上财朱杰金融计量经济学4_Returns

If f (x ) is a convex function, then f (E [x ]) 6 E [f (x )] If f (x ) is a concave function, then f (E [x ]) > E [f (x )] Example, f (x ) = e x is a convex function, thus we have e E [x ] 6 E [ e x ] . (6) (5)
The simple gross return between date t
1 and t is de…ned as 1 + Rt . k to date t is
The gross return over k periods from date t
1 + Rt (k ) = (1 + Rt )(1 + Rt 1 ) (1 + Rt k +1 ) Pt Pt 1 Pt k +1 Pt = = . Pt 1 Pt 2 Pt k Pt k The simple net return over k periods is just Rt (k ). These multiperiod returns Rt (k ) are called compound return.
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Returns
We need to specify the return horizon in order to compare di¤erent returns. Annualized return: Annualized Rt (k )g = [ ∏ (1 + Rt
Financial Econometrics
Zhu, Jie
Shanghai University of Finance and Economics
October 2012
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Returns
Denote by Pt the price of an asset at date t with no dividend. The simple net return Rt between dates t Rt = Pt Pt 1 1. 1 and t is de…ned as (1)
(X
σ4 x
µ )4
].
(14)
The quantity K (x ) 3 is called the excess kurtosis. For normal distributions, K (x ) = 3. The positive excess kurtosis means that the random sample from the distribution tends to contain more extreme values compared to the normal distribution.
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Distribution of Returns
Now consider a joint distribution function for fRi 1 , ..., RiT g, F (Ri 1 , ...RiT ; θ). We may rewrite F (Ri 1 , ...RiT ; θ) as the product of conditional distributions: F (Ri 1 , ...RiT ; θ) = F (Ri 1 )F (Ri 2 jRi 1 ) F (RiT jRi ,T
Consider a collection of N assets at date t, each with return Rit , where t = 1, ..., T . The most general model is its joint distribution function: F (R11 , ..., RN 1; R12, ...RN 2 ; ...; RN 1, ..., RNT ; x; θ), (20)
where x denotes state variables and θ represents the parameter vector. In practice, the model of (20) is too general. The CAPM considers the joint distribution of the cross section of returns, fR1t , ..., RNt g. Other models focus on the dynamic process of individual asset returns, fRi 1 , ..., RiT g, the time series of returns.
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Estimation
Estimation of skewness and kurtosis. Let fx1 , ..., xT g be a random sample of X with T observations. The sample mean: 1 T b (15) µ= ∑ xt . T t =1 The sample variance: b σ2 = b S= b K = 1 T ∑ (xt T t =1 b µ )2 . b µ )3 . b µ )4 . (16)
The estimated variances 6/T b S t = p6/T and
The Jarque-Bera normality test statistic: JB =
b b S and K 3 are distributed normally and have and 24/T respectively, thus the t-statistics are b t = pK 3 . 24/T b b S2 (K 3)2 + . 6/T 24/T (19)
= log(1 + Rt ) + log(1 + Rt = rt + rt 1 + + rt k +1 .
1) +
(1 + Rt + log(1 + Rt k +1 )
1)
k +1 ))
(8)
2. rt has no constrained lower limit - make it easier to apply to statistical analysis. The disadvantage of log return: For a portfolio, it holds that Rpt = ∑ wi Rit , but it only holds
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Financial Econometrics
Skewness and Kurtosis
The …rst two moments can uniquely determine a normal distribution. The third central moment measures the symmetry of X with respect to its mean. The standardized third central moment is called skewness: (X µ )3 S (x ) = E [ ]. (13) σ3 x The standardized fourth central moment is called kurtosis, it measures the tail behavior of X : K (x ) = E [
Annualized Rt (k ) Rt (k ) is called the arithmetic average. We can show that Rt (k )g 6 Rt (k ).
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Jensen Inequality
1 , ..., Ri 1 ). 1 , ..., Ri 1 )
= f (Ri 1 ) ∏ f (Rit jRi ,t
t =2
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Distribution of Returns
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Returns
Continuous compounding (log return): rt = log(1 + Rt ) = log Pt = pt Pt 1 pt
1.
(7)
The advantage of log return: 1. It is easy to calculate multiperiod return: rt (k ) = log(1 + Rt (k )) = log((1 + Rt )(1 + Rt
1 , ..., Ri 1 ). 1 , ..., Ri 1 )