Estimating CAPM Beta with Irregular Asset Observations
Summary
The note considers CAPM beta estimation for an asset that trades infrequently while a related price index is available monthly. It questions a proposed procedure that regresses price levels on the index, interpolates monthly asset prices, converts those estimates to returns, and then regresses asset returns on market returns. A high price-level fit does not establish that the resulting return beta is valid.
The answer recommends modeling returns rather than price levels and explains that interpolation assigns arbitrary values to unobserved prices, understating the variance over missing intervals. It proposes maximum likelihood estimation under an explicitly parameterized joint stochastic model. For nonsynchronous observations, the likelihood should account for the conditional distribution of missing prices given observations on both sides; that distribution is wider than one based on synchronous data. The note does not prescribe a specific stochastic model or provide an empirical comparison, so model choice and assumptions remain essential.
Key ideas
- CAPM beta is based on returns, not a regression of price levels.
- Interpolating unobserved prices can understate asset price variance and impose arbitrary intermediate values.
- Maximum likelihood can accommodate irregular observations when a joint stochastic process is specified.
- The model should account for uncertainty in missing prices using information from observations before and after the gap.
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Full text
# Estimating Beta from unevenly spaced price history
# Estimating Beta from unevenly spaced price history
I have a certain non-stock asset that has 1 transaction every 1 to 8 months. I also have a price index of that class of asset compiled by another party on monthly basis. If I regress $price = \alpha' + \beta' index$, the $R^2$ is 0.975 to 0.999 for every asset.
How do I obtain an estimate of $\beta$ of CAPM $R_i = R_f + \beta(R_m-R_f)$
Right now, I am blindly guessing it through the following steps: 1. Regress $price = \alpha' + \beta' index$, obtain $\alpha'$ and $\beta'$ 2. Generate interpolated monthly price $\widehat{price}_t$ by plugging in the monthly index 3. Generate monthly asset return $\hat{R}_{t+1}=\frac{\widehat{price}_{t+1}}{\widehat{price}_t}-1$ 4. Generate monthly market return $\hat{R}_{m_{t+1}}=\frac{index_{t+1}}{index_t}-1$ 5. Regress $\hat{R_{t}}$ on $\hat{R}_{m_{t}}$ and obtain $\beta$ of CAPM
Is this method valid? If not, what would be the proper convention?
Possibly relevant references: Eckner, Andreas (2012). A Framework for the Analysis of Unevenly Spaced Time Series Data. Scholes, M. and J. Williams (1977). Estimating betas from nonsynchronous data.
Possibly relevant questions: How do you estimate the volatility of a sample when points are irregularly spaced? How to interpolate gaps in a time series using closely related time series?
## Answer by André Levy (score 2)
https://quant.stackexchange.com/a/11034
No, it's not. First, what you ought to be regressing are returns, not prices. Second, by interpolating you're underestimating the variance of the asset price in the interval between index price observations. Through your choice of interpolation method, you're essentially picking an arbitrary price in the middle.
What you ought to be doing is maximum likelihood estimation (MLE). You'll have to assume a parameterized family of joint stochastic processes and estimate the parameters given the price observations. Whenever you don't have synchronous data, you'll have a probability distribution for the missing price conditional on all other data points (in its future and in its past). Hence the distribution you'll be using to maximise the likelihood of the observed price will be wider than otherwise.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.