RISK: CAPM/Sharpe Ratio:FF3F/Z-Ratio (Parts 1-5)

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RISK: CAPM/Sharpe Ratio:FF3F/Z-Ratio (Parts 1-5)

Post by zalzel » Sat Mar 15, 2008 3:58 pm

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:


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, for any tilted portfolio, the Z-Ratio is incalculable, and the Sharpe Ratio is irrelevant! 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.

(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
Last edited by zalzel on Thu Mar 20, 2008 2:36 pm, edited 5 times in total.
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RISK: CAPM/Sharpe Ratio:FF3F/Z-Ratio (Part 2)

Post by zalzel » Sat Mar 15, 2008 9:16 pm

2. The Z-Ratio and Market Efficiency

In the FF3F model, any portfolio can be described by its degree of exposure to Market (volatility), SmB, and HmL risks. Three coefficients: Beta, S, and V, respectively, are employed to describe the degrees of exposure. For example, the Market portfolio is: Beta = 1, S = 0, V= 0 (or 1/0/0 for shorthand).

Any portfolio’s return (above that of the risk-free asset) is:
Beta* Equity Risk Premium + S* SmB Premium + V* HmL Premium

(SmB and HmL premiums into the future can be estimated by what they appear to have been in the past, tempered by any considerations one believes will impact them going forward. The historical premiums can be downloaded at: http://mba.tuck.dartmouth.edu/pages/fac ... actors.zip)


Any portfolio’s risk is:
Sum[Beta*S.D., S*SmB risk, V*HmL risk].

Any portfolio’s Z-Ratio is:
[portfolio return – risk-free return]/[portfolio risk – S.D. of risk-free return]


We know the Z-Ratio for any combination of the Market portfolio with the risk free asset; it is the same as the Sharpe Ratio. What about a tilted portfolio combined in the same proportion with the risk free asset? Not being able to determine values for SmB and HmL risk, we cannot calculate the Z-Ratio (and, as has been explained previously, the Sharpe Ratio is irrelevant). Nevertheless, we can say something about it (Z-Ratio).

If (if, if, if, if, if, etc., etc., etc…) the Market is (Fama, 1970) Efficient (see appendix below), any tilted portfolio/risk-free asset combination must have a Z-Ratio less than the same Market portfolio/risk-free asset combination. That is simply because the definition of 'efficient market' being employed makes it tautologic that the Market portfolio lies on the 'Efficient Frontier' (of all equity portfolios mixed in all possible combinations with the risk-free asset). That is, it is the most efficient possible portfolio, and all (tilted) sub-portfolios must lie below the efficient froniter. How far below? That cannot be calculated and is, thus, open to conjecture. Thus, (in an efficient market) the Z-Ratio for the Market portfolio represents an upper bound on what the Z-Ratio may be for a tilted portfolio.

There is no need, at this point, to make a vigorous defense of Fama’s Efficient Market Hypothesis (EMH). The model that is in the process of being described allows adjustment for one’s personal belief in the strength, or lack thereof, of EMH. For example, those who believe that any historical Small/Value premiums were due to behavioral factors (i.e. that they were due to human psychology rather than risk) are free to conclude that:
SmB risk = HmL risk = zero, and thus that any tilted portfolio’s Z-Ratio can be calculated using S.D. as the sole measure of risk. That is, the Z-Ratio collapses to the Sharpe Ratio. Those who take a more moderate view on the possible contribution of behavioral factors to any historical S/V premiums can adjust their estimate for the Z-Ratios accordingly. As a quick reminder, it is the Z-Ratio that one truly desires. It incorporates all dimensions of risk, not just volatility.

To be continued…

Z.

