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.
Dimson’s model is widely used to formulate the relationship between appraisal returns and economic returns 1
yt −μ=w0rt +w1rt−1 +···+wmrt−m, (1)
where yt denotes appraisal returns, μ = 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 =α+ βjfj,t +εt, (2)
i=1
where α and βj are constant, and fj,t denotes the j-th factor returns. Finally, putting above two models
together, we have
Kmm yt=α+ wiβjfj,t−i+ wiεt−i. (3)
j=1 i=0 i=0
In this assignment, you are asked to estimate α and βj 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 =θ0at +θ1at−1 +···+θmat−m, θ0 =1. (4)
Then they reparameterize the innovations of Equation (4) to retrieve the economic returns. Specifically,
they consider
m
yt = wirt−i,
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.
1
where
and
θi
wi = m θ , i = 0,1,2,…,m, (5)
i=0 i
m rt=y ̄+at· θi , (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=α+ βjXj,t+ wiεt−i. (7)
j=1 i=0
m
Xj,t = wifj,t−i. j = 1,2,…,K, (8)
i=0
• Lin (2017):
Lin (2017) suggests estimating of α and βj in Equation (3) from
where
and
mKm
yt =α+ βi,jfj,t−i + θiut−i, (9)
i=0 j=1 i=0
m
βj = βij, j = 1,…,K, (10)
i=0
βij
wi= m β ,∀j. (11)
i=0 ij
2
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 α and βj, 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
wˆi, 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 α and βj, 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 α and βj 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 α and βj, (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 α and βj 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.
3
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.
4