Final Exam Solution 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 ’ran out of time’ on the last page and still make sure you submit on time.
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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 specific as possible.
Momentum trading strategies: positive autocorrelations at horizons of less than one year.
Reversal trading strategies: negative autocorrelation at horizons of more than one year or less than one month (short-term reversal).
(b) Assume returns follow the process:
rt+1 = 0.01+0.8×xt +0.05×εt+1,
xt+1 = 0.7×xt +0.25×xt−1 −0.01×εt+1, where εt+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 coefficients.
From the first equation, we can get xt = rt+1−0.01−0.05εt+1 0.8
If we put it back into second equation, we can see rt is an ARMA(2,2) process which follows,
rt+2 = 0.0005 + 0.7rt+1 + 0.25rt − 0.0125εt − 0.043εt+1 + 0.05εt+2
ii. What is the standard deviation of expected returns, that is, V ar (Et (rt+1))? Solution:
Et (rt+1) = 0.01 + 0.8 × xt
Var(Et(rt+1))=0.82 ×Var(xt)
V ar(xt) = 0.72V ar(xt) + 0.252V ar(xt) + 0.012 + 2 × 0.7 × 0.25 × 0.7V ar(xt) 1−0.25
V ar(xt) = 0.000828 and V ar (Et (rt+1)) = 0.023
(c) Let πt be year-on-year (YoY) inflation where t measures months. Thus, the February 2021 value of πt is the price-level at the end of February 2021 divided by the price level at the end of February 2020. You believe monthly deseason- alized inflation is stationary and follows an AR(1) process. What process does the YoY inflation πt follow? Write down the ARMA process for πt including as much as possible about the coefficients in this equation.
rt+1 = φ0 + φ1rt + εt+1
11 rt−i = 12φ0 + φ1 11 rt−i−1 + 11 εt−i
(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 first observations in the sample.
The log-likelihood function is:
lnp(r1, r2, …, rT ; Θ) = lnp(rt|rt−1, …, r1; Θ) + lnp(r1, r2; Θ)
1 T (rt − φ0 − φ1rt−1 − φ2rt−2)2
πt = 12φ0 + φ1πt−1 + 11 εt−i
This process is captured by ARMA(1,11) with coefficients shown above.
(ln(2π) + ln(σε2) + σε2 ) + lnp(r1, r2; Θ)
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 define all variables and pa- rameters.
Zt+1 − μ = φ(Zt − μ) + εt+1 where Zt = (et, mbt)′ and μ = (μ1, μ2)′
φ φ φ= 11 12
The residual vector εt = (εet , εmbt ) has a 2×2 covariance matrix Σ. We could assume the residuals are jointly normal and i.i.d., but this is not needed for consistency. You do need your residuals to be stationary.
(b) Explain in words how you would estimate the parameters of this VAR.Solution: Estimate the model via OLS. That is, run the following two regressions:
et+1 = φ10 + φ11et + φ12mbt + εe,t+1
mbt+1 = φ20 + φ21et + φ22mbt + εmb,t+1
OLS is consistent even if the errors are non-normal and/or not i.i.d. In the case of heteroskedasticity and autocorrelation, the standard errors of the regression coeffcients must be estimated accordingly using, e.g., Newey-West standard errors.
(c) Recall from Homework 4 that we can write log returns as
rt+1 =κ×mbt+1 −mbt +et+1, (1)
where 0 < κ < 1 is a log-linearization constant (e.g., κ = 0.96). Iterate this equation forward to get an expression for mbt as a function of the infinite sum of future et+1 and rt+1. Using this equation, why are firm market-to-book ratios different across time and stocks?
mbt = Et ∞j=1 κj−1et+j − Et ∞j=1 κj−1rt+j
Firm market-to-book ratios are different across time and stocks either because future expected return on equity or expected future returns are different.
(d) Using the VAR you wrote down in 2.a and the equation above, derive the formulas for the following expectations:
i. Et (rt+1), Et (rt+2).
ii. E ∞ ρjr ,where|ρ|<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?
Et (rt+1) = κEt (mbt+1) − mbt + Et (et+1)
Et (rt+2) = κEt (mbt+2) − Et(mbt+1) + Et (et+2)
In the VAR, the eigenvalues of φ need to all be less than 1.