Appendix:
Fama, 1970 wrote:
An ‘efficient’ market is defined as a market where there are large numbers of rational, profit ‘maximisers’ actively competing, with each trying to predict future market values of individual securities, and where important current information is almost freely available to all participants. In an efficient market, competition among the many intelligent participants leads to a situation where, at any point in time, actual prices of individual securities already reflect the effects of information based both on events that have already occurred and on events which, as of now, the market expects to take place in the future. In other words, in an efficient market at any point in time the actual price of a security will be a good estimate of its intrinsic value.
Edit1:
- Better description of 'Efficient Frontier'
- specified lower bound for the Z-Ratio of tilted portfolios

- Better isolation of the Fama quote.

Edit2: correction of errors made in Edit1. That is, no specified lower bound for Z-Ratio of tilted portfolios. :?
Edit3: correct typo in (paranthetic) portion of behvaviorist definition.
Edit4: clarified all equations. In particular Sum[Mkt risk, Small risk, value risk] is meant to explicitly reflect that the sum of the risk is not simply obtained by arithmetic addition. (Thanks to Rodc for requesting clarification).
Last edited by zalzel on Thu Mar 20, 2008 2:44 pm, edited 4 times in total.
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RISK: CAPM/Sharpe Ratio:FF3F/Z-Ratio (Part 3)

Post by zalzel » Sat Mar 15, 2008 11:36 pm

3. What Does the Investor Truly Wish to Know?

What does an individual investor truly wish to know when considering a particular portfolio? He wishes to be able to weigh and balance the expected return against the risk posed. He wants the Z-Ratio! To facilitate focus and clarity, let’s (for now) consider only that investor who plans to use his portfolio for retirement funding.
The investor must balance his desired retirement withdrawal plan (i.e. desired portfolio consumption in retirement), against his tolerance for the risk required to build up and maintain sufficient capital to fund such a withdrawal (consumption) plan.
Adapted from: http://www.diehards.org/forum/viewtopic ... highlight=

Expected return is straightforward enough (though not necessarily simple) to estimate and/or calculate. One can look to what the same portfolio has done in the past, or, if that isn't available, calculate expected return from the FF coefficients (Beta, S, V) and estimated risk (market, SmB, HmL) premiums. What about risk (Z-Ratio)? We will first need a definition of risk.
Given a specified consumption (i.e. withdrawal) plan, a portfolio's risk is the probability of it reaching zero value (or a specified minimal value) before the consumption plan is completed (e.g. death of the portfolio holder).
Adapted from: http://www.diehards.org/forum/viewtopic ... 883#173883

To be continued…

Z.
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RISK: CAPM/Sharpe Ratio:FF3F/Z-Ratio (Part 4)

Post by zalzel » Sun Mar 16, 2008 2:56 am

E. Inherent Portfolio Risk = Z

Let us look at a simulated return presentation that is typical of many we see on the Bogleheads forum (in blue color). My added comments appear in black.

Adapted from http://www.diehards.org/forum/viewtopic ... 227#167227
Trev H wrote: 1970-2007 with backtested data (yearly rebalancing):

30% US Cap Weighted Market
30% EAFE/EM
40% InterTerm Treasury
=====================
CAGR: 10.68
StDev: 10.80
Sharpe: 0.5


Note: It appears that Trev H used a risk-free rate of return of 5.3% to calculate Sharpe ratios.
This is a (global) market equity portfolio/risk free treasury combination that represents one point on the Frontier of all global market/treasury portfolios. (Because this is a market portfolio, it lies on the upper edge of the frontier). The equity component has FF coefficients of 1/0/0 (Beta/S/V). (I won’t, at this time, go into the subtle difference between this global Beta and the U.S. equity market Beta.) Thus, for this market portfolio with no SmB and HmL exposure, the Z-Ratio is .5 (same as the Sharpe Ratio).
15% US Cap Weighted Market
15% US Small Value
30% EAFE/EM
40% InterTerm Treasury
=======================
CAGR: 11.24
StDev: 10.54
Sharpe: 0.56
This is similar to the above portfolio except it has some S/V tilt. I don’t know the source of Trev H’s numbers, so I cannot be confident of the S and V coefficients. It doesn’t matter for our present purposes. The Sharpe Ratio was calculated (I am guessing) as:
(11.24 – 5.3)/10.54 = .56.
As has now been repeated more than once, this Sharpe ratio tells us little about this portfolio’s actual Return/Risk relationship (i.e. Z-Ratio).

