Fundamentals of creating a diversified portfolio based on numbers?

Question

There are a number of articles and posts across the internet describing how to diversify your portfolio. But, each claim typically focuses on how to allocate your portfolio rather than describing quantitatively the underlying reasons for that allocation, for example, they might tell you to allocate a certain percentage of your portfolio in each of the following categories

• domestic stocks small cap
• domestic stocks mid cap
• domestic stocks large cap
• international stocks small cap
• international stocks mid cap
• international stocks large cap
• domestic bonds (government/corporate/municipal)
• international bonds
• etc.

There may also be recommendations for diversification by sector, e.g.

• Energy
• Consumer discretionary
• Financials
• Materials
• Utilities
• Industrials
• Telecom
• etc.

Many folks (and robo-advisors) recommend specific portfolio allocations across all of the above. My question is, how do they quantitatively come up with their allocation recommendations that they believe will give the optimal gain/risk ratio? I'd like to create my own allocation based directly on the numbers (or convince myself that a specific recommendation is quantitatively sound) rather than relying on opaque recommendations.

Answer 1

Good question. There are plenty of investors who think they can simply rely on intuition, and although luck is always present it is not enough to construct a proper portfolio.

First of all there are two basic types of portfolio management: Passive and Active. The majority of abnormal gains are made with active portfolio management although passive managers are less likely to suffer loses.

Both types must be created with some kind of qualitative and quantitative research, but an active portfolio requires constant adjustments (Market Timing) to preserve the desired levels of risk and return.

The topic is extremely broad and every manager has his own preferred methods of quantitative analysis. I will try to list here some most common, in my opinion, ways of stock-picking and portfolio management.

Roy's Criterion:

The best portfolio is that with the lowest probability that the return will be below a specified level.

This is achieved by maximising the number of standard deviations between the return on the portfolio and minimum specified level:

Max k = (Rp-Rl)/Sp

Where (Rp) - return on portfolio, (Rl) - specified minimum return, (Sp) - standard deviation of portfolio return.

Kataoka's Criterion:

Maximise the minimum return (Rl) subject to constraint that the chance of a return below (Rl) is less than or equal to a specified value (a).

Maximise (Rl) Subject to Prob (Rp < Rl) =< a

For example, assume that the specified value is 20% - this will be met provided (Rl) is at least 0.84 standard deviations below (Rp). Therefore the best portfolio is the one that maximises (Rl) where:

Rl = Rp-0.84*Sp

Telser's Criterion:

Maximise expected return subject to the constraint that the chance of a return below the specified minimum is less than or equal to some specified minimum (a)

Maximise (Rp) subject to Prob (Rp < Rl) =< a

Assuming same data as previously:

Rl =< Rp-0.84*Sp and select the portfolio with highest expected return.

Security Selection

Now let's look at some methods of security selection. This is important when a manager believes some shares are mispriced.

The required return on security 'i' is given by:

Ri = Rf+(Rm-Rf)Bi

Where (Rf) - is a risk-free rate, (Rm) - return on the market, (Bi) - security's beta.

The difference between the required return and the actual return expected is known as the security's alpha (Ai).

Ai = Rai - Ri, where (Rai) is actual return on security 'i'.

Stock Picking

One way of stock-picking is to select portfolios of securities with positive alphas.

Alpha of a portfolio is simply the weighted average of the alphas of the securities in the portfolio.

Ap = {(n*Ai)

Where ({) is sigma (sorry for such weird typing, haven't figured out yet how to type proper-looking formulas), (n) - share of 'i'th security in portfolio.

So another way of stock-picking is ranking securities by their excess return to beta (ERB):

ERB = (Ri - Rf)/Bi

The greater the ERB the more desirable the security and the greater the proportion it will make up of the portfolio. Thus portfolios produced by this technique will have greater proportion of some securities than the market portfolio and lower proportions of other securities.

The number of securities depends on a cut-off rate (C*) for the ERB, defined so that all securities with ERB>C* are included in portfolio while if ERB

The cut-off rate for a portfolio containing the first 'j' securities is given by (i'm inserting an image cut from Word below):

Here comes the tricky part:

Basically what you do is first calculate ERB for each security, then calculate Cj for each security mix (gradually adding new securities one by one and recalculating Cj each time). Then you select an optimum portfolio by comparing Cj of each mix to ERB's of it's securities. Let me show you a simple example:

Say you have securities A,B,C and D

you calculated ERB's: ERB(a)=6, ERB(b)=6.5, ERB(c)=5, ERB(d)=4

also you calculated: C(a)=4.1, C(ab)=4.8, C(abc)=4.9, C(abcd)=4.5.

