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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 EX 1 X 2 EX 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 axb. ab 2 (b a) E( X ) , (X ) 2 12 1 f ( x) , a xb ba xa 0, x a F ( x) , a xb b a bx 1, Zvi Wiener BKM Ch 5 slide 22 Uniform Distribution 1 1 ba 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-0.52. 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