Learning outcomes. After watching this video you will be able to, one, understand style analysis and how it is carried out. Now we move on to something called style analysis, which is very interesting. Introduced for the first time by Professor William Sharpe who also by the way won the Nobel Prize for coming up with the capital asset pricing model. Now, the style analysis model is very popular in the practitioner world. The basic idea here is to say, now how much of the variation of my particular fund can be explained simply as a combination of a bunch of passive index investments. To be more formal, suppose I have a bunch of indices on the right hand side represented in this equation by F_1 through F_n. What I want to do is, I want to fit an equation r_p = [b_1 F_1 + b_2 F_2 +... + b_n F_n]+ e_p. It looks like a regression equation, but this is not really a regression equation. The difference is number one, here Sharpe wants to constrain the betas to all lie between zero and one, and he wants the sum of the bs to be constrained to one. In other words, he's thinking of the bs really as portfolio weights. The idea is to the extent possible, can we figure out a set of weights which could have been employed by a passive investor to largely mimic the variation in returns of this fund? Of course, you're never going to do a perfect job of mimicking the fund using a combination of these indices on the right hand side. The idea is to find the best possible combination exposed, that is, after the fact. The way to do this is to essentially minimize the variance of that error term and figure out the right combination of the bs that would have minimized the variance of the e. And the idea is there's an R-squared type measure here, which basically says most of the variation on this active managed fund could have been essentially replicated by a certain combination of index funds on the right hand side, then the active fund manager is not doing much. If there is a significant part of the variation which cannot be explained by a combination of passive investments in the right hand side, then we begin to give some credit to the active fund manager. Now this style analysis is a, easy, b, interesting, and c, very very useful. Now it can be used in a variety of ways and I'm going to show you an example very soon as to how it can be used. In particular, it can be used very easily to check if funds follow the philosophy they say they will follow. For example, today in the United States as well as in several countries, lot of funds essentially align themselves along something called a style grid. On one axis, you have say growth and value as the polar opposites. On another axis, you have size that is small to large as the polar opposites. Obviously, you can have, let's say nine quadrants on that three by three, and basically say, "Well, I'm a large cap value fund. I'm a small cap growth fund etc." Now it is all well for a fund to say that they follow a certain philosophy, but it is up to you, the investor, or up to you, the analyst, to actually figure out if the fund is actually following the particular style that they're saying that they will follow. Now, here what I've done is essentially I've taken a couple of U. S. stock mutual funds. These are large mutual funds, one of them is 32 billion in size, the other is about half that size, about 16 billion, still very large. One is the Vanguard fund, other one is the Fidelity Dividend Growth fund. Now the interesting thing is, the Vanguard fund advertises itself, I plucked this off their website, they say they are a large growth fund. The Fidelity fund on the other hand says they're a large blend fund, blend being, they're a blend of growth and value. And Morningstar, which is a well-known mutual funds rating agency, happens to give the Vanguard fund a five star rating and the Fidelity fund a three star rating. Now my quest is very simple. I want to look at the battery of measures we've all studied until now and see if we can employ some of those measures and come up with a measure that replicates what Morningstar is doing. In other words, are our measures doing what they're supposed to do? Do they lead to the same conclusions as the Morningstar ratings? I have a bunch of data which is slightly dated now but it will serve the purpose. My data goes from January '94 all the way through November 2005, which is about 143 months of data. I'm going to use all our performance measures, and if you sit down and collect the returns of these funds and calculate each of these measures that we have discussed, what you've seen in the table is that really, yes, there is some slight edge of the VP fund, which is the Vanguard fund, over the Fidelity fund in terms of slight increase in Sharpe ratio or slight increase in say, the one factor alpha etc. But nothing really pops out at you from this table. In other words, it's practically impossible to conclude just from this set of numbers that somehow we have this superlative performance of the Vanguard fund as opposed to the mediocre performance of the Fidelity fund. I try to use style analysis now, and since these are both equity funds, I use as my passive indices or styles as Sharpe calls them, on the right hand side, I use six particular styles. The first one is a large cap growth index, which is essentially within the S&P 500, all the high market to book stocks. Market value to book value stocks. And the large cap value, I have essentially that half of the S&P 500 which is the low market to book, or conversely the high book to market stocks. Similarly, mid-cap growth mid-cap value, small cap growth and small cap value. Essentially, I have six indices on the right hand side as regress source if you will, in the Sharpe style analysis. Now what I have on the right hand side in the graph is that, I have derived the optimal bs if you will, the optimal rates, that could have largely replicated the variance in fund returns for both funds. First, the Vanguard fund, and the Vanguard fund it turns out has exposure to large growth, to small growth, to mid-cap growth, and large value. And what did they say they were? They said they were a large cap value fund. Sorry, they said there were a large growth fund. With style analysis, what we look at with the Vanguard fund and the Fidelity fund is that the weighting of different passive benchmarks that could have potentially replicated the performance of these two funds is shown in these two graphs. Now with the Vanguard fund, what we see is it has exposures, not just a large growth as it purports to be, but also to other things like large values, small growth, mid-cap growth, etc. However, with the Fidelity fund, which says it is a large blend fund, you will see that the largest exposures are to large value and large growth. In other words, if you pardon me the pun, the Fidelity fund shows greater fidelity to its investment objective rather than the Vanguard fund. The other thing we could do is to actually conduct a rolling style analysis, that is to say, we take x number of months of returns and roll the window across which you determine the style of the fund up to the very end of the data, and you see how a fund style has evolved across time. And what you see of course is that in the Vanguard case, the style has been changing a lot over time. More importantly, it has never really been an exposure only to large growth, it has been exposed to several other styles if you will over time whereas, with the Fidelity blend fund, what you see is, although it had exposures to other kinds of styles in the beginning of the sample period towards the latter part of the period, the exposure is almost exclusively large growth and large value as it should be. Now we could potentially use this style exposure at every point, because we have a style exposure at every month, beyond a certain month, until the end of the period as the appropriate benchmark for this particular fund, and try to benchmark the fund to its own style benchmark. In other words, we are now benchmarking both these funds not against S&P or any other such index but we are saying, I have my own special weighted set of passive benchmarks that I'm going to call the benchmark for VPMCX and I have my, again, similar set of a style benchmark for FDGFX and I'm going to benchmark the fund performance against its own style benchmark. And when you do that, some things start to pop out right there. And what you see is, if you look at the Vanguard fund against its style benchmark, for the first time, statistically, significantly, there is an out-performance relative to its style benchmark. In fact you can do this more formally and try to run a regression of each of these fund returns against the S&P, against the Fama and French three-factors, which is the market factor, the small minus big factor, and the high minus low factor as you know, and of course, now the new addition, the last row of this table, which is the rolling style benchmark. And what you see is, results are practically statistically insignificant in the S&P case and the Fama-French three-factor case. But when you look at the rolling style analysis, what you see is of course, that the Vanguard fund has an out-performance of 1% per month as opposed to the Fidelity fund of 0.61% per month. And the performance is much more statistically significant with the Vanguard fund. It is of course, also significant with the Fidelity fund, which earned it three stars if you remember, with the Morningstar guys, but clearly Vanguard emerges as the outperformer among the two. And now you see why Morningstar would actually give the former fund a five star rating and the second fund a three star rating. That is the clear use of style analysis which I said is a very interesting analysis.