[SOLVED] R-finance: Private asset modeling STA457H1S assignment

Background
Modern finance theories are built for marked-to-market and liquid assets. However, the returns on private or alternative assets are usually appraised and illiquid. Appraisal returns lead to the stale-pricing bias and cause difficulties on analyzing private assets.
Dimsons model is widely used to formulate the relationship between appraisal returns and economic returns 1
yt =w0rt +w1rt1 ++wmrtm, (1)
where yt denotes appraisal returns, = E(yt), rt denotes economic returns, wi (0, 1), i = 0, 1, 2, . . . , m,
and mi wi=1.
In the literature, the multifactor model is usually used to model economic returns. Specifically, we consider
K
rt =+ jfj,t +t, (2)
i=1
where and j are constant, and fj,t denotes the j-th factor returns. Finally, putting above two models
together, we have
Kmm yt=+wijfj,ti+ witi. (3)
j=1 i=0 i=0
In this assignment, you are asked to estimate and j from a given appraisal and factor returns using
different methods.2 Specifically, we considers the following methods: Getmansky et al. (2005):
Getmansky et al. (2005) suggest retrieving economic returns from appraisal returns via reformulating a MA(q) models. They first assume that yt follows a MA(m) model
yt =0at +1at1 ++matm, 0 =1. (4)
Then they reparameterize the innovations of Equation (4) to retrieve the economic returns. Specifically,
they consider
m
yt = wirti,
i=0
1You may think of economic returns as the marked-to-market and liquid private asset returns.
2Read Lin (2017) and Pedersen et al. (2014) for more background information.
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where
and
i
wi =m , i = 0,1,2,,m, (5)
i=0 i
mrt=y +ati , (6)
i=0
where y =nt=1 yt is the sample mean of the appraisal returns.
Pedersen et al. (2014):
Pedersen et al. (2014) suggest estimating factor loadings by rearranging equation (3) as follows.
where
Km yt=+ jXj,t+ witi. (7)
j=1 i=0
m
Xj,t =wifj,ti. j = 1,2,,K, (8)
i=0
Lin (2017):
Lin (2017) suggests estimating of and j in Equation (3) from
where
and
mKm
yt =+i,jfj,ti + iuti, (9)
i=0 j=1 i=0
m
j =ij, j = 1,,K, (10)
i=0
ij
wi= m ,j. (11)
i=0 ij
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3. Problems
Question 1 (Getmansky et al. 2005, JFE):
1. Calculatetheappraisalweightsw0,w1,,wm usingEquation(5);
2. Retrieve the (unsmoothed) economic returns using Equation (6);
3. Estimate factor loadings by regressing the unsmoothed economic returns against quarterly factor returns3 ;
4. Discuss your results, including (1) the statistical significance of and j, and (2) conduct diagnostics on regression residuals and comment on the results.
Question 2 (Pedersen et al. 2014, FAJ):
1. Calculate the smoothed quarterly factor returns Xj,t, j = 1, . . . , K using the the appraisal weights
wi, i = 1, . . . , m estimated in Question 1 and Equation (8);
2. Estimate factor loadings using by regressing appraisal returns (yt) on smoothed factor returns (Xj,t);
3. Discuss your results, including (1) the statistical significance of and j, and (2) conduct diagnostics on regression residuals and comment on the results.
Question 3 (Lin 2017):
1. Estimate Eqn. (9) using quarterly factor returns; (Hint: arima in R.)
2. Calculate and j for j = 1,,K using Equation (10);
3. Calculate the appraisal weights wij implied by different factors using Equation (11);
4. Discuss your results, including (1) the statistical significance of and j, (2) compare the appraisal weights implied by different factors, and (3) conduct diagnostics on regression residuals and comment on the results.
Question 4 (Lin 2017):
1. Estimate Eqn. (9) using monthly factor returns; (Hint: arima and the mls function in midasr in R.) 2. Compare and comment on your estimates of and j with those of Question (3).
3For simplicity, you may calculate the quarterly factor returns as the sum of the monthly factor returns within a given quarter.
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3. Data
1. Appraisal returns (yt): Quarterly Cambridge US private equity (PE) index returns (available at Course Materials)
2. Factor returns (fj,t): Fama-French benchmark factors (available online)
4. Reading
1. Dimson, Elroy, 1979, Risk Measurement When Shares Are Subject to Infrequent Trading, Journal of Financial Economics, Vol 7, 197-226.
2. Getmansky, Mila, Andrew W. Lo, and Igor Makarov, 2005, An econometric model of serial correlation and illiquidity in hedge fund returns, Journal of Financial Economics, Vol 74, 529-609.
3. Lin, Jenwen, Direct Estimation of Factor Exposures from Appraisal Returns (March 9, 2017). Available at SSRN: https://ssrn.com/abstract=2935424 or http://dx.doi.org/10.2139/ssrn.2935424
4. Pedersen, Niels, Sebastien Page, and Fei He, 2014, Asset Allocation: Risk Models for Alternative Investments, Financial Analysts Journal, Vol 70, No. 3.
5. STA457H1S Course slides on July 28, 2017.
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