Case 008Regression and model reviewCore

An illiquid mid-cap has a beta of 0.55 from daily returns but 0.85 from weekly returns. Explain the gap, and compute a Dimson beta from lag coefficients of 0.55, 0.22 and 0.08.

The situation

Vardhira Pharma is a thinly traded mid-cap: on many days only a few hundred shares change hands, and the last trade can be hours old at the close. A risk analyst regresses its returns on the market index and gets a beta of 0.55 from three years of daily data but 0.85 from weekly data over the same period.

A second regression puts today's, yesterday's and the day before's market returns on the right-hand side of the daily regression. The coefficients come out at 0.55, 0.22 and 0.08.

Your task

Explain why the two betas differ, compute the Dimson beta, and say which number you would use to hedge a position or to set a cost of equity.

Quick check

Why is the daily beta lower than the weekly one?

Worked solution

Try it on paper, then open one step at a time.

30-second answerThe answer to give first

The Dimson beta is 0.55 + 0.22 + 0.08 = 0.85, matching the weekly beta. Vardhira's price responds to the market late because it trades rarely, so a same-day regression sees only the first 0.55 of its sensitivity. Summing the lagged coefficients recovers the rest. Use about 0.85 for a hedge held longer than a few days and for a cost of equity; 0.55 would under-hedge a Rs 10 crore position by about Rs 3 crore of index.

Step 1Why does an illiquid stock look less sensitive on daily data?

Picture a village shop that updates its prices only when the delivery van comes, every few days. Prices in town rise on Monday, and the shop's prices follow on Wednesday. Vardhira's closing price is often the last trade of the afternoon, or of yesterday, so it cannot reflect today's market move until someone next trades it. This is non-synchronous tradingWhen a stock trades less often than the market index, its recorded closing price is stale, so its measured returns lag the market.: the response is real, but it arrives on day 1 and day 2, and a regression of same-day returns attributes it to noise.

Step 2How does the Dimson correction work?

Give the regression a chance to see the late response. Regress today's stock return on today's, yesterday's and the day before's market returns, then add the slopes. Here 0.55 arrives the same day, 0.22 a day later and 0.08 two days later, a total of 0.85, the same as the weekly beta. The weekly beta agrees because a week is long enough for most of the catch-up to fall inside the same return. The Dimson method keeps the daily data, and its larger sample, while fixing the timing problem.

The relationship
rt=α+β0mt+β1mt−1+β2mt−2+εtβDimson=0.55+0.22+0.08=0.85r_t = \alpha + \beta_0 m_t + \beta_1 m_{t-1} + \beta_2 m_{t-2} + \varepsilon_t \qquad \beta_{\text{Dimson}} = 0.55 + 0.22 + 0.08 = 0.85
r_tVardhira's return on day t
m_tmarket return on day t
\beta_kresponse to the market move k days ago
What it says in wordsPut lagged market returns into the regression and add their coefficients, so the late part of the response is counted.
Where Vardhira's beta hides: in the days after the market moves0.20.40.60.80.55Same day0.22One day later0.08Two days later0.550.220.08Dimson beta 0.85Sum of the threeweekly beta 0.85daily beta 0.55Coefficients from one regression on today's and the two previous days' market returns
Vardhira's same-day coefficient is 0.55, but it adds 0.22 a day later and 0.08 two days later, so the Dimson beta is 0.85, equal to the weekly beta; the daily beta misses the part of the response that arrives late.
The market rises 1% on day 0: when each stock catches upLiquid stockVardhiraDay 0Day 1Day 2+0.85% at once+0.55%running total 0.55%+0.22%running total 0.77%+0.08%running total 0.85%Same destination, reached late: a same-day regression only sees the first step.
After a 1% market rise, a liquid stock with beta 0.85 moves 0.85% the same day, while Vardhira moves 0.55%, then 0.22% and 0.08% over the next two days, reaching the same total late.
Step 3Which beta do you use, and what does the wrong one cost?

Use 0.85 for anything held over more than a few days. A Rs 10 crore position hedged at 0.55 carries only Rs 5.5 crore of index short instead of Rs 8.5 crore, leaving Rs 3 crore of market exposure unhedged; a 10% market fall costs about Rs 30 lakh that the hedge was meant to stop. For a cost of equity, with an illustrative risk-free rate of 7% and equity premium of 6%, the two betas give 10.3% and 12.1%, a gap that moves a valuation materially. Treat those rates as placeholders and confirm current figures before using them.

State the limits. Each lag coefficient comes with its own standard error, and adding three noisy estimates gives a noisier total, so check that the lags are individually meaningful before adding them. Add a lead term as a sanity check: a stock should not respond to tomorrow's market, and a significant lead points to a data alignment problem, such as time zones or a misdated file. The Scholes and Williams estimator is an alternative built for the same problem.

Where candidates lose it

The common loss is concluding the stock is defensive because its daily beta is 0.55. The low number is a measurement artefact of stale prices, and hedging or valuing on it understates the real market exposure.

The second is saying weekly data is always right. Weekly betas fix the timing but use one fifth of the observations, so they are noisier; the Dimson regression gets the timing right while keeping the daily sample.

What the interviewer asks next

  • How many lags would you include, and how would you decide?
  • Would you expect the same problem for the index itself? What does that do to the betas of liquid stocks?
  • The one-day lead coefficient comes out at 0.15 and significant. What do you check first?

Company names and figures are illustrative.