Z-Ratio = [11.24 – 5.3]/[Sum(10.54, S*SmB risk, V*HmL risk) (= 5.94/Risk

Let’s give the quantity in the denominator of the Z-Ratio a name. Recall, it represents the value of inherent portfolio risk. Let’s call it “Z”.

We don’t know the values for the last two terms in the above equation, so we cannot calculate Z directly. All we can say with confidence is that, if the market is Fama-efficient, the Z-Ratio is less than the Z-Ratio of the market portfolio above. Thus the Z-Ratio is less than .5 (or, to put it another way, .5 is greater than Z-Ratio).

Thus,
.5 is greater than (g.t.) 5.94/Z
Z is g.t. than 5.94/(.5)
Z is g.t. than 11.88

“Z is g.t. 11.88”! What does that mean? It means that this portfolio has an inherent risk greater than any portfolio with Z less than or equal to 11.88. It means, that this portfolio has inherent risk greater than the market portfolio (above) that has a Z of 10.54. How much g.t. 11.88 might Z be? More on that later...

What do we do with this Z? Turn from consideration of of inherent portfolio risk to the real risk posed to a real investor.

To be continued...

Z.

Edit1: Changed all instances of 'absolute' to 'inherent'.

Edit2: See edit4, part 2.
Last edited by zalzel on Thu Mar 20, 2008 2:48 pm, edited 1 time in total.
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Post by Rodc » Sun Mar 16, 2008 2:52 pm

Just skimming at this point, but doesn't this formula: Volatility Risk + SmB Risk + HmL Risk assume zero diversification of risk?

In general in the mean/variance world you have a formula like:

Variance(aX + bY) = a^2*Var(X) + b^2*Var(Y) + 2*a*b*COV(X,Y).

If the correlations in the risk factors are (nearly) zero the last term drops out, leaving

Variance(aX + bY) = a^2*Var(X) + b^2*Var(Y)

Standard deviation is the square root of each side, so you get sort of a Pythagorean Theorem. The important thing is that risks do not simply add: this of course is the whole point behind diversifying a portfolio.

Shouldn't something like this hold here as well?
We live a world with knowledge of the future markets has less than one significant figure. And people will still and always demand answers to three significant digits.

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RISK: CAPM/Sharpe Ratio:FF3F/Z-Ratio (Part 5)

Post by zalzel » Sun Mar 16, 2008 4:31 pm

E.2. The Nature of Z (What is it?)

Before looking into the use(s) of Z in portfolio decision-making, let us try to understand it better. Z can be quantified (in theory, and often in reality) for any portfolio of equities. It is the inherent risk of that portfolio, and within a FF framework for asset pricing, that inherent risk is:

Z = Sum[Beta* Market risk, S*SmB risk, V*HmL risk].

a. Volatility component

We are very familiar with Beta*Market risk. It is the measure of portfolio risk in CAPM. ‘Market risk’ is the S.D. of returns of the market portfolio. In both CAPM and FF, the market portfolio has a Beta of 1.0. Tilted portfolios in FF all have a Beta very close to 1. Beta ranges more widely in CAPM. (While instructive, this last point is of no particular import for current considerations).

b. Non-volatility component

As required by Fama-EMH, and the historic behavior of small and high BTM (book-to-market) stocks not explained by CAPM, FF risk include two additional terms to capture the non-volatility risk of Small and high BTM stocks. The unit of measure for non-volatility risk is identical to that for volatility risk: percent/yr. Percent/yr of what exactly? Presumably, percent/yr of (average) yearly portfolio return. Z would appear to be an enlarged ‘virtual S.D.’ that encompasses non-volatility risk in addition to volatility risk. Z, it seems, is an ‘implied S.D’. It implies the S.D. that would be seen if the non-volatility risk in question had manifested itself. The non-volatility component of Z is thus a measurement of risk that, while not having yet manifested itself, may well do so at any time in the future. Non-volatility risk is ‘lurking' risk.


c. Is Non-Volatility (Lurking) Risk ‘Real’?