Then you check:

ERB(a),ERB(b),ERB(c) are greater than C(a), but C(a) only contains security A so C(a) is not an optimum mix.

ERB(a),ERB(b),ERB(c) are greater than C(ab), but C(ab) only contains securities A and B

ERB(a),ERB(b),ERB(c) are greater than C(abc), and C(abc) contains A B and C so it is an optimum.

ERB(d) is lower than C(abcd) so C(abcd) is not an optimum portfolio.

Finally the most important part:

Below is a formula to find the share of each security in the portfolio:

Here you simply plug in already obtained values for each security to find it's proportion in your portfolio.

I hope this somehow answers your question, however there is a lot more than this to consider if you decide to manage your portfolio yourself.

Some of the most important areas are:

1. Market Timing
2. Hedging
3. Stocks vs Bonds

Good luck with your investments!

And remember, the safest portfolio is the one that replicates the Global Market.

Answer 2

Most of the “recommendations” are just total market allocations.

Within domestic stocks, the performance rotates. Sometimes large cap outperform, sometimes small cap outperform. You can see the chart here (examine year by year):

https://www.google.com/finance?chdnp=1&chdd=1&chds=1&chdv=1&chvs=maximized&chdeh=0&chfdeh=0&chdet=1428692400000&chddm=99646&chls=IntervalBasedLine&cmpto=NYSEARCA:VO;NYSEARCA:VB&cmptdms=0;0&q=NYSEARCA:VV&ntsp=0&ei=_sIqVbHYB4HDrgGA-oGoDA

Conventional wisdom is to buy the entire market. If large cap currently make up 80% of the market, you would allocate 80% of domestic stocks to large cap.

Same case with International Stocks (Developed). If Japan and UK make up the largest market internationally, then so be it.

Similar case with domestic bonds, it is usually total bond market allocation in the beginning.

Then there is the question of when you want to withdraw the money. If you are withdrawing in a couple years, you do not want to expose too much to currency risks, thus you would allocate less to international markets. If you are investing for retirement, you will get the total world market.

Then there is the question of risk tolerance. Bonds are somewhat negatively correlated with Stocks. When stock dips by 5% in a month, bonds might go up by 2%. Under normal circumstances they both go upward. Bond/Stock allocation ratio is by age I’m sure you knew that already.

Then there is the case of Modern portfolio theory. There will be slight adjustments to the ETF weights if it is found that adjusting them would give a smaller portfolio variance, while sacrificing small gains. You can try it yourself using Excel solver.

There is a strategy called Sector Rotation. Google it and you will find examples of overweighting the winners periodically. It is difficult to time the rotation, but Healthcare has somehow consistently outperformed. Nonetheless, those “recommendations” you mentioned are likely to be market allocations again.

The “Robo-advisors” list out every asset allocation in detail to make you feel overwhelmed and resort to using their service. In extreme cases, they can even break down the holdings to 2/3/4 digit Standard Industrial Classification codes, or break down the bond duration etc.

Some “Robo-advisors” would suggest you as many ETF as possible to increase trade commissions (if it isn’t commission free). For example, suggesting you to buy VB, VO, VV instead a VTI.

Answer 3

Your question is a complex one because knowledge of the investor's beliefs about the market is required. For almost any quantitative portfolio, one must have a good estimate of the expected return vector and covariance matrix of the assets in question. The expected return vector, in particular, is far from estimable. No one agrees on it and there is no way to know who is right and who is wrong.

In a world satisfying the conditions of the CAPM, you can bypass this problem because the main implication of the CAPM is that the market weights are optimal. In that case the answer to your question is that you should determine the market weights of the various assets and use those along with saving in a risk-free account or borrowing, depending on your risk tolerance. This portfolio has the added benefit that you don't need to rebalance much...the weights in your portfolio adjust at the same rate as the market weights.

Any portfolio that has something besides this also includes some notion of expected return aside from CAPM fair pricing. The question for you, then, is whether you have such a notion. If you do, you can mix your information with the market weights to come up with a portfolio. This is what the Black-Litterman method does, for example: get the expected return vector implied by market weights and the covariance matrix, mix with your expected return vector, then use mean-variance optimization to come up with your final weights.