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ECO 4185 | Financial Econometrics

Section B00, Winter 2022

1. (8 points) The price of an asset changes from 195.35 $ to 209.05 $ during a period of 762 days.

(a) Give the simple net return and the log return for the period.

(b) Give the annualized simple net return.

2. (10 points) The skewness of a sample of 240 monthly returns is -0.03 and the kurtosis is 4.41. Under the null hypothesis H0 that the returns are iid Gaussian, the distribution of the Jarque- Bera test statistic follows a chi-squared distribution with two degrees of freedom,

JB = + ~ x2 (2).

(a) Is the test with the Jarque-Bera statistic one-sided or two-sided? Justify. (b) Test H0 with a size of 5 % using the Jarque-Bera test statistic.

F ( α )	0 . 010	0 . 025	0 . 050	0 . 100	0 . 500	0 . 900	0 . 950	0 . 975	0 . 990
α	0 . 020	0 . 051	0 . 103	0 . 211	1 . 386	4 . 605	5 . 992	7 . 378	9 . 2 10

Table 1: Some values of the distribution function of a law x2 (2).

3. (26 points) Consider the following ARMA(2,1) process, where et is Gaussian white noise: γt = 1.3 γt − 1 - 0.4 γt − 2 + et + 0.6 et − 1 , et ~ N(0, 0.2).

(a) Show that one of the two roots of the characteristic polynomial is 2 and ﬁnd the other. Is

the process stationary? Justify.

(b) Find the values w1 and w2 of the MA(o) expansion of the process:

γt = et + w1 et − 1 + w2 et − 2 + w3 et − 3 + . . .

(c) Give the system of equations that allows to determine the autocovariance coeﬃcients 80 , 81 and 82 of the process and explain how to obtain the coeﬃcients 8l for d = 3, 4, . . . as a function of the coeﬃcients 8k , k = 1, . . . , d - 1.

4. (20 points) You consider an ARMA(1,2) model,

γt = o0 + o1 γt − 1 + et - 91et − 1 - 92et − 2 , et ~ D(0, 72 ),

where et is a white noise process, with the objective of predicting the return of a ﬁ nancial asset. You get the following point estimates for the ﬁtted model:

0 = 0.65

1 = 0.55

9 ˆ 1 = -0.40

9ˆ2 = 0.25

= 1.25

(a) Compute the point estimate of the unconditional mean u 三 E(γt).

(b) Suppose that the value of the last two observations is γ n = 2.15 and γ n − 1 = 3.95 and

that the estimated value of the corresponding innovations is n = 2.51 and n − 1 = -0.68. Calculate the point predictions γˆn(1) and γ ˆn(2) as well as their standard deviation.

(c) The table below contains four point predictions γˆk − 1 (1) obtained with an ARMA(1,2) model and an AR(1) model in a prediction exercise with moving window. The observed values of γ t are listed on the last line. Propose a selection criterion and indicate which of the two models is favored according to those prediction results.

k	≠ + 1	≠ + 2	≠ + 3	≠ + 4
AR M A ( 1 , 2 )	1 . 26	- 0 . 56	1 . 04	- 0 . 69
A R ( 1 )	1 . 33	- 0 . 61	1 . 18	- 0 . 72
γ k	1 . 23	- 0 . 52	1 . 16	- 0 . 76

Table 2: Results of the prediction exercise.

5. (16 points) Consider an economy with only two risky assets and one risk-free asset. The number of shares, the price and the expected value and standard deviation of the return of the risky assets are available in the table below. The correlation coeﬃcient between the two risky assets is 1/3. Suppose the Sharpe-Lintner version of the CAPM is satisﬁed.

A sse t	Nb d e Shares	P rice	E ( R )	′ V a r( R )
A	10 0	1 . 5	15	15
B	15 0	2 . 0	12	9

Table 3: Description of the market for an economy with two risky assets.

(a) Calculate the expected return and the standard deviation of the market portfolio. (b) Calculate the market beta of asset A. Reminder: Cov(X, Y) = E[(X - uX )(Y - uY )].

(c) Calculate the return of the risk-free asset and construct the capital market line.

(d) Calculate the weight of each asset for an eﬃcient portfolio with a targeted risk 7p = 5.

6. (20 points) You want to build an investment portfolio and decide to use the following version of

the CAPM

model,

Ri,t = γf,t + βi(Rm,t - γf,t) + ei,t ,

ei,t ~iid N(0, 7 i ),

as a tool to estimate the joint returns distribution of diﬀerent assets.

(a) What is the advantage of using this approach to estimate the joint distribution of returns?

(b) Explain how to proceed to obtain point estimates of the parameters of the previous equation.

Specify the variables to be used and the appropriate econometric methods.

(c) Consider two risky assets A and B with relative parameters to the CAPM equation βA = 1.20, βB = 0.45, 7 ∈A = 0.17, 7 ∈B = 0.10.

Suppose that eA,t and eB,t are not correlated. Calculate the variance-covariance matrix of assets A and B if the risk (standard deviation) of the market portfolio is 0.15.

(d) Give the proportion of the market risk of an equally weighted portfolio of asset A and B.

(e) Name a practical limitation of the CAPM model when used to estimate the joint returns

distribution of diﬀerent assets. Suggest an alternative approach and explain why it should have been used instead.