The (arguably) most serious challenge to the FF pricing model is that it has not satisfied those who demand definition and demonstration of the unique risks inherent to small and high BTM stocks. ‘Where’s the risk?’ they ask. Indeed, the historical performance of small and high BTM stocks makes them appear attractive precisely because it is difficult to discern how an investor in such stocks might have been harmed (over and above the harm suffered by the general market investor) to such a degree as to warrant such very handsome extra premiums.

Robert T, at http://www.diehards.org/forum/viewtopic ... 750#167750 presents a list of reasons to believe that small and high BTM stocks carry risk that warrant a/some premium. There is, however, no accompanying framework for quantifying the premiums to be demanded for these risks. Again, skeptics want to be shown the real-world damage done by these ‘risks’ that justifies the significant premiums they have enjoyed (over and above the market premium). To be clear, the skeptics referred to here are those that believe that SmB and HmL premiums have not been appropriate compensation for risk, but rather, ample reward/bonus for those not detered by irrational considerations (e.g. fear, greed). Such skeptics believe that the possibility/probability that historical SmB and HmL premiums will persist, offers the delicious possibility of a free (and ample) lunch! That is, anyway... for those who have no fear that there may be poison in the soup!

To be continued…

Z.

Edit1: Italicization added. General improvement in readability. No change in meaning.
Edit2: See edit4, part 2.
Last edited by zalzel on Thu Mar 20, 2008 2:50 pm, edited 1 time in total.
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Corrrelations

Post by Robert T » Sun Mar 16, 2008 7:26 pm

.
Zalzel,

Just three thoughts/observations:

First, I’m not convinced that the risk of 'value' has not shown up in the last 80 years, so think that the historical data are important (we have had a WW, ‘great’ depression, many financial crises, oil embargos etc). I agree, the future in unknowable and 'black swans' exists - but the historical data is still instructive.

Second, IMO SD’s are not 100% additive, their correlation affects the resulting total portfolio SD (point raised by Rod). Consider the example below:

Code: Select all

                     Vanguard	   Vanguard	     70:30		
                       TSM     Int. Trsy (IT)	  TSM:Int.Trsy		
             1993      10.6        11.4             10.9	
             1994      -0.2        -4.3             -1.4	
             1995      35.8        20.4             31.2	
             1996      21.0         1.9             15.3	
             1997      31.0         9.0             24.4	
             1998      23.3        10.6             19.5	
             1999      23.8        -3.5             15.6	
             2000     -10.6        14.0             -3.2	
             2001     -11.0         7.6             -5.4	
             2002     -21.0        14.2            -10.4	
             2003      31.4         2.4             22.7	
             2004      12.5         3.4              9.8	
             2005       6.0         2.3              4.9	
             2006      15.5         3.1             11.8	
             2007       5.5        10.0              6.8	
					
Annualized Return      10.3         6.6              9.6
Standard deviation     16.9         6.9             11.8

Source: Vanguard website
  • The resulting SD of the 70:30 portfolio is closer to (but not exactly):

    0.7*SD TSM + (correlation coefficient TSM:IT)*0.3*SD = Portfolio SD
    0.7*16.9 + (-0.09)*0.3*6.9 = 11.7

    Rather than:

    0.7*SD TSM + 0.3*SD IT = Portfolio SD
    0.7*16.9 + 0.3*6.9 = 13.9

    IMO, the same applies to TSM, SmB, HmL. Here is an example, using the FF research factor data series from 1927-2007.

Code: Select all

1927-2007

        SD      Correl with Mkt
Mkt   20.1             1.00
SmB   14.5             0.41 
HmL   14.1             0.12

Source: FF research factors, Ken French website
  • The SDs of SmB and HmL are relatively large, but the correlations are relatively small.

