**A. CAPM and the Sharpe Ratio**

In the Capital Asset Pricing Model (CAPM), price volatility of an investment portfolio is the sole measure of that portfolio's risk. It alone determines the expected return premium (over that for a risk free asset) that an investor will require/demand for him his capital at such risk. The greater is the price volatility (risk) of such a portfolio, the greater is the premium demanded by the investor. The behavior of a portfolio's volatility relative to that of the the Market as a whole, is termed Beta. Thus, for a portfolio with a Beta of 1.0, the investor would demand the Market risk premium.

When comparing two different portfolios, the investor would like to compare their risk-adjusted returns. Specifically he would like to know such a thing as: for a given level of risk, which portfolio offers the higher expected return? While the future is uncertain, the investor can learn much from studying the past. How did a particular portfolio’s returns, and standard deviation of returns, compare to a risk free asset? Sharpe, of course, provided his famous ratio for doing just that.

Sharpe Ratio (simplified) = (portfolio return - risk-free return), divided by, (S.D. portfolio return - S.D. of risk-free return)

**B. Fama-French**

Precisely because a three-factor model better accounted/explained (for) the return behavior of Small and Value stocks than did the one-factor (volatility) CAPM, Fama and French (1992) proposed two additional ‘factors’, distinct from, and independent of, volatility, into their pricing model. They labeled these factors SmB (Small minus Big) and HmL (High minus Low Book-to-Market).

Because they (FF) believe that the Market is Efficient, they believe that these two factors must reflect underlying risk. That is, for exposure to SmB and HmL ‘factors’ to account for a premium over and above the volatility premium, these factors must be reflecting two additional, distinct and independent, dimensions of risk other than volatility. Thus, the Fama-French 3-factor (FF3F) model states/predicts that a equity portfolio’s returns are overwhelmingly determined by that portfolio’s exposure to three dimensions of risk:

1) Volatility (Market) risk

2) SmB risk

3) HmL risk

**C. Non-Volatility Risk**

In the CAPM and FF3F models, quantitation of volatility risk is straightforward- it is Standard Deviation. In the CAPM no further quantitation of risk is necessary as volatility risk is the only risk recognized. That is not, however, the case with the FF3F model.

As we have seen, the FF3F model ascribes return, in addition to volatility risk, to two non-volatility risks: SmB and HmL. Standard Deviation is silent on the magnitude of non-volatility risks!

**D. The Z-Ratio**

1.Definition:

1.Definition:

In CAPM, the Sharpe ratio (discussed above), reflects the risk/return relationship of a portfolio. Is there an analogous measure of risk/return for portfolios considered in the context of the FF3F model? Recall, the Sharpe Ratio is Return/Risk. Risk in CAPM is Standard Deviation (SD), therefore the Sharpe Ratio is simply Return/SD. However, in FF3F, the denominator of Return/Risk is: Volatility Risk + SmB Risk + HmL Risk.

Let us call:

**Z-Ratio = [Return - risk free return], divided by, [Sum(Volatility + SmB + HmL Risks) - volatility of risk free asset]**

Unfortunately, the Z ratio can be calculated for only one portfolio- the Market portfolio. For the Market Portfolio: SmB Risk = HmL Risk = Zero, and,

**Z-Ratio (Market) = (Market Return - risk-free return)/(Market SD + 0 + 0 - risk-free SD) = Sharpe Ratio.**

For any tilted portfolio, SmB and HmL risks cannot be calculated (at this time). Hence,

**However, while not calculable, Z-Ratio for a tilted portfolio can be estimated based on a portfolio's historic behavior in combination with one's view of Market Efficiency. Risk/Reward relationships can be estimated for tilted portfolios.**

*for any tilted portfolio, the Z-Ratio is incalculable, and the Sharpe Ratio is irrelevant!*(To be continued)

Z.

Edit1:

Slight tweaking of Sharpe and Z-Ratio equations.

Clarification of last two sentences.

Edit2: Only to change Subject line.

Edit3: Only to change Subject line.

Edit4: Only to change Subject line.

Edit5: Only to change subject line.

Edit 6: Clarified definition of Beta and Z-ratio