【Asset Allocation】Diversification

Recap

In our first post, we stressed the importance of:

1. Minimizing Loss ($)

2. Minimizing Frequency of Loss

3. Minimizing Volatility

 

To cut to the point, there are three main approaches to achieving these goals:

1. Diversification

2. Asset Allocation

3. Trend Following

 

Strategy 1. Diversification

Two Stock Porfolio

Let's say, you, investor is only given two options: APPLE stock and cash. APPLE stock is expected to yield 10% return this year and has 20% risk, whereas cash yields 0% return and has 0% risk. Based on this information, you decided to invest 50% in APPLE stock and keep the remaining 50% as cash. We can write this down in an expression such that:

 

Expected Return

$$E(R_p) = w_{apple} E(R_{apple}) + w_{cash} E(R_{cash})$$

 

where,

$E(R_p)$ = Expected Portfolio Return

$E(R_x)$ = Expected Return on Investment

$w_x$ = Weight of an Investment

 

If we substitute our expected return from apple and expected weights:

 

$$E(R_p) = 0.5 (0.1) + 0.5 (0) = 5\%$$

 

as cash does not yield any return.

 

Standard Deviation

The risk of portfolio is represented by standard deviation of returns.

 

$$\sigma^2_p = w^2_x \sigma^2_x + w^2_y \sigma^2_y + 2w_x w_y cov_{xy}$$

 

Because Correlation is $\frac{cov_{xy}}{\sigma_x \sigma_y}$, this equation can also be expressed as:

 

$$\sigma^2_p = w^2_x \sigma^2_x + w^2_y \sigma^2_y + 2w_x w_y \sigma_x \sigma_y \rho $$

 

If we substitute the values, we obtain:

 

$$\sigma^2_p = 0.5^2 (0.2)^2 + 0.5^2_y (0)^2_y + 2 (0.5) (0.5) (0) = 0.01$$

$$\sigma_p = \sqrt{0.01} = 0.1 $$

 

Standard Deviation of 0.1 equals 10%. You should have noticed by now that by adjusting the weight of risky asset, APPLE stock, and mixing it with an uncorrelated (since COVxy and $\rho$ are 0) asset actually cut down the portfolio risk by half.

 

Let's tweak this example a bit.

 

Now, assume the selection expanded to APPLE stock, which has expected return of 10% and risk of 20%, and MICROSOFT stock, which has expected return of 20% and risk of 40%. Because they are both technology companies, their stocks seem to move together with Correlation of 0.8. You decided to continue investing with 50%-50% strategy.

Expected Return

$$E(R_p) = w_{apple} E(R_{apple}) + w_{microsoft} E(R_{microsoft})$$

$$E(R_p) = 0.5 (0.1) + 0.5 (0.2) = 15% $$

 

Standard Deviation

$$\sigma^2_p = w^2_x \sigma^2_x + w^2_y \sigma^2_y + 2w_x w_y \sigma_x \sigma_y \rho$$

$$\sigma^2_p = 0.5^2 (0.2^2) + 0.5^2 (0.4^2) + 2 (0.5)(0.5)(0.2)(0.4)(0.8) = 0.082$$

$$\sigma_p = \sqrt(0.082) = 28.6%$$

 

While the expected return substantially increased, the standard deviation or risk increased as well.

 

Now, assume the selection changed to APPLE stock, which has expected return of 10% and risk of 20%, and P&G stock, which has expected return of 20% and risk of 40%. Because they are from different industries, their stocks seem to have a low correlation 0.2. You decided to continue investing with 50%-50% strategy.

Expected Return

$$E(R_p) = w_{apple} E(R_{apple}) + w_{P\&G} E(R_{P\&G})$$

$$E(R_p) = 0.5 (0.1) + 0.5 (0.2) = 15% $$

 

Standard Deviation

$$\sigma^2_p = w^2_x \sigma^2_x + w^2_y \sigma^2_y + 2w_x w_y \sigma_x \sigma_y \rho$$

$$\sigma^2_p = 0.5^2 (0.2^2) + 0.5^2 (0.4^2) + 2 (0.5)(0.5)(0.2)(0.4)(0.2) = 0.058$$

$$\sigma_p = \sqrt(0.058) = 24.08%$$

 

Even though P&G stock had same level of risk as MICROSOFT, the portfolio risk turned out to be lower. This is due consequence of investing in two stocks that are less correlated.

 

Risk & Return for Different Values of $\rho$

The takeaway here is:

1. Correlation ranges from -1 to 1, and lower the correlation between securities, the safer your porfolio.

2. More number of securities with diverse correlation, the safer your porfolio.

 

Given this framework, we can expand to more number of stocks in our porfolio.

 

Diversified Portfolio

Now imagine repeating the same computation (1) calculating expected return and (2) calculating portfolio risk for a lot more number of stocks available in the market. You could start from having 2 stocks in the portfolio then move all the way to having all stocks in the portfolio.

 

Univerise: $Stock_1 \cdots Stock_n$

Plot: $(E(R_1),\sigma_1) \cdots (E(R_n),\sigma_n)$

Calculate $E(R_p)$ and $\sigma_p$ of all combinations

 

Then, we obtain:

The portfolio positioned at the left end is called "Global Minimum-Variance Portfolio", because it has the least possible volatility within the universe. All the portfolios that form the left-frontier of this plot starting from the "Global Minimum-Variance Portfolio" are called "Efficient Frontier/Portfolios". This is based on our very intuition where we would always choose the portfolio with highest return at the given level of risk.

 

By mixing various securities, you can achieve formulating a portfolio that offers highest return at a given risk level. Which portfolio to choose entirely depends on your risk appetite (ability or willingness to take risk) and leverage ratio. These are topics for discussion in the future.

 

In Practice

"Can't experts pick better stocks?"

"Can't I narrow down to healthier, better stocks?"

 

Your experts may be a good stock picker, and yes you can narrow down to healthier or better stocks. These are definitely the key skills that will give your porfolio an edge over others. 

 

But one undeniable fact is that such approach is very subjective and unclear. What is a healthier stock? What is a material information? When will the stocks go up? More so, such unstandardized and subjective approaches tend to yield ambivalent returns over time. 

 

A more practical approach would be to establish an evidence-based set of rules, such as:

1. Price of most products with inherent value (gold, companies, bonds and houses) tend to increase over time.

2. You cannot guess exact timing of the price increase.

3. Diversification reduces the volatility of your portfolio.

 

This gives us a clear direction: Diversify your porfolio with companies that you deem are "good" or "healthly" but are not correlated with each other. Other option is to invest in Exchange Traded Funds (ETFs) such as VOO, SPY, or QQQ, which are a large basket of companies with specific set of rules (i.e., top 500 market capitalizations).