    A 1,0.2,0.4 Mkt, SmB and HmL loaded portfolio would have a SD closer to:

    1*SD Mkt + (correlation coefficient Mkt:SmB)*0.2*SD SmB + (correlation coefficient Mkt:HmL)*0.4*SD HmL = Portfolio SD

    20.1 + 0.41*0.2*14.5 + 0.12*0.4*14.1 = 22.0

    [although this ignores the SmB:HmL correlation]

    than to

    1*SD Mkt + 0.2*SD SmB + 0.4*SD HmL = Portfolio SD

    20.1 + 0.2*14.5 + 0.4*14.1 = 28.6

    The SD of the actual 1*Mkt +0.2*SmB + 0.4*HmL series was 22.8. IMO, this implies that expectations about correlations among Mkt, SmB, and HmL matter to ‘estimated’ expected return/SD characteristics of the portfolio.
Third, as indicated earlier, ‘total risk’ is also IMO measured against other ‘non-market’ risks such as the correlation of portfolio returns to personal earning (in the accumulation stage). So if human capital and financial capital returns and SD were included in one portfolio then the SD may give a closer indicator of ‘total risk’.

Robert

Edit:

Just a note when the 'risk of value' showed up (IMO):

Code: Select all

       Standard deviation
             of returns

       1927-2007   1930-57

Mkt       20.1       25.0
SmB       14.5       14.8 
HmL       14.1       15.7

The relative standard deviations look similar (table above), but the correlations differed significantly (as in the table below).

Code: Select all

         
       Correlation coefficients
               with Mkt

       1927-2007   1930-57
           
Mkt       1.00       1.00
SmB       0.41       0.57 
HmL       0.11       0.69

IMO the risks showed up when the correlations of Mkt and HmL increased significantly (to 0.69 compared to the 0.11 for the full period), and this is reflected in the SD and Sharpe ratio differences across the two periods (as shown below).

Here are the return/SD characteristics of a 1, 0.2, 0.4 Mkt, SmB and HmL loaded portfolio (tilted) relative to the Mkt portfolio for the two time periods:

Code: Select all

                  Annualized      Standard        Sharpe
                   Return         Deviation        Ratio

1927-2007
Mkt                10.06            20.11         0.409
Tilted             12.31            22.84         0.421

1930-56
Mkt                 8.43            11.42         0.429
Tilted              9.98            30.52         0.322

Source: Derived from the Ken French data
.

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SmB and HmL risks defined?

Post by zalzel » Sun Mar 16, 2008 8:50 pm

Hi Robert.

I will not comment on your first point, as that is one for the behaviorists to address. I have not (yet) become one (behaviorist), but I don't exclude the possibility that before all is said and done I will have done so. :wink:

I understand your third point and have given it consideration. It does not impact an analysis (such as the current one) which is based on Fama-EMH. Those not satisfied with that, may consider the extent to which the point you raise is relevant to a portfolio earmarked for retirement funding.

Your second point is, of course, correct. It is not simply a matter of opinion as to whether or not S.D.s are additive. The 70:30 TSM:Int. Trsy data you provide is an excellent example of MPT (Modern Portfolio Theory) at work. This, however, is not germane to the current analysis.

That brings us to:
Robert wrote:

Code: Select all

1927-2007

        SD      Correl with Mkt
Mkt   20.1             1.00
SmB   14.5             0.41 
HmL   14.1             0.12

Source: FF research factors, Ken French website
1*SD Mkt + (correlation coefficient Mkt:SmB)*0.2*SD SmB + (correlation coefficient Mkt:HmL)*0.4*SD HmL = Portfolio SD = 20.1 + 0.41*0.2*14.5 + 0.12*0.4*14.1 = 22.0
The SD of the actual 1*Mkt +0.2*SmB + 0.4*HmL series was 22.8.


This again (only?) reveals the beauty of mixing less-than-perfectly correlated assets. Are you saying that SmB and HmL risks are their respective Standard Deviations?

Z.

Edited
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Post by Robert T » Mon Mar 17, 2008 2:47 am

.
Zalzel,
Are you saying that SmB and HmL risks are their respective Standard Deviations?
If the factor returns are added next to their standard deviations (from the earlier table), the standard deviation’s relative to the respective returns seem comparable. i.e. the standard deviations of SmB and HmL returns are not dramatically smaller than that of the equity premium relative to the size of the three historical return premiums. The implication is that if the standard deviation is viewed as a reasonable measure of risk for the equity premium (as I think is being suggested based on observations of returns and SD), why shouldn’t the same be true for SmB and HmL given the numbers below?

Code: Select all

1927-2007
                            Annualized    Average        Standard
                              Return      Returns       Deviation                                     
Equity premium (Rm-Rf)         6.16         8.22           20.45
Size premium (SmB)             2.59         3.54           14.47     
Value premium (HmL)            4.18         5.15           14.11

Source: Derived from the Research Factors on Ken French's website

The above views the three premiums separately and SmB and HmL are only investible as a long-short strategy (i.e. taking out beta). If SmB and HmL exposure is gained through common index mutual funds, then the risk (higher additive standard deviation) shows up if correlations between Rm-Rf, SmB, and HmL increase (and was trying to show the impact of when this happens in the earlier post). If correlations remain low then we are back to:
"the beauty of mixing less-than-perfectly correlated assets." [I would swap the word assets for 'risks']
Robert
.

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Return, SD for 0/.2/.4 portfolio?

Post by zalzel » Mon Mar 17, 2008 3:34 am

Hi Robert.

Do you have a (1927-2007) correlation for SmB and HmL?
Would you kindly provide 1927-2007 return and SD for a simulated 0/.2/.4 (Beta,S,V) portfolio?

Thanks,

Z.
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Two more Q for Robert

Post by zalzel » Mon Mar 17, 2008 4:45 am

Hi Robert.

Two more questions please. I'm actually more interested in these two than the two I posed you above, so please feel free (if you wish) to tackle these first.

Scenario:
Period: 1927-2007
Assume the equity market is Fama-efficient.
Maximal (equities) tilt: 1/.4/.8 (Beta/S/V)
Consider all possible equity/U.S. 30d T-bill portfolios.

1)
Select all the portfolios with a total risk of (say) 10%/yr.
Of those, select the one portfolio (equity/T-bill) with the highest return.
What are the FF loading factors (Beta, S, V) for the equities part of that portfolio?

2)
Select all the portfolios with a S.D. of (say) 10%/yr.
Of those, select the one portfolio (equity/T-bill) with the highest return.
What are the FF loading factors (Beta, S, V) for the equities part of that portfolio?

Thanks,

Z.

Edited
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Correlation, factor loads, optimizations

Post by Robert T » Mon Mar 17, 2008 9:36 am

.
Zalzel,
Do you have a (1927-2007) correlation for SmB and HmL?

Code: Select all

Correlation Matrix:1927-2007

Rm-Rf = equity premium
SmB = size premium
HmL = value premium

FF Research Factors

 	           Rm-Rf    SmB   HmL 
Rm-Rf          1.00		
SmB            0.41    1.00	
HmL            0.12    0.13  1.00


FF Benchmark Factors
			
               Rm-Rf    SmB   HmL 
Rm-Rf          1.00		
SmB            0.43    1.00	
HmL            0.11    0.08  1.00
  • Research factors are rebalanced annually
    Benchmark factors are rebalanced quaterly
Would you kindly provide 1927-2007 return and SD for a simulated 0/.2/.4 (Beta,S,V) portfolio?

Code: Select all

                                             Annualized
                                              Return         SD   
Simulated from the FF research factors         12.31        22.84
Simulated from the FF benchmark factor         12.17        22.76

Market (from the FF research factors)          10.06        20.11
Market (from the FF benchmark factors)         10.24        20.01   
Two more questions please. I'm actually more interested in these two than the two I posed you above, so please feel free (if you wish) to tackle these first.

Scenario:
Period: 1927-2007
Assume the equity market is Fama-efficient.
Maximal (equities) tilt: 1/.4/.8 (Beta/S/V)
Consider all possible equity/U.S. 30d T-bill portfolios.

1)
Select all the portfolios with a total risk of (say) 10%/yr.
Of those, select the one portfolio (equity/T-bill) with the highest return.
What are the FF loading factors (Beta, S, V) for the equities part of that portfolio?
  • Total risk to whom?
2)
Select all the portfolios with a S.D. of (say) 10%/yr.
Of those, select the one portfolio (equity/T-bill) with the highest return.
What are the FF loading factors (Beta, S, V) for the equities part of that portfolio?
  • Using FF research factors:

    Equity:T-bills = 38:62
    Within equities the size and value loads were 0.25:0.80 respectively.

    Annualized return = 8.20
    Standard deviation = 10.00

    Using FF benchmark factors:

    Equity:T-bills = 38:62
    Within equities the size and value loads were 0.32:0.80 respectively.

    Annualized return = 8.39
    Standard deviation = 10.00

    The above were derived using the ‘solver’ function in excel. They were done relatively quickly so will check for errors again later.
Robert
.

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zalzel
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Re: Correlation, factor loads, optimizations

Post by zalzel » Mon Mar 17, 2008 11:41 am

Hi Robert-
Robert T wrote:
  • Total risk to whom?
Let's say an individual whose only withdrawals will be to supplement his retirement funding. His planned initial withdrawal rate is 2.5%. Please recall that the scenario assumes that the market is Fama-efficient. (Hint: No calculation is required.)
Robert wrote:
  • Using FF research factors:
    Within equities the size and value loads were 0.25:0.80 respectively.

    Using FF benchmark factors:
    Within equities the size and value loads were 0.32:0.80 respectively.

    The above were derived using the ‘solver’ function in excel. They were done relatively quickly so will check for errors again later.
Your answer to question 2 is very similar to what I came up with. I performed no calculation, nor did I use a computer. It took me less than a few seconds.

Z.

Edit: Change in bold.
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Robert T
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Post by Robert T » Mon Mar 17, 2008 12:11 pm

.
Zalzel,
Let's say an individual whose only withdrawals will be to supplement his retirement funding. His planned initial withdrawal rate is 3%. Please recall that the scenario assumes that the market is Fama-efficient. (Hint: No calculation is required.)
What about ‘non-market’ risks such as tracking error regret (not being able to stay the course etc…)? IMO this can be significant and varies by individual.
Your answer to question 2 is very similar to what I came up with. I performed no calculation, nor did I use a computer. It took me less than a few seconds.
If you knew the answer, then why ask? (seems like wasted effort and time on my part [more than a few seconds]…probably my last post on this…).

Robert
.

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zalzel
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V/S ratio of 2.5

Post by zalzel » Mon Mar 17, 2008 12:35 pm

Hi Robert-
Robert T wrote:
Zalzel wrote:Let's say an individual whose only withdrawals will be to supplement his retirement funding. His planned initial withdrawal rate is 3%. Please recall that the scenario assumes that the market is Fama-efficient. (Hint: No calculation is required.)
What about ‘non-market’ risks such as tracking error regret (not being able to stay the course etc…)? IMO this can be significant and varies by individual.
You have, please allow me to say, missed the forest for the trees. The clear answer to the question (#1) posed is: 1/0/0. That is, the market portfolio.
Robert wrote:
Zalzel wrote:Your answer to question 2 is very similar to what I came up with. I performed no calculation, nor did I use a computer. It took me less than a few seconds.
If you knew the answer, then why ask?
Because I did not know the answer. My hunch was that the portfolio that would shake out would be the maximally tilted one. Thus, my answer was: 1/.4/.8- the maximal allowed tilt. That is very similar to the 1/.32/.8 (benchmark) Excel came up with.

