Fall-02http://pluto.mscc.huji.ac.il/

~mswiener/zvi.htmlInvestments

Zvi Wienertel: 02-588-3049 mswiener@mscc.huji.ac.il

History of Interest Rates and Risk Premiums, BKM Ch 5

Zvi Wiener BKM Ch 5 slide 2

Factors Influencing Rates

SupplyHouseholdsInstitutions

DemandBusinesses

Government’s Net Supply and/or DemandFederal Reserve Actions

Zvi Wiener BKM Ch 5 slide 3

Q0 Q1

r0

r1

Funds

Interest Rates

Supply

Demand

Interest Rates

Supply

Q0 Q1

r0

r1

Funds

Demand

Level of Interest Rates

Zvi Wiener BKM Ch 5 slide 4

Fisher effect: Approximationnominal rate = real rate + inflation premium

R = r + i or r = R - iExample 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

Real vs. Nominal Rates

Zvi Wiener BKM Ch 5 slide 5

P

DPPHPR0

101

HPR = Holding Period Return

P0 = Beginning price

P1 = Ending price

D1 = Dividend during period one

Inappropriate for complex securities!

Rates of Return: Single Period

Zvi Wiener BKM Ch 5 slide 6

Ending Price = 48

Beginning Price = 40

Dividend = 2

HPR = (48 - 40 + 2 )/ (40) = 25%

Rates of Return: Single Period Example

Zvi Wiener BKM Ch 5 slide 7

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.

Characteristics of Probability Distributions

Zvi Wiener BKM Ch 5 slide 8

Symmetric distribution

mean

s.d. s.d.

Normal Distribution

Zvi Wiener BKM Ch 5 slide 9

Subjective returns

p(s) = probability of a stater(s) = return if a state occurs1 to s states

Measuring Mean: Scenario or Subjective Returns

rprE s

s

s1

)(

Zvi Wiener BKM Ch 5 slide 10

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 11

Geom. Arith. Stan.Series Mean% Mean% Dev.%Sm Stk12.6 18.8 39.7Lg Stk 11.1 13.1 20.2LT Gov 5.1 5.4 8.1T-Bills 3.8 3.8 3.3Inflation 3.1 3.2 4.5

Annual Holding Period ReturnsFrom Figure 6.1 of Text

Zvi Wiener BKM Ch 5 slide 12

Risk RealSeries Premiums% Returns%Sm Stk 15.0 15.6Lg Stk 9.3 9.9LT Gov 1.6 2.2T-Bills --- 0.6Inflation --- ---

Annual Holding Period Risk Premiums

and Real Returns

Zvi Wiener BKM Ch 5 slide 13

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 14

Home Assignment

: איזה משני האפיקים הבאים רווחי יותר למשקיע

ריבית מחושבת רבע שנתית או אפיק %5': אפיק א

ריבית צמודה למדד מחושבת חצי %3.5ב':

שנתית. שני האפיקים ברמת סיכון זהה.

.%1.5האינפלציה השנתית שהתממשה היתה

Zvi Wiener BKM Ch 5 slide 15

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

xx

i

i

xfxF )()(

If X is continuous then

x

duufxF )()(

Note thatdx

xdFxf

)()(

Zvi Wiener BKM Ch 5 slide 16

MomentsMean = Average = Expected value

dxxxfXE )()(

Variance

dxxfXExXV )()()( 22

VarianceDeviationdardS tan

Zvi Wiener BKM Ch 5 slide 17

Chapter 5 Weblinkshttp://www.bloomberg.com/markets/wei.htmlReturns on various equity indexes can be located here. http://app.marketwatch.com/intl/default.aspReturns on various equity indexes can be located here. http://www.quote.com/quotecom/markets/snapshot.aspReturns on various equity indexes can be located here. http://www.bloomberg.com/markets/rates.htmlCurrent rates on U.S. and international government bonds can be located on this site.

http://www.bondmarkets.comThe 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 18

http://www.investinginbonds.comThe 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.pdfThe 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 19

221121 ),( XEXXEXEXXCov

Its meaning ...

3

3

1XEXE

21

2121

),(),(

XXCov

XX

Skewness (non-symmetry)

4

4

1XEXE

Kurtosis (fat tails)

Zvi Wiener BKM Ch 5 slide 20

Main properties

)()( XbEabXaE

)()( XbbXa

)()()( 2121 XEXEXXE

),(2)()()( 2122

12

212 XXCovXXXX

Zvi Wiener BKM Ch 5 slide 21

Portfolio of Random Variables

XwXwY TN

iii

1

N

iiiX

TTp wwXEwYE

1

)()(

N

i

N

jjiji

T wwwwY1 1

2 )(

Zvi Wiener BKM Ch 5 slide 22

Uniform DistributionUniform distribution defined over a range of values axb.

bxaab

xf

,1

)(

12

)()(,

2)(

22 ab

Xba

XE

xb

bxaab

ax

ax

xF

,1

,

,0

)(

Zvi Wiener BKM Ch 5 slide 23

Uniform Distribution

a b

ab 1

1

Zvi Wiener BKM Ch 5 slide 24

Normal DistributionIs defined by its mean and variance.

2

2

2

)(

2

1)(

x

exf

22 )(,)( XXE

Cumulative is denoted by N(x).

Zvi Wiener BKM Ch 5 slide 25

-3 -2 -1 1 2 3

0.1

0.2

0.3

0.4

Normal Distribution66% of events liebetween -1 and 1

95% of events liebetween -2 and 2

Zvi Wiener BKM Ch 5 slide 26

Normal Distribution

-3 -2 -1 1 2 3

0.2

0.4

0.6

0.8

1

Zvi Wiener BKM Ch 5 slide 27

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 50

3.715 3.09 2.326 2.000 1.96 1.645 1.282 1 0

Zvi Wiener BKM Ch 5 slide 28

Central Limit Theorem

The mean of n independent and identically distributed variables converges to a normal distribution as n increases.

n

iiX

nX

1

1

nNX

2

,

Zvi Wiener BKM Ch 5 slide 29

Lognormal DistributionThe normal distribution is often used for rate of return.

Y is lognormally distributed if X=lnY is normally distributed. No negative values!

2

2

2

))(ln(

2

1)(

x

ex

xf

22

2

22222 )(,)(

eeXeXE

222 )(ln)(,)(ln)( XYXEYE

Zvi Wiener BKM Ch 5 slide 30

Lognormal DistributionIf r is the expected value of the lognormal variable X, the mean of the associated normal variable is r-0.52.

0.5 1 1.5 2 2.5 3

0.1

0.2

0.3

0.4

0.5

0.6