MPT

Theory?

There's a typo (or error) where he states that Risk = (w_m^2 \sigma_m ^2 + [ w_a^2 \sigma_a^2 + 2 w_m w_a \rho_{am} \sigma_a \sigma_m] ) a '^2' is missing after the last sigma_m. Other than that, it's fairly intuitive. Joe

Harry Markowitz won the Nobel prize for his work in this field. It (MPT) speaks to risk, reward, and the value of diversifying. It's worth understanding, and in a roundabout way, answers questions like "how about this 20% return I'm promised?" (uh, the risk shoots off the scale).

I have to say, from studying this in school some time back, the Wiki appears to do a good job condensing this the one web page. I'm having some wonderful Deja Vu.

JOE

I have always been very uncomfortable with the idea expressed in this paragraph:

"It is further assumed that investor's risk / reward preference can be described via a quadratic utility function. The effect of this assumption is that only the expected return and the volatility (i.e. mean return and standard deviation) matter to the investor. The investor is indifferent to other characteristics of the distribution of returns, such as its skew. Note that the theory uses a historical parameter, volatility, as a proxy for risk, while return is an expectation on the future."

The notion that volatility is the only risk I care about is nonsense. Lets consider two mythical stocks: LOVOL has a price of exactly $10 for five years, then goes bankrupt and has a price of $0 forever more. HIVOL goes down 20% one day and up 26% the next trading day and repeats this forever. Which would you regard as the risky stock? Which would you rather own?

I see why volatility is attractive to analysts, you can measure it, get lots of data on it, and do all kinds of mathematics on it. But as the only "proxy for risk", it is minimally useful at best.

Can someone tell me why I am misguided about this?

-- Doug

A three sentence tangent that may or may be an appropriate analogy: I can put sensors on your hip, knee and ankle joints and observe that the path they travel when you walk is described by certain coefficients modifying a basic sine function. Then when you run, those coefficients change to another, fixed (you are a consistent runner) set of numbers. Whether you care about this doesn't matter, you follow these equations regardless.

Back to finance. I take a stock and make some observations about its behavior over time in comparison to the stock market (an index such as the S&P). I take a second stock, and comparing it to stock one, find a high correlation, higher than either to the S&P. Odds are I have chosen two companies in the same sector, perhaps Pfizer and Merck. Other stock pairs will have other levels of correlation, say those of Merck and IBM (i.e. low to negative correlation). At some point, adding stocks does not reduce the risk of the portfolio, and the portfolio is considered diversified. Your examples are contrived, and therefore, not applicable. Risk and volatility are near synonyms when it comes to investing. The fact that the market average return has been 10.5% over some period is meaningless without understanding the standard deviation of returns. And even while we can all parrot "past performance is no guarantee", understanding the nature of past returns provides some value looking forward. And in the end, risk turns out to be the chance that a return will be more or less than expected, which quickly becomes the description of volatility.

JOE

An important lesson that derives from MPT, besides the intuitive relationship between risk and reward, is that the riskiness of a portfolio has to do with the covariance of its assets, not with the number of stocks you have in the portfolio. You could have a portfolio with 1000 stocks, but if their returns are perfectly correlated, you get no diversification at all. On the contrary, if you have a portfolio of just two stocks but with negatively correlated returns, then you may end up with a relatively safe (low risk) portfolio. For instance, if you have a portfolio of 30 banks doing the same business, operating in the same region and under similar management, you may have higher risk than a portfolio of -for example- just one bank and a credit collection bureau: when non-performing loans increase, the typical bank's balance sheet and income statement suffer, while for the credit collection bureau more non-performing loans means more business. Notice that in this case, both the bank and the credit collection bureau are in the same sector (finance), so you may have diversification even if the two companies are in the same sector; they just need to have their incomes negatively correlated -or at least uncorrelated-.

Another important lesson -derived from this one- is that a stock's riskiness -i.e., its standard deviation with respect to the market changes- does not mean that this stock is less attractive investment than an alternative stock with the same expected return and a lower standard deviation. It all depends of the overall standard deviation of the portfolio. You may have an asset with very high return volatility, but if the asset reduces the overall volatility of the portfolio

-because its returns are negatively correlated with the rest of the portfolio- then it would make sense to add this individually "risky" asset to your portfolio.