I, certainly, and perhaps you (hopefully), have learned something from your effort. That is, that with an upper bound on V of .8, a tilt to Small of greater than S=.32 is counterproductive. Thus, a V/S ratio of 2.5 seems to be desirable for maximal efficiency, if S.D. only is to be used as the guage of risk. That is not, however, IMO, the most important point to be made from your calculations.


Thanks,

Z.

P.S. It is also noted that, with the ‘solver’ function in excel (as reported by Robert), a high value V of .8 is not too high for consideration. Such a V is not difficult to attain with a real-world portfolio. (See, for example: http://www.diehards.org/forum/viewtopic.php?t=14939.

Edited to add P.S.
Last edited by zalzel on Tue Mar 18, 2008 1:45 pm, edited 1 time in total.
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stratton
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Post by stratton » Mon Mar 17, 2008 3:15 pm

It appears this whole exercise is to try and boil risk down to one number. This over simplifies the issue and actually increases risk. I don't see how this can be simplified to one number.

Using the backtesting spreadsheet as an example using any of the single measurement values by themselves is foolish. Especially with the questionable data since it has been collected from many sources. Still it the backtest spreadsheet gives you:

CAGR
Std Deviation
Sharpe Ratio
Sortino Ratio
Correlation w/US market
Correlation w/Intl market
Draw downs

Using any one of those values as a risk meausre in isolation is risky. Using all of them together is still risky and requires some judgement otherwise you'll end up with whacky automated MVO type output. Just look at some of the wierd posts about having 100% equities with 100% EM. Ughh!

Like any tool backtesting, MVO analysis, monte-carlo analysis, correlation matrixes, MPT etc. can be abused.

Paul

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Paul's (stratton) view?

Post by zalzel » Tue Mar 18, 2008 12:38 pm

Hi Paul.
stratton wrote:It appears this whole exercise is to try and boil risk down to one number.
The point of this whole exercize is to come to a better understanding of risk.
Using the backtesting spreadsheet as an example using any of the single measurement values by themselves is foolish. (snip)

Std Deviation
Sharpe Ratio
Sortino Ratio
Draw downs
(snip)

Bolding added.
I agree. I do note with interest that none (not one) of the parameters you specified encompasses the concept of 'non-volatility risk'. Out of curiosity (if you don't mind), would you share your view as to whether historical small/value premiums are appropriate compensation for risks taken, or have they been (largely) free lunches?

Z.
"What we can't say we can't say, and we can't whistle it either." | Frank P. Ramsey

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Re: Paul's (stratton) view?

Post by stratton » Thu Mar 20, 2008 11:33 am

zalzel wrote:I do note with interest that none (not one) of the parameters you specified encompasses the concept of 'non-volatility risk'. Out of curiosity (if you don't mind), would you share your view as to whether historical small/value premiums are appropriate compensation for risks taken, or have they been (largely) free lunches?
Huh? The only one that has anything to do with volatility is Std deviation which is a measure of volatility. None of the others I mentioned have anything to do with volatility.

After reading SmallHi's post it appears you're confused on lot of MPT and FF stuff so I'm going to bow out on this subject.

Paul

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Re: Paul's (stratton) view?

Post by zalzel » Thu Mar 20, 2008 2:26 pm

stratton wrote:
zalzel wrote:
stratton wrote:Using the backtesting spreadsheet as an example using any of the single measurement values by themselves is foolish. (snip)

Std Deviation
Sharpe Ratio
Sortino Ratio
Draw downs
(snip)
I agree. I do note with interest that none (not one) of the parameters you specified encompasses the concept of 'non-volatility risk'.
Huh? The only one that has anything to do with volatility is Std deviation which is a measure of volatility. None of the others I mentioned have anything to do with volatility.
You're joking.... aren't you?

Z.
"What we can't say we can't say, and we can't whistle it either." | Frank P. Ramsey

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