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History of Interest Rates and Risk
Premiums, BKM Ch 5
Zvi Wiener
tel: 02-588-3049
Fall-02
[email protected]
http://pluto.mscc.huji.ac.il/~mswiener/zvi.html
Investments
Factors Influencing Rates
Supply
Households
Institutions
Demand
Businesses
Government’s Net Supply and/or Demand
Federal Reserve Actions
Zvi Wiener
BKM Ch 5
slide 2
Level of Interest Rates
Interest Rates
Supply
r1
r0
Demand
Q0 Q1
Zvi Wiener
BKM Ch 5
Funds
slide 3
Real vs. Nominal Rates
Fisher effect: Approximation
nominal rate = real rate + inflation premium
R = r + i or r = R - i
Example r = 3%, i = 6%
R = 9% = 3% + 6% or 3% = 9% - 6%
Fisher effect: Exact
r = (R - i) / (1 + i)
2.83% = (9%-6%) / (1.06)
Empirical Relationship:
Inflation and interest rates move closely together
Zvi Wiener
BKM Ch 5
slide 4
Rates of Return: Single Period
P
1  P0  D1
HPR 
P0
HPR = Holding Period Return
P0 = Beginning price
P1 = Ending price
D1 = Dividend during period one
Inappropriate for complex securities!
Zvi Wiener
BKM Ch 5
slide 5
Rates of Return:
Single Period Example
Ending Price =
48
Beginning Price =
40
Dividend =
2
HPR = (48 - 40 + 2 )/ (40) = 25%
Zvi Wiener
BKM Ch 5
slide 6
Characteristics of Probability
Distributions
1) Mean: most likely value
2) Variance or standard deviation
3) Skewness
* If a distribution is approximately normal,
the distribution is described by
characteristics 1 and 2.
Zvi Wiener
BKM Ch 5
slide 7
Normal Distribution
s.d.
s.d.
mean
Symmetric distribution
Zvi Wiener
BKM Ch 5
slide 8
Measuring Mean:
Scenario or Subjective Returns
Subjective returns
s
E (r )   p s r s
1
p(s) = probability of a state
r(s) = return if a state occurs
1 to s states
Zvi Wiener
BKM Ch 5
slide 9
Basic Statistics
Mean
Independence
Standard Deviation
Conditioning
Variance
Joint distribution
Volatility
CDF, PDF
Skewness
Regression
Correlation
Arithmetic, Geometric
Quantile
Normal, Student, …
Zvi Wiener
BKM Ch 5
slide 10
Annual Holding Period Returns
From Figure 6.1 of Text
Geom.
Series
Mean%
Sm Stk 12.6
Lg Stk
11.1
LT Gov
5.1
T-Bills
3.8
Inflation 3.1
Zvi Wiener
Arith.
Mean%
18.8
13.1
5.4
3.8
3.2
BKM Ch 5
Stan.
Dev.%
39.7
20.2
8.1
3.3
4.5
slide 11
Annual Holding Period Risk
Premiums
and Real Returns
Risk
Series
Premiums%
Sm Stk
15.0
Lg Stk
9.3
LT Gov
1.6
T-Bills
--Inflation
--Zvi Wiener
BKM Ch 5
Real
Returns%
15.6
9.9
2.2
0.6
--slide 12
Home Assignment
Required:
• problems 2, 8, 11, 16, 17 (3rd ed).
• problems 2, 8, 11, 14, 15 (5th ed).
• visit http://research.stlouisfed.org/publications/mt
• closely follow financial news!
Zvi Wiener
BKM Ch 5
slide 13
‫‪Home Assignment‬‬
‫איזה משני האפיקים הבאים רווחי יותר למשקיע‪:‬‬
‫אפיק א'‪ 5% :‬ריבית מחושבת רבע שנתית או‬
‫אפיק ב'‪ %3.5 :‬ריבית צמודה למדד מחושבת חצי‬
‫שנתית‪ .‬שני האפיקים ברמת סיכון זהה‪.‬‬
‫האינפלציה השנתית שהתממשה היתה ‪.%1.5‬‬
‫‪slide 14‬‬
‫‪BKM Ch 5‬‬
‫‪Zvi Wiener‬‬
Random Variables
Distribution function of a random variable X
F(x) = P(X  x) - the probability of x or less.
If X is discrete then
F ( x )   f ( xi )
xi  x
x
If X is continuous then F ( x) 
dF ( x)
Note that f ( x) 
dx
Zvi Wiener
BKM Ch 5
 f (u)du

slide 15
Moments
Mean = Average = Expected value
  E( X ) 

