Can someone explain this equation? I feel stupid.
I'm reading a paper "An Alternative Future" (C. Asness) and I feel like I'm reading Chinese.
"We can start with a simple and familiar example, a traditional, actively managed equity portfolio. This is a collection of stocks, usually a relatively small subset of those in a stock index, that an active manager believes will outperform the others in the index. Define these portfolio weights for the active manager as A and the weights in the index as I. The weights that make up A and I separately sum to 100%".
We can represent the active manager's holdings A as:
A = I + [A-I]"
Q:
- Why is he talking about weights in the index? Why it is relevant?
- Why do we have "portfolio weights"? Shouldn't A (Manager Portfolio) = (Stock * weight in portfolio) + all the other (stock * weight in portfolio)?
- Can someone explain in plain English the equation?
I'm reading this paragraph assuming that he explains what is Portfolio Weight (stock value / portfolio value) but it seems that I'm wrong.
Thanks a lot.
It's been a while since I studied this but A and I refer to vectors where elements refer to % holdings in each stock. Think of I as the % of each stock which makes up an index in value terms. There is no multiplication by dollar amounts as academics only care about % returns.
It seems like a redundant / obvious equation, equivalent to saying 1 = 2 + 1-2, but my guess is the equation is being set up this way in order to then derive a more insightful equation / theorem.
Suppose A is a long-only portfolio, which invests in stocks with weights a_k representing the percentage of the portfolio in dollars allocated to stock k. One meaningful benchmark for A's performance is "the index" I - if your universe is liquid US equities then this might be the S&P500 or Russel2k - which has its own weights i_k (proportional to market cap, or something close to it).
The decomposition is just saying that your portfolio A can be viewed as the combination of two portfolios: the index/benchmark portfolio I, plus a long-short portfolio B = (A-I) which has weights b_k = a_k - i_k. I call the latter portfolio long-short because (1) if A and I are not identical, then some of the b_k are negative, so B must be short some stocks, and (2) the b_k sum to zero, meaning there is no net exposure to stocks and the long + short legs are balanced in dollar terms. A related concept is the idea of "alpha" exposures vs. "beta", which roughly corresponds to contrasting B vs. I in our notation.
A quick example: let A be a portfolio which is 100% TSLA. Holding this is making two bets: one bet that the market in general will go up (long index portfolio I), and one bet that TSLA in particular will outperform the rest of the market (B is long TSLA and short every other stock in the index).
For context, Cliff's motivating gripe is that the asset management industry, at least at that time, would charge fees based on the total return of A. But exposure to I is very easy and cheap to come by - today I can buy a Vanguard ETF to achieve the index's return basically for free. So why would I pay fees to A for returns that come from just holding I? Many funds take very little so-called "active risk", so their B portfolio has only a tiny contribution to the return, but they might still charge relatively high fees. Cliff's firm sought to disrupt this paradigm by selling exposure to B at a fair price, which requires investors understand the distinction in the first place.
Some things to think about, in case you are interested in learning more:
. What if A can take leverage, so that sum(a_i) > 100%? How would you modify the decomposition?
. What if B is a portfolio that goes long stocks with high volatility and short stocks with low volatility? Is the decomposition still "fair"?
So is he saying to charge fees only for alpha?
I'm just starting out with financial economics, but it seems like this leads down the road of obsessing over beta. I'm sure there's literature about this, but does trailing beta approximate future beta? Like if I'm an O&G specialist and oil prices are trending up in a bear market, a regression will tell me my betas are very low. So at the end of the year, I would rake in crazy fees because my systematic risk was understated.
Yes, exactly.
You are asking a good question, which is essentially: how do we decide on the benchmark factors? If your oil fund is structurally long, then taking the benchmark to be an energy sector index might be reasonable, in which case you may not have delivered alpha in your example even though the return was positive. If your fund is sometimes long and sometimes short oil in aggregate, and being long was a tactical timing decision, then you deserve to be paid for that bet paying off.
It is always possible to add more factors and "explain away" more and more of the return. What's important is: as the investor, which risks am I paying you to take? As long as both parties are clear up front about what "doesn't count" there is no ambiguity about how to attribute pnl ex-post.
A multi-manager platform handles this by making PMs hedge out various factor exposures ex-ante, but modulo riskmodel failures this is similar to removing any factor-aligned pnl at the end - taking on factor risk in such a setup gives you no upside but can create downside via drawdown limits etc, so the PM is incentivized to hedge out the factors themselves anyway.
I'm amused that Cliff said "we can start with a simple and familiar example" and despite going 30 times through the paragraph I still had no idea what he was talking about. But thanks for the explanation, now I get the point.
I'm relatively new in terms of HF knowledge but I'm trying to improve on that. I'm slightly surprised by the amount of how math-inclined tend to be many explanations about some common things. I think that happens because the more those guys see the world quantitatively, the easier it becomes to put numbers on opportunities and risks (or maybe viceversa?).
Great explanation, thx again!
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