Outline regression and call option model techniques for gauging a fund manager's market timing skill, and work out the return that a manager earns from timing.

FRM® Part II · Risk Management and Investment Management · Portfolio Performance Evaluation · FRM2.IM.7.h

The idea

Market timing means shifting between the market and the risk-free asset according to forecasts, so beta should be higher when the market is about to rise. The Treynor-Mazuy regression adds a squared term to the market model. Rp − Rf = a + b(Rm − Rf) + c(Rm − Rf)² + e. A positive, statistically significant c is consistent with timing skill. The Henriksson-Merton regression uses a dummy variable D, equal to 1 when the market beats T-bills and 0 otherwise. The regression then adds one interaction term. Rp − Rf = a + b1(Rm − Rf) + b2 × D × (Rm − Rf) + e. A positive, statistically significant b2 is consistent with timing ability under the model, because it means a higher beta in up markets. In option terms, a perfect timer earns the larger of Rm and Rf, which is T-bills plus a call on the market struck at the risk-free payoff. The value of timing per dollar is that call's price, for example from Black-Scholes. Beware that a positive intercept does not measure timing, and that noise or misspecification can give a positive estimate.

Formulas

Treynor-Mazuy market timing regression

Rp − Rf = a + b × (Rm − Rf) + c × (Rm − Rf)² + e

Rp: the fund's return
Rf: the T-bill rate
Rm: the market return
a: the intercept
b: the market beta
c: the timing coefficient on the squared market excess return
e: the residual

A positive, significant c suggests timing skill. Write returns as decimals when squaring, and do not read the intercept as timing.

Henriksson-Merton market timing regression

Rp − Rf = a + b₁ × (Rm − Rf) + b₂ × D × (Rm − Rf) + e

Rp: the fund's return
Rf: the T-bill rate
Rm: the market return
a: the intercept
b₁: the down-market beta
b₂: the extra beta in up markets
D: 1 when the market beats T-bills, 0 otherwise
e: the residual

A positive, significant b₂ is consistent with timing ability. In a down month D is 0, so the up-market increment does not apply.

Return of a perfect market timer

timer return = max(Rm, Rf)

Rm: the market return
Rf: the T-bill return

Equal to Rf plus a call on the market struck at the T-bill proceeds, so timing is worth the call's price. In down years the timer earns the T-bill rate, not zero.

A practice question

From the question bank, with the reason each wrong option is wrong.

In the Treynor-Mazuy approach to measuring market timing, which finding indicates timing skill?

A significantly positive coefficient on the squared market excess return. - correct
A significantly positive intercept (alpha).Why not: Alpha reflects security selection and other effects, not timing.
A beta above one.Why not: A high beta is risk-taking and says nothing about changing beta ahead of the market.
A significantly negative coefficient on the squared market excess return.Why not: A negative quadratic term indicates perverse timing: beta falling when the market rises.

The regression Rp - Rf = a + b(Rm - Rf) + c(Rm - Rf)² + e has a positive c if beta rises when the market's excess return is high, the sign of good timing.

A harder one

A Treynor-Mazuy regression for a fund gives a = 0.1%, b = 0.8 and c = 2.0 for monthly excess returns written as decimals: Rp - Rf = a + b(Rm - Rf) + c(Rm - Rf)². If the market excess return in a month is +5%, the fund's expected excess return is closest to:

4.1%Why not: Omits the quadratic term c x (Rm - Rf)² that captures timing.
14.1%Why not: Uses c x (Rm - Rf) without squaring the market excess return.
4.5%Why not: Omits the intercept a = 0.1% from the sum.
4.6% - correct

0.001 + 0.8 x 0.05 + 2.0 x 0.05² = 0.001 + 0.04 + 0.005 = 0.046, or 4.6%.

The hardest tier

A perfect market timer holds the market portfolio over one year if it will beat T-bills and T-bills otherwise. The market's annual volatility is 18% and the continuously compounded risk-free rate is 4%. Using the Black-Scholes call value on the market (no dividends) with exercise price equal to the T-bill proceeds, the value of this timing ability as a percentage of the portfolio is closest to:

7.2% - correct
9.2%Why not: Uses a call struck at the current market level rather than at the T-bill proceeds.
3.6%Why not: Takes N(d1) - 0.5 as the option value, which is not the Black-Scholes price.
53.6%Why not: Uses N(d1), the option's delta, as the call value in percent.

With strike = S x e^(rT) the discounted strike equals S, so d1 = sigma/2 = 0.09 and d2 = -0.09. Call = S x [N(0.09) - N(-0.09)] = 0.0717 x S, about 7.2% of the portfolio.

In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.

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