Final Exam 2021, MGMTMFE 407 Empirical Methods in Finance Prof. . Lochstoer
You are only free to consult your lecture notes, homeworks, and the textbook (Tsay) when answering this exam. You are not allowed to discuss the exam with anyone else. Please be clear and concise. Good luck!
1. Time: 180 minutes, beginning when you download exam from CCLE. Please upload exam back to CCLE **within** 180 minutes of your starting time. If you run out of time, simply write íran out of timeíon the last page and still make sure you submit on time.
2. There are a total of 4 longer questions (100 points in total). Please answer all questions. To get credit you must show your work.
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3. Write your answers on blank sheets of paper, number each page.
4. When done, you have to scan your pages and save the Öle as a pdf Öle with title: EmpiricalYOURFULLNAME.pdf. YOURFULLNAME = last name and Örst name as in LochstoerLars or ZhangDanyu. The Örst page should have your name, student id, and the sentence “I acknowledge my obligations under the UCLA Honor Code” with your signature following.
5. Upload your answers to CCLE.
1. Autocorrelation and ARMA models
(a) Describe how Momentum- and Reversal-based trading strategies relate to re- turn autocorrelations. Be as speciÖc as possible.
(b) Assume returns follow the process:
rt+1 = 0:01+0:8xt +0:05”t+1;
xt+1 = 0:7xt +0:25xt 1 0:01”t+1;
where “t+1 is an i.i.d. standard Normal shock.
i. Give the most parsimonious ARMA process for returns implied by the above equations. That is, give the order of the AR and MA terms, as well as the values of the AR and MA coe¢ cients. p
ii. Whatisthestandarddeviationofexpectedreturns,thatis, Var(Et(rt+1))?
(c) Let t be year-on-year (YoY) ináation where t measures months. Thus, the February 2021 value of t is the price-level at the end of February 2021 di- vided by the price level at the end of February 2020. You believe monthly deseasonalized log ináation is stationary and follows an AR(1) process. What process does the log of YoY ináation t follow? Write down the ARMA process including as much as possible about the coe¢ cients in this equation.
(d) Write down the conditional log-likelihood function for an AR(2) process as- suming the residuals are Normally distributed, where you condition on the two Örst observations in the sample.
2. VAR models, return predictability and the present-value restriction
(a) Write down a VAR(1) that has two state-variables: log return-on-equity (et) and the log market-to-book ratio (mbt). Clearly deÖne all variables and para- meters.
(b) Explain in words how you would estimate the parameters of this VAR.
(c) Recall from Homework 4 that we can write log returns as
rt+1 =mbt+1 mbt +et+1; (1) where 0 < < 1 is a log-linearization constant (e.g., = 0:96). Iterate this
equation forward to get an expression for mbt as a function of the inÖnite sum 2
of future et+1 and rt+1. Using this equation, why are Örm market-to-book ratios di§erent across time and stocks?
(d) Using the VAR you wrote down in 2.a and the equation above, derive the formulas for the following expectations:
i.E(rP),E(r ): t t+1 t t+2
ii. Et 1j=1 jrt+j, where jj < 1.
iii. Explain why you can get expected returns from a VAR that only uses
market-to-book ratios and return-on-equity (the log of 1+earnings over lagged book equity). That is, what are the statistical and economic re- strictions we are using?
3. Volatility models
(a) Currentmarketvolatilityishigherthanthehistoricalaverageofmarketvolatil- ity from a long sample. Given this information and the stylized facts about market return volatility, what can you say about expected future market volatility?
(b) Explain what írealized varianceíis. Use both a mathematical expression and an intuitive description.
(c) A Variance Swap is a very popular over-the-counter derivative contract. In the contract, the Öxed leg pays a Öxed dollar amount every month, while the áoatinglegpaysanamountproportionaltotherealizedvarianceineachmonth based on daily data. Assume any variance risk premium and the risk-free rate are both zero.
In particular, the payo§ to a long variance position in a 2-month swap is:
Payofft+2 = RVt+1 + RVt+2 2 Ft;
where RVs is the realized variance in month s and Ft is the Öxed payment decided at the start of the swap (at time t). Recall that swaps have zero value at inception, thus Ft is set so that Et (Payofft+2) = 0. Assume RV follows an AR(1) process. Derive a formula for the fair time t swap rate, Ft.
(d) Consider the GARCH(1,1) process 2t+1 = 0:1 + 0:07"2t + 0:922t . Using the fact that2t Et 1["2t],writethisGARCH(1,1)asanARMAprocessin"2t includ- ing the values of the ARMA coe¢ cients. Why is it not technically appropriate to assume the residuals in this ARMA process are Normally distributed?
4. Factor models
You are evaluating a long-short equity hedge fund and are given the below regression results:
Re =0:03+1:5MKT 0:3HML 0:2SMB +0:2"; (2) fund;t t t t t
where the factors are the FF3 factors and where " is a standard Normal error term. Assume all coe¢ cients are signiÖcant.
(a) What investment ístylesíwould you say characterizes this fund?
(b) What is the Information Ratio of this fund? In this calculation, assume the relevant benchmark has returns
Rb;t =1:5MKTt 0:3HMLt 0:2SMBt: (3)
(c) Assume the maximal Sharpe ratio one can obtain by investing in these three factors (MKT, HML, and SMB) is 0.7. What is the maximal Sharpe ratio one can obtain by combining these factors with the fund?
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