-free rate of the sample
average

26. • The continuously compounded annual return on a
stock is normally distributed with a mean of 25%
and standard deviation of 33%. With 95.45%
confidence, we should expect its actual return in
any particular year to be between which pair of
values?

27. • With probability 0.9545, the value of a normally
distributed variable will fall within 2 standard
deviations of the mean; that is, between −41.0%
and 91.0%. Simply add and subtract 2 standard
deviations to and from the mean.

28. • During a period of severe inflation, a bond offered a
nominal HPR of 86% per year. The inflation rate was
76% per year.
What was the real HPR on the bond over the year?
2 decimal places
Find the approximation rreal ≈ rnom − i. .

29. rnominal − i
=
0.86 − 0.76
= 0.058, or 5.68%
1 + i 1.76
rnominal − i = 86% − 76% = 10% ≈rreal
Clearly, the approximation gives a real HPR
that is too high. WHY???

30. • So normally the approximation is ok to use unless
inflation is above say 10% …but remember it IS an
approximation!!

31. Probability distribution of price and one-year holding period return for a 30-year
U.S. Treasury bond (which will have 29 years to maturity at year-end) assuming
100 PAR – note 100 not 1000
Economy Probability YTM Price
Capital
Gain
Coupon
Interest HPR
Boom 0.20 11.0% $ 74.05 $25.95 $8.00 17.95%
Normal growth 0.50 8.0 100.00 0.00 8.00 8.00
Recession 0.30 7.0 112.28 12.28 8.00 20.28

32. 5-32
Average Excess Returns Around the
World: 1900-2015