Quantitative Analysis, Risk Management, Modelling, Algo Trading, and Big Data Analysis

2-Asset Portfolio Construction


When you start your journey in the world of finance and money investment, sooner or later, you hit the concept of a portfolio. The portfolio is simply a term used to describe shortly a set of assets you are or your are planning to hold in order to make a profit in a given horizon of time.

If you are a fan of stocks, your desire is to purchase some shares of the company you like or you anticipate it will bring you a profit. Make a note here. I said- the company. Single, not plural. If so, having some initial capital set aside for a particular choice of yours, say $C=\$100,000$, you decided to buy IBM stock for $\$100$. Great! You own a fraction of the IBM company. Regardless the fact whether you are a beginner in investing/trading or you are more advanced in that field, your purchase of IBM shares is now called a long position, i.e. you expect the value of the stock to go up. We will skip the scenario of short-selling (i.e. making the profit when the price of the stock goes down) for the simplicity of this article. All right, you waited a couple of days and, luckily, two wonderful events took place in meanwhile. Firstly, the IBM stock is now not worth $\$100$ any longer (the demand for that stock was so high that there were more buyers wanting to purchase a piece of IBM that their buying demand pushed the stock price higher), and now IBM is worth $\$120$ per share. And guess what?! You can’t believe it! You’re smiling just because now you can sell your part (share) of the IBM company to someone else (a buyer who is now willing to pay $\$120$ for the same share of IBM, i.e. $\$20$ more!). So, you do it! Secondly, in meanwhile, the company of IBM decided that a part of its net income would be distributed (given to, divided among) all shareholders in the form of dividends that value had been announced by IBM to be $\$5$ per one share hold. As a consequence of these two events, your net return occurred to be equal:
$$
R = \frac{S_t+D_t}{S_{t-1}} – 1 = \frac{\$120+$5}{\$100} – 1 = 0.25 \ \ \mbox{or} \ \ 25\%
$$
what in terms of your capital gain simply means you became richer by extra $\$25,000$ according to
$$
\mbox{Capital updated} \ = C(1+R) = \$100,000(1+0.25) = \$125,000 .
$$

To be completely honest, this situation would take place under one condition of the market’s operations of buying and selling the IBM stock. You may ask- what’s that? It is commonly denoted and referred to as the situation of perfect financial market(s) – i.e. an assumption of no taxes, no transaction costs, no costs of writing and enforcing contracts, no restrictions on investment in securities, no differences in information across investors, investors take prices as given because they are too small to affect them by their action of buying/selling (i.e. there exists a good liquidity of the stock in the market) – is taken as default. Oh, it’s a rough assumption! It is, indeed. But it has been assumed with a clear purpose of simplifying the basic understanding of what you can gain/lose in the processes of trading. For a curious reader, I refer to my article of Number of Shares for Limit Orders which provides with a broader feeling on how extra fees of trading operations influence your initial capital, $C$, that you wish to invest in a single stock.

Let’s now consider that you want to split your initial capital of $C$ between two choices: IBM and NVDA (NVIDIA Corporation). From some resources you have found out that the latter stock is more volatile than IBM (i.e. its price jumps more rapidly; it’s more risky) therefore you decided to invest:

  1. $\$75,000$ in IBM (0.75 share of portfolio)
  2. $\$25,000$ in NVDA (0.25 share of portfolio) .

Now, also having information that the expected return in 1-year horizon is equal 13% and 26% for IBM and NVDA, respectively, and their corresponding volatilities are 30% and 60%, firstly we may calculate the fundamental portfolio measures based on these two assets (stocks) we wish to hold (invest in and hold).

The Expected Portfolio ($P$) Return
$$
E(R_P) = \sum_{i=1}^{N} w_iE(R_i)
$$
what in our case study returns the following expectation for two assets ($N=2$),
$$
E(R_P) = 0.75\times 0.13+0.25\times 0.26 = 0.1625 \ \ \mbox{or} \ \ 16.25\% .
$$

The Volatility of the Return of the Portfolio


Before reaching the exact formula for the volatility of the expected return of our 2-asset portfolio, we need to consider the basic mathematics standing for it. Firstly, have in mind that for any variance for a random variable $R_i$ (here, the expected return of the asset $i$),
$$
Var(w_iR_i) = w_i^2 Var(R_i) .
$$
Secondly, keep in mind a high-school formula of $(a+b)^2=a^2+b^2+2ab$. Got it? Great! Let’s move forward with the variance of the sum of two variables, say $a$ and $b$:
$$
Var(a+b) = E[a+b-E(a+b)]^2 \\
= E[a+b-E(a)-E(b)]^2 \\
= E[a-E(a)+b-E(b)]^2 \\
= E[a-E(a)]^2 + E[b-E(b)]^2 + 2E[a-E(a)]E[b-E(b)] .
$$
Where does it take us? Well, to the conclusion of the day, i.e.:
$$
= Var(a) + Var(b) + 2Cov(a,b)
$$
where $Cov(a,b)$ is a covariance that measures how $a$ and $b$ move together. At this point, we wish to define correlation between two variables $a$ and $b$ as:
$$
Cov(a,b) = Vol(a)\times Vol(b)\times Corr(a,b)
$$
what for two same assets (variables) perfectly correlated with each other (i.e. $corr(a,a)=1$) would return us with:
$$
Cov(a,b)=Vol(a)\times Vol(b)\times 1 = Var(a) .
$$
Keeping all foregoing mathematics in mind, we may define the variance of the portfolio return as follows:
$$
Var(P) = Var\left(\sum_{i=1}^{N} w_iR_i\right) = \sum w_i^2Var(R_i)+\sum_{i=1}^{N}\sum_{i\ne j,\ j=1}^{N} w_iw_jCov(R_i,R_j) .
$$
where again,
$$
Cov(R_i,R_j) = Var(R_i)Var(R_j)Corr(R_i,R_j) ,
$$
describes the relation between the covariance of expected returns of an asset $i$ and $j$, respectively, and their corresponding variances and mutual (linear) correlation (factor). The latter to be within $[-1,1]$ interval. The formula for $Var(P)$ is extremely important to memorize as it provides the basic information on whether a combination of two assets is less or more risky than putting all our money into one basket?

Therefore, 2-asset portfolio construction constitutes the cornerstone for a basic risk management approach for any investments which require, in general, a division of the investor’s initial capital $C$ into $N$ number of assets (investments) he or she wishes to take the positions in.

Coming back to our case study of IBM and NVDA investment opportunities combined into the portfolio of two assets, we find that:
$$
Var(P) = (0.75)^2(0.30)^2 + (0.25)^2(0.60)^2 + 2(0.75)(0.25)(0.30)(0.60)\times \\ Corr(\mbox{IBM},\mbox{NVDA}) = 0.11
$$
at assumed, ad hoc, correlation of 0.5 between these two securities. Since,
$$
Vol(P)=\sqrt{Var(P)} ,
$$
eventually we find that our potential investment in both IBM and NVDA companies – the 2-asset portfolio – given their expected rates of returns at derived volatility levels for a 1-year time-frame is:
$$
E(P) \ \ \mbox{at}\ \ Vol(P) = 16.25\% \ \ \mbox{at}\ \ 33.17\% .
$$
It turns to be more rewarding ($16.25\%$) than investing solely in IBM (the expected return of $13\%$) at a slightly higher risk ($3\%$ difference).

We will come back to the concept of 2-asset portfolio during the discussion of risk-reward investigation for N-asset investment case.

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