xf
(
x
)
dx


Variance
  V (X ) 
2

 x  E ( X )
2
f ( x)dx

  S tan dard Deviation  Variance
Zvi Wiener
BKM Ch 5
slide 16
Chapter 5 Weblinks
http://www.bloomberg.com/markets/wei.html
Returns on various equity indexes can be located here.
http://app.marketwatch.com/intl/default.asp
Returns on various equity indexes can be located here.
http://www.quote.com/quotecom/markets/snapshot.asp
Returns on various equity indexes can be located here.
http://www.bloomberg.com/markets/rates.html
Current rates on U.S. and international government bonds can be
located on this site.
http://www.bondmarkets.com
The site above is from the bond market association. General
information on a variety of bonds and strategies can be accessed
on line at no charge.
Zvi Wiener
BKM Ch 5
slide 17
http://www.investinginbonds.com
The site above is from the bond market association. General
information on a variety of bonds and strategies can be accessed
on line at no charge. Current information on rates is also available
on the www.investinginbonds.com site.
http://www.stls.frb.org/
The site listed above contains current and historical information
on a variety of interest rates. Historical data can be downloaded
in spreadsheet format and is available through the Federal
Reserve Economic Database (FRED)
http://www.stls.frb.org/docs/publications/mt/mt.pdf
The site listed above contains current and historical information
on a variety of interest rates. Historical data can be downloaded
in spreadsheet format and is available through the Federal
Reserve Economic Database (FRED)
Zvi Wiener
BKM Ch 5
slide 18
Cov( X 1 , X 2 )  E X 1  EX 1  X 2  EX 2 
( X1, X 2 ) 
Cov( X 1 , X 2 )
 1 2
Skewness (non-symmetry)
Kurtosis (fat tails)
Zvi Wiener
Its meaning ...
 
 
BKM Ch 5
1

3
1

4

E  X  E X 

3
E  X  E  X 

4

slide 19
Main properties
E (a  bX )  a  bE ( X )
 (a  bX )  b ( X )
E( X 1  X 2 )  E( X 1 )  E( X 2 )
 ( X 1  X 2 )   ( X 1 )   ( X 2 )  2Cov( X 1 , X 2 )
2
Zvi Wiener
2
2
BKM Ch 5
slide 20
Portfolio of Random Variables
N
Y   wi X i  w X
T
i 1
N
E (Y )   p  w E ( X )  w  X   wi  i
T
T
i 1
N
N
 (Y )  w w   wi ij w j
2
T
i 1 j 1
Zvi Wiener
BKM Ch 5
slide 21
Uniform Distribution
Uniform distribution defined over a range of
2
values axb.
ab 2
(b  a)
E( X ) 
,  (X ) 
2
12
1
f ( x) 
, a xb
ba
xa
0,
x  a

F ( x)  
, a xb
b  a
bx
1,
Zvi Wiener
BKM Ch 5
slide 22
Uniform Distribution
1
1
ba
a
Zvi Wiener
b
BKM Ch 5
slide 23
Normal Distribution
Is defined by its mean and variance.
f ( x) 

1
 2
e
( x )2
2 2
E( X )  ,  ( X )  
2
2
Cumulative is denoted by N(x).
Zvi Wiener
BKM Ch 5
slide 24
Normal Distribution
66% of events lie
between -1 and 1
0.4
0.3
95% of events lie
between -2 and 2
0.2
0.1
-3
Zvi Wiener
-2
-1
1
BKM Ch 5
2
3
slide 25
Normal Distribution
1
0.8
0.6
0.4
0.2
-3
Zvi Wiener
-2
-1
1
BKM Ch 5
2
3
slide 26
Normal Distribution
symmetric around the mean
mean = median
skewness = 0
kurtosis = 3
linear combination of normal is normal
99.99 99.90 99
97.72 97.5 95
90
84.13
3.715 3.09 2.326 2.000 1.96 1.645 1.282 1
Zvi Wiener
BKM Ch 5
50
0
slide 27
Central Limit Theorem
The mean of n independent and identically
distributed variables converges to a normal
distribution as n increases.
1 n
X   Xi
n i 1
 2 

X  N   ,
n 

Zvi Wiener
BKM Ch 5
slide 28
Lognormal Distribution
The normal distribution is often used for rate
of return.
Y is lognormally distributed if X=lnY is
normally distributed. No negative values!
f ( x) 
E( X )  e

2
2
1
x 2

(ln(x )   ) 2
e
,  (X )  e
2
2 2
2   2 2
e
2   2
E (Y )  E (ln X )   ,  (Y )   (ln X )  
2
Zvi Wiener
BKM Ch 5
2
2
slide 29
Lognormal Distribution
If r is the expected value of the lognormal
variable X, the mean of the associated normal
variable is r-0.52.
0.6
0.5
0.4
0.3
0.2
0.1
0.5
Zvi Wiener
1
1.5
BKM Ch 5
2
2.5
3
slide 30