A final lesson is that, through well- designed portfolio diversification , you may get rid of all non-systematic risk of investing in individual assets, but you still have a systematic risk that is the market risk. Therefore, if your main concern is reducing the riskiness of a portfolio, then you should not only invest in U.S. stocks but also in stocks of other world markets, in particular of those markets with lower correlation with the U.S. A practical example is: if you assume that very high oil prices have a negative effect on the U.S. economy and therefore on U.S. stocks, you should invest in the stock markets of OPEC countries as well, because these economies will benefit from increasing oil prices (in fact, this is what happened in the last few years: the markets of many OPEC countries, like Saudi Arabia or UAE, were booming with the high oil prices).

An important caveat of the standard MPT (and therefore portfolio design based on MPT) is that it assumes that the matrix of variances-covariances remains unchanged over time. This means that, when you design the portfolio, you just collect data on the returns of different stocks in the portfolio and then compute the covariance matrix for the whole portfolio. This assumption is relatively safe when there is no relevant structural change, in these cases the statistical relationships do not change in a significant way. Of course, when there is srtuctural change, everything gets screwed up and what you computed as the covariance matrix (riskiness of the portfolio) becomes irrelevant.

joetaxpayer wrote:

The question I'm asking is "why"? How does volatility deal with risks like bankruptcy/default, inflation, currency, dividend cuts, or any of the other real world things you need to worry about when investing?

If you are using "average" to mean "arithmetic mean", then I agree. An arithmetic mean of market return is meaningless. The geometric mean is the meaningful (pun intended) figure.

Why is it a risk when a return is higher than expected?

I know my examples were contrived. The point I was trying to make is that real risk is the unexpected. A stock that is predictably volatile is not risky. A stock that is not volatile still carries risk.

-- Doug

I believe this is the result of a logical progression. Although inflation would impact any asset, even TIPs, which is why they yield so low. But the starting point is the risk free rate, the one year t-bill. Then it's presumed that one would command a higher return to compensate for any of the risks you mention.

Even bonds, in a normal yield curve one says that the higher yield for say a 5 yr note is due to uncertainty in the future.

I don't know that "predictably volatile" can exist, I need to think about this.

I fear that some of my logic may appear to be circular. I offer that criticism of my own post/explanation. Hoping others would kick in their thoughts.

JOE

Can you amplify this thought? It seems you are saying that variances and covariances of assets are relatively constant, barring a structural change. What is a structural change in this context?

-Will

This is the assumption of MPT. The assumption makes sense in periods of relative stability, but not when there is structural change. Structural change is defined as a fundamental change in the relationship between different variables (for an econometrician, the technical name is "location shift"). For stocks, this means -for instance- that the relationship between real estate and financial sectors stocks change and becomes more (or less) significant. When the econometric relationships change in a fundamental way, this means that the variance-covariance matrix between different stocks of a portfolio changes. Therefore, if you design a portfolio in such a way that, for a given expected return, minimizes risk -given by the variance-covariance matrix- such portfolio could not be the one with the lower return-adjusted risk once structural change happens.

An example of how structural change affects stocks or portfolios or funds is Long-Term Capital Management. This hedge fund was designed given a variance-covariance matrix which implied relatively low risk for the portfolio as a whole. However, because of the developments in

So I think you're saying that the variances and covariances are stable until they're not. How often does structural change occur? Or conversely, how long are variances and covariances stable? A few days? A few months? A few years? A few decades?

-Will

Also, I don't see that a changing covariance matrix makes MPT useless (as some prior posters appeared to imply).

A changed matrix means the efficient frontier has to be redrawn and one needs to switch your portfolio to one on the new efficient frontier.

If the matrix changes slowly enough, you can just "move" your portfolio every few years to get back to the frontier. Sure, there will be (mostly tax) costs associated with that, but it's far from making MPT useless.

Are the changes slow? They certainly weren't for LTCM. How often must this adjustment occur? How far from optimum do you get in the meantime?

-Will

No, changes are usually not slow but they often take place quite suddenly. Nobody predicted the Mexican or East Asian crisis till the moment they actually started (December 1994 and July 1997, respectively). The turmoil hit the markets quite suddenly. If you had a MPT designed portfolio before these crisis, the whole portfolio risk- adjusted structure was changed by them and the ex-ante portfolio was no longer the minimum risk -adjusted by returns- portfolio after these crisis. When you face such a change, then the liquidity position of your portfolio could become completely completely inadequate. That's why it is always better for an investor to play it safe and hold extra liquidity, even if liquidity has a clear opportunity cost -in terms of the loss of higher returns of alternative investments-. If you are forced to sell stocks or bonds or other financial instruments in a moment of crisis, you risk yourself to suffer huge losses.

Will Trice wrote: