Need expected Portfolio Avg, StDev values

Factual Questions
1

As part of our family “When can we retire?” investigations, I’ve built a small program that uses a Monte Carlo simulation technique to measure the likelihood of our running out of money while we’re still alive. IE, something that answers the old “what are the odds that we’re going to end up eating cat food” question. Not that there’s anything bad with cat food.

The program is in pretty good shape, but running it brings up the question of expected returns / deviations of various investment portfolios. I’ve gotten some pretty good estimates for certain portfolio types from various web sites (mostly brokerage house sites), but am looking for more estimates - specifically, I’m looking for estimates for various combinations of stock/bond portfolios. Help!

The numbers I’m looking for are constant-dollar estimates (IE, after inflation). The various sites I’ve checked out seem to pretty consistently give the following estimates based on historical values (“I-bonds” being the “inflation-plus-x%” bonds that you can purchase from the US Treasury):

Stocks: 7.5% avg post-inflation return, 0.16 standard deviation (*)
Bonds: 3.0% avg post-inflation return, 0,07 standard deviation
I-Bonds: 1.0% avg post-inflation return, 0.00 standard deviation

Note that the above are all “100% this” portfolios, which of course everyone says are a bad idea. Most everyone recommends a percentage mix of investment classes - say, 50% stocks, 40% bonds, 10% cash-equivalents. Most of the brokerage sites specifically state that, because of the less-than-perfect correlation between investment classes, the standard deviation of a mixed-investment portfolio drops faster than the average return, making a mixed-investment portfolio generally safer. Many sites even have neat-looking graphs showing this relationship (all of which are apparently generated stylistically rather than using actual values - lots of nice, smooth curves there). Unfortunately, I haven’t been able to find a site that givens actual numerical values for mixed-investment portfolios that I can use in my model.

Help! Does anyone have estimates (or even better, pointers to web sites that have them) for historical mixed-portfolio return average / standard deviation values? Something like 25% bonds / 75% stocks or 50% bonds / 50% stocks. Actually, I’ll cheerfully take anything I can get.

(This is a straightforward “General Questions” topic, but I’m assuming that it will deteriorate pretty quickly into the opinion area, so I’m posting it here to start. Mods, feel free to adjust as necessary.)

(*) Historical-return figures, of course, aren’t necessarily good estimates of future returns. Some sites, for example, are assuming a lesser average future return on stocks over the next twenty years or so because they feel that current prices are too high. The nice thing about using historical figures, though, is that there isn’t as much of a range of opinion about them as there are about future ones :slight_smile:

2

I strongly recommend you move this to GQ; you’ll get a better quality set of answers. But as you posted here, I’ll give you my absolutely useless opinion:

I find it hard to believe that stocks have such a low SD. 0.16% seems like practically a guarantee of high performance. That means (assuming a standard distribution, which may be a stretch), that you have a 99% chance (3 SDs) of getting a return between 7% and 8%. I don’t know nothing about nothing when it comes to stocks and expected returns, but this sounds mighty fishy to me.

3

[QUOTE=Bill H.]
I find it hard to believe that stocks have such a low SD. 0.16% seems like practically a guarantee of high performance. That means (assuming a standard distribution, which may be a stretch), that you have a 99% chance (3 SDs) of getting a return between 7% and 8%.

[QUOTE]
Sorry about that - I mixed up my terms when I listed the two values. I should have entered something more like the following to keep the values consistent:

Stocks: 0.075 avg post-inflation return, 0.16 standard deviation
Bonds: 0.030 avg post-inflation return, 0,07 standard deviation
I-Bonds: 0.010 avg post-inflation return, 0.00 standard deviation

And it sure looks like you’re right about my posting this in the wrong forum. Mods, could someone do me a favor and move it?

4

Sure.

5

The biggest problem with extrapolating historic returns into the future is that the past returns do not account for changes in the stock market’s valuation. If the market has a great year this year and the P/E multiple expands further, it will raise the historic average return, and therefore the expected return. This would mean that high returns beget higher returns–clearly a nonsensical conclusion.

That said, a summary of 200 years of historic returns can be found here:

http://www.aimrpubs.org/ap/issues/v2002n1/pdf/p0020035a.pdf

Also keep in mind that the high returns the U.S. has experienced might be anomalous. Dimson, Marsh and Staunton show that the average stock market return for the only 16 countries with complete histories for the period 1900-2002 was just 4.1%.

A better method for determining expected returns might be to use the Gordon Growth Formula:

E® = current dividend yield + projected real dividend growth rate + projected change in P/D multiple

The current dividend yield of the S&P 500 is 1.8%. Throughout the 20th century, dividends of U.S. corporations grew at a real rate of just 0.6%, and in many countries the dividend growth rate has been negative. Even if we use the somewhat unconservative assumption that the P/D (or P/E) ratio will remain at its current high level, we’re left with an expected real return for stocks of just 2.4%.

Alternatively, another method for forecasting equity returns is given by

E® = E/P

Using data for the S&P 500 this would give us an expected return of about 3.6%.
In general, or an N-asset class portfolio, to do the type of analysis you’re suggesting you will need estimates for N expected returns, N standard deviations, and (N[sup]2[/sup]-N)/2 correlation coefficients.

For a 3-asset portfolio, the expected return is just

E® = w[sub]1[/sub]E®[sub]1[/sub] + w[sub]2[/sub]E®[sub]2[/sub] + w[sub]3[/sub]E®[sub]3[/sub]

where w is the portfolio weighting of the asset, such that w[sub]1[/sub] + w[sub]2[/sub] + w[sub]3[/sub] = 1.0

Note that in order to maintain your portfolio weightings, you will likely need to rebalance, which would involve selling an asset that has performed well recently in order to purchase more of an asset that has performed poorly–buy low, sell high. This can create a “rebalancing bonus”, leading to a slightly higher return than the above formula would indicate.

The expected standard deviation of the portfolio is given by

SD[sub]p[/sub] = [(w[sub]1[/sub])[sup]2/sup[sup]2[/sup] + (w[sub]2[/sub])[sup]2/sup[sup]2[/sup] + (w[sub]3[/sub])[sup]2/sup[sup]2[/sup] + 2r[sub]12[/sub]w[sub]1[/sub]w[sub]2[/sub]SD[sub]1[/sub]SD[sub]2[/sub] + 2r[sub]13[/sub]w[sub]1[/sub]w[sub]3[/sub]SD[sub]1[/sub]SD[sub]3[/sub] + 2r[sub]23[/sub]w[sub]2[/sub]w[sub]3[/sub]SD[sub]2[/sub]SD[sub]3[/sub]][sup]1/2[/sup]

where r[sub]ij[/sub] is the correlation coefficient between assets i and j. You can approximate this by downloading monthly returns from Yahoo finance, adjusting the data for splits/dividends, then using the Excel’s CORREL function to determine the correlation coefficient.

Once you’ve done this you’ll see that some asset weightings will have a lower SD for the same expected return, and that some weightings will have higher returns for the same SD. These are the portfolios that lie on the “efficient frontier”, optimizing the risk/return tradeoff.

A word of caution is in order: on October 19, 1987 the S&P 500 fell 23% in a single day. Since the SD of daily returns is about 1%, this would be a 23-sigma event–an incredibly unlikely event in a Gaussian distribution. But these types of event happen regularly in the financial markets. Clearly, kurtosis is a factor here, so any assumptions of sustainable withdrawal rates based on a normal distribution could prove to be disastrous.

6

Thanks for the info (and the link), cynic.

The problem that I’m having at the moment is that the various sites that I’ve looked at have all claimed that a specific asset weighting is on that efficient frontier, but none of them seem to give any hard numbers (as in expected returns/standard deviations) as to where that efficient frontier actually is. It’s sort of like a harbor pilot telling you that he’ll keep your ship in the main channel but he won’t tell you where that channel is. :slight_smile:

In a former life (early 70s) I worked as a programmer for the finance department at the U of P, and one of our studies involved that phenomena. I (dimly) recall that we decided that the market returns for the recent past had the properties of a non-symmetric stable Paretian distribution with a Kurtosis of 1.6 - but there wasn’t much we could do statistically with non-symmetric stable Paretian distributions, so we cheerfully ignored it and kept assuming normal distributions. :slight_smile:

7

Everybody has their own Expected Return estimates. Further, people measure volatility over different time periods as well. Also, correlation can be measured over different time periods. Technically, you don’t need to know their actual estimates–you just need to know what the optimal asset mix is and its spot on the risk/return graph. Then draw a line through that point and the risk-free rate point. Now, you can pick whatever expected return/risk portfolio you want by investing a portion in the risk-free asset (treasuries) and the rest in the optimal portfolio.

8

For the reasons that daniel801 has given, I’ve always looked at mean-variance analysis more as a tool to demonstrate the benefits of diversification rather than a tool for selecting a specific asset allocation. The results you obtain from an MVO are extremely sensitive to the inputted values, which can vary considerably over time.

But if you must, you can download a 3-asset MVO from here:

http://home.golden.net/~pjponzo/Efficient-Frontier-2.htm

Just right-click on the spreadsheet and click on “Save Target As…”

That website belongs to Dr. Peter Ponzo, a retired math professor whose website has tons of information about Monte Carlo simulations, mean-variance analysis, sustainable withdrawal rates, correlation coefficients, asset allocation, and just about everything else related to personal finance. I think you’ll be able to find everything you need right there.

9

Thanks for that new reference, cynic. I do believe there’s plenty of gold in there - I just have to start mining it.