Problem Solving

I'm trying to figure out how to solve the following issue:

I'm trying to create an equation where we can estimate the actual returns for the stock and bond portions of a clients portfolio given limited information.

Example: Client's Return for a 90% stock/ 10% bond portfolio is 10%

The return of the stock index is 10% (remember this isn't the client's returns because their stock/bonds are commingled in such a way that we can't determine their stock vs bond performance).
The return of the bond index is -5% (same caveat as above).

We now compare the clients return to a weighted average of the index returns to see if they outperformed or underperformed. We'll call this Alpha and if it's positive that's good and it means we outperformed and if it's less that we underperformed.

(10% * .90) + (-5% * .10) = 8.5% is the expected return if they were invested in the indexes.

Now let say our portfolio returned 15% so we did significantly better than we would expect so our Alpha is the difference 15% - 8.5% = 6.5%.

Since our portfolio is 90% in stocks it would be a safe assumption to say that most of the Alpha came from the stock portion of the portfolio, however, I can't figure out a way to do that mathematically and be able to get back to the overall return figure for the entire portfolio. The only way I can get it to work is to evenly allocate the 6.5% to both the stock and bond categories.

An example makes it more clear.

Since we outperformed by 6.5% if I add that to both stocks and bonds the figures work out.
Estimated stock return = 10% + 6.5% = 16.5%
Esimated bond return = -5% + 6.5% = 1.5%
And the math works when you roll the figures back up which is crucial:
16.5% * .9 + 1.5% * .10 = 15%

The problem is that this Alpha is unlikely to be evenly attributed to each category considering the portfolio is 90% stocks so I want a way to allocate it appropriately, but I also need it to still add up to the 15% afterwards.

You would think simply weighting it would work, but see below.

10% + (6.5% *.90) = 15.85% This is allocating 90% of the Alpha to stocks
-5% + (6.5% *.10) = 4.35% This is allocating 10% of the Alpha to bonds

The problem is it fails when you try to get back to the portfolio return:
(15.85% * .90) + (4.35% * .10) = 14.7% (and this needs to be exactly 15% because we're estimating the return of his stocks and bonds so they need to equal his total return when you weight them out). The equation needs to work every time. It can't be some sort of filler number you find out and plug in at the end.

I don't know if this is for class or for a client. However, I will say, since I'm in the business, that one couldn't responsibly attribute performance this way and report it to a client.

This because there are an infinite number of combinations of stock and bond returns, appropriately weighted by 90/10 that would yield the overall portfolio result and therefore infinite alphas that could foot to the total.

For example, a 12% stock return and a 42% bond return would yield 15% overall using 90/10. The corresponding alphas of 2% and 47% also weighted 90/10 would foot to the overall 6.5% alpha.

Or, you could assume the stocks and bonds both earned 15%. So the stocks would have a 5% alpha and bonds 20%. Weighted 90/10 also yields 6.5% overall alpha.

The point is, who knows what those underlying returns are and their alphas.

I could go on. If this were a multiperiod calculation with rebalancing to 90/10 then you'd also have to consider correlation between the stock and bond holdings.

I would look up the phrase "performance attribution" and the CFA Institute.

With the caveat that I don't work in this industry and am guessing what some of these terms mean, I think I see what's happening mathematically: you're trying to solve for two variables at once with only one equation. (The index performance doesn't help UNLESS you it gives you an equation tying your stocks and bonds to the index of stocks and bonds.)

Suppose your stock return is x and your bond return is y, with x and y representing multiples of our original allocation. (For instance, if you had a 5% return on stocks, x = 1.05, and if you had a 10% loss on bonds, y = .9.)

If you had a 15% return overall, you know that .9x + .1y = 1.15. You've got an infinite number of solutions here, so you're stuck. (Visually, this is just a line. You could have x positive and y negative, x negative and y positive, or both x and y positive.)