Define the vector ee = (1, 0) and emb = (0, 1).
Thus, eeZt+k = et+k and embZt+k = mbt+k
The k-period ahead forecast at time t is:
Et(eeZt+k) = ee(μ + φk(Zt − μ)) and Et(embZt+k) = emb(μ + φk(Zt − μ)) Where μ = (I2 − φ1)−1φ0
Therefore, we could get the cases for k=1 and k=2, which are,
Et (et+1) = ee(μ + φ(Zt − μ)), Et (et+2) = ee(μ + φ2(Zt − μ)).
Similarly we could get Et(mbt+1) and Et(mbt+2).
Et (mbt+1) = emb(μ + φ(Zt − μ)), Et (mbt+2) = emb(μ + φ2(Zt − μ)). Finally we could get,
Et(rt+1)=κemb(μ+φ(Zt −μ))−mbt +ee(μ+φ(Zt −μ))
Et(rt+2) = κemb(μ + φ2(Zt − μ)) − emb(μ + φ(Zt − μ)) + ee(μ + φ2(Zt − μ))
=E ∞ ρj(κ×mb −mb +e
t t+j t+j−1 t+j
) =emb((κ−1) ρ μ+(κρφ−ρ)(I2 −ρφ)−1(Zt −μ))
+ee( ρ μ+ρφ(I2 −ρφ)−1(Zt −μ))
We need to assume that κ is less than 1, limj→∞κjmbt+j = 0. We are
simply imposing the present value restriction in the initial approximation.
3. Volatility models
(a) Current market volatility is higher than the historical average of market volatil- 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?
The expected future volatility is likely to stay high in the near-term since market return volatility is clustered and persistent, but in the long-term it will revert back (in expectation) to the unconditional mean.
(b) Explain what ’realized variance’ is. Use both a mathematical expression and
an intuitive description.
σˆ2 = ni=1(rt,i−r ̄t)2
RV is simply the sample variance over time t. With high frequency data, this
can give an accurate description of variance over a short time interval. It assumes that errors are not autocorrelated.
(c) A Variance Swap is a very popular over-the-counter derivative contract. In the contract, the fixed leg pays a fixed dollar amount every month, while the floating leg pays an amount proportional to the realized variance in each month based on daily data. Assume any variance risk premium and the risk-free rate are both zero.
In particular, the payoff 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 fixed 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. Solution:
If we assume RVt = φ0 + φ1RVt−1 + εt Et(Payofft+2) = Et(RVt+1) + Et(RVt+2) − 2 × Ft =φ0 +φ1RVt +φ0 +φ1(φ0 +φ1RVt)−2×Ft =0 Ft = 12((φ1 +φ21)RVt +2φ0 +φ1φ0)
(d) Consider the GARCH(1,1) process σ2 = 0.1+0.07ε2 +0.92σ2. Using the fact t+1 t t
that σt2 ≡ Et−1 [ε2t ], write this GARCH(1,1) as an ARMA process in ε2t includ- ing the values of the ARMA coefficients. Why is it not technically appropriate to assume the residuals in this ARMA process are Normally distributed? Solution:
ε 2t = σ t2 + κ t
ε2t+1 = 0.1 + 0.07ε2t + 0.92(ε2t − κt) + κt+1
ε2t+1 = 0.1 + 0.99ε2t + κt+1 − 0.92κt
This shows ε2t follows an ARMA(1,1) process. It is not appropriate to assume the residuals are normally distributed because it may drive ε2 to be negative.
4. Factor models
You are evaluating a long-short equity hedge fund and are given the below regression results:
Re =0.03+1.5×MKT −0.3×HML −0.2×SMB +0.2×ε, (2) fund,t t t t t
where the factors are the FF3 factors and where ε is a standard Normal error term. Assume all coefficients are significant.
(a) What investment ’styles’ would you say characterizes this fund? Solution:
The fund invests in growth companies and large companies.
(b) What is the Information Ratio of this fund? In this calculation, assume the relevant benchmark has returns
Rb,t = 1.5×MKTt −0.3×HMLt −0.2×SMBt. (3)
Information Ratio= α =0.15
(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?
Maximum Sharpe ratio=
0.72 + 0.152 = 0.716
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