paper/essay/report写作
AECO 480/580(9671/9672) Financial
Econometrics
II. Linear Models and CAPM
Zhongwen Liang
Department of Economics
University at Albany, SUNY
March 23, 2018
Hedging
- We can also use linear regression to study optimal hedging.
- Suppose a large institutional investor holds a huge well-diversified portfolio of Japanese stocks that has returns following closely that of the N225 Stock Index return or rate of change.
- Suppose in September 1999, the investor was nervous about an imminent big fall in Japan equity prices, and wished to protect his portfolio value over the period September to mid-October 1999.
- He could liquidate his stocks. But this would be unproductive since his main business was to invest in the Japanese equity sector.
- Besides, liquidating a huge holding or even a big part of it would likely result in loss due to impact costs.
- Thus, the investor decided to hedge the potential drop in index value by sellinghNikkei 225 Index futures contracts.
- If the Japanese stock prices did fall, then the gain in the short position of the futures contracts would make up for the loss in the actual portfolio value.
- The investors original stock position has a total current valueUVt. For example, this could be 10 billion Yen.
- Suppose his stock position value is a constant factorf the N225 Index valueSt.
- Then,Vt+ 1 =fSt+ 1 , and the portfolio return rate Vt+ 1 /Vt= St+ 1 /St, as mentioned in the last paragraph.
- In essence, the investor forms a hedged portfolio comprisingfStYen, andhnumber of short positions in N225 Index futures contracts.
- The contract with maturityThas notional traded priceFt,T and an actual price value ofU 500 Ft,Twhere the contract is specified to have a value ofU500 per notional price point.
- At the end of the risky period, his hedged portfolioUvalue change would be:
Pt+ 1 Pt=f(St+ 1 St)h 500 (Ft+ 1 ,TFt,T).
- In effect, the investor wished to minimise the risk or variance ofPt+ 1 Pt= P.
- Now, simplifying notations, from the above equation
P=fSh 500 F.
- So,
var(P) = f^2 var(S) +h^2 5002 var(F)
2 h 500 fcov(S,F).
- The FOC for minimisingvar(P)with respect to decision variablehyields:
2 h( 5002 )var(F) 2 ( 500 f)cov(S,F) = 0 ,
or a risk-minimising optimal hedge of
h=
fcov(S,F)
500 var(F)
.
- This is a positive number of contracts sinceStandFt,T would move together and recall that at maturityTof the futures contract,ST=FT,T.
- hcan be estimated by substituting in the sample estimates of the covariance in the numerator and of the variance in the denominator.
- It can also be estimated through the following linear regression employing OLS method:
S=a+bF+e,
whereeis the usual residual error that is uncorrelated with
F.
- We run this regression and the results are shown in the next table.
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February 1, 2011 13:41 9in x 6in b1016-ch03 Financial Valuation and Econometrics
Two-Variable Linear Regression 65
by substituting in the sample estimates of the covariance in the numerator and of
variance in the denominator. It can also be estimated through the following
ar regression employing OLS method:
1S b1F e,
where is the usual residual error that is uncorrelated with 1F. We run
regression and the results are shown in Table 3.2. Theoretically,
cov(1S,1F)/var(1F) /f. (Recall that earlier in the chapter, when
dealing with two-variable linear regression, cov(X,Y )/var(X .) The OLS
estimate is thus the risk-minimising or optimal hedge ratio or estimate of
estimate is then found as r of the futures con-
cts to short in this case.
Table 3.2, is 0.71575. With a 10 billion portfolio value and spot
N225IndexonSeptember1,1999at17479, 10 b/ Number
of futures contract to short in this case is estimated at:
f/ N225 futures contracts.
Table 3.2. Regression of Change in Nikkei Index (SPOTCHANGE) on Change in Nikkei
Futures Price (FUTCHANGE):1S =a+b1F +e.
Variable Coefficient Std. Error t-Statistic Prob.
FUTCHANGE 0.715750 0.092666 7.723968 0.
C 4.666338 24.01950 0.194273 0.
R-squared 0.688436 Mean-dependent var. 1.
Adjusted R-squared 0.676897 S.D.-dependent var. 227.
S.E. of regression 129.3249 Akaike info criterion 12.
Sum squared resid. 451573.2 Schwarz criterion 12.
Log likelihood 181.1206 Hannan-Quinn
criterion
12.
F-statistic 59.65968 Durbin-Watson stat. 2.
Prob (F-statistic) 0.
Dependent variable: SPOTCHANGE
Method: Least squares
Sample: 2 30
Included observations: 29
One of the earliest studies to highlight use of least squares regression in optimal hedging is Louis
H. Ederington, The hedging performance of the new futures markets, Journal of Finance 34
- Theoretically,
b=
cov(S,F)
var(F)
= 500 /fh.
- The OLS estimatebis thus the risk-minimising or optimal hedge ratio or estimate of 500/fh.
- hestimate is then found asbf/500 number of the futures contracts to short in this case.
- From Table,bis 0.71575. With aU10 billion portfolio value and spot N225 Index on September 1, 1999 at 17479, f= 10 b/ 17479 =572115.
- Number of futures contract to short in this case is estimated at:
h = bf/ 500 = 0. 71575 572115 / 500
819 N225 futures contracts.