# Portfolio Construction

February 23, 2018

# What is a Portfolio?

A portfolio is a set of financial securities or assets that an investor holds which together produce a financial outcome (profit and loss) over time.

When investors construct a portfolio, they are usually faced with multiple important decisions. The main question is: **what should the asset allocation of the portfolio be?** Does the investor want to hold shares, bonds, currencies, futures, or any two or more of those categories of assets combined? Perhaps the investor would like to exclude certain asset classes from his portfolio, or only focus on one asset class (for example, stocks). The choice is entirely up to the discretion of the investor or the manager of the portfolio.

While constructing the portfolio, the investor or manager needs to take into account the costs associated with the construction itself (regardless of the risks), such as transaction costs, commissions, spreads and so on. Those costs are ultimately deducted from either the amount invested or from the profit earned (or from both).

Another consideration is the tax accrued on the investment activities, such as capital gain tax for stocks, or taxes on dividends or interest payments. Those taxes will eventually lower your net profit.

# Calculating the Return of Your Portfolio

# Return on Stocks

The return on stock can be calculated as follows:

$$ R = \frac{(P_1-P_0+D)}{P_0} $$The above equation sums the two returns possible on a stock (appreciation in value and dividends), where:

*P _{0}* → the initial price

*P _{1}* → the exit price

*D* → dividends.

## Example: Return on Stocks

For example you bought a stock at the beginning of the period for $250 then you sold it at the end of the period for$300, and during your holding period you got a dividend of $20 per share.

The return on the stock is ($300 -$250 + $20) /$250 = 0.28 or 28%.

## Example: Return on Bonds

The return on bonds is done by calculating the interest payments on the bond multiplied by the number of years.

For example, if you hold a bond with a principal amount of $1000 which offers a 3% interest payment a year, and it matures in 5 years. Then the interest payment each year is$30. Total interest payment in $30 x 5=$150.

When the bond holder buys and holds the bond from the issue date till maturity, then he will have made no gains on the price fluctuations. The example above depicts such case. The return includes only interest payment and no capital gains or losses. It is calculated as follows:

$$ \text{Return} = \frac{\text{Total Interest Payments}}{\text{Purchase Value of the Bond}} $$In case the investor buys the bond at a discount, then he will have made extra earnings with the discount.

$$ \text{Return on Bonds} = \frac{\text{Interest Payment} + \text{Capital Gain}}{\text{Purchase Value}} $$Note if the investor had a capital loss then you would subtract the capital loss from interest payment to find the return.

## Return on Portfolio

If we have a portfolio of two assets, the return on the portfolio is calculated by adding the weighting return of each asset. To calculate this, we use the following equation:

$$ \text{Expected Portfolio Return} = W_1 \times E(R_1) + W_2 \times E(R_2) $$*W _{1}* → the weight (percentage) of the first asset in the portfolio

*W _{2}* → the weight (percentage) of the first asset in the portfolio

*E(R _{1 } )* → is the expected return on the first asset

*E(R _{2 } )* → is the expected return on the second asset

**Example**: let’s say you have a portfolio of two assets: asset A and asset B. 70% of your portfolio is invested in asset A and 30% in asset B. The expected return on asset A is 12% and on asset B is 15%.

Therefore, the expected return on the portfolio is calculated as follows:

E(R_{P})= (0.7 x 0.12) + (0.3 x 0.15) = 0.084 + 0.045 = 0.129 = 12.9%

# Diversification vs. Concentration

One of the methods to reduce risk for investors is to diversify their portfolio. That is, they need to select assets that perform well in different business conditions. An important rule for diversification is that the selected assets should not have high correlation with one another.

In other words, **assets in the portfolio need to have different exposure**. For example, if you hold a stock of a high-tech company, and you want to diversify, you should select a stock from an entirely different industry, such as a transportation or agriculture, for example. This way you are exposed to different industries, and if something were to happen in one industry you would still offset the losses with the good performance in other industries.

But while diversification does work to protect you from risk, it eventually reduces your returns, since any returns you have made in one asset can be reduced by losses in another. Excessive diversification usually leads to greatly reduced returns, to the point that such diversification is not advised.

For that reason, a degree of concentration in your portfolio is prudent. Concentration means investing more in one area than in other areas based on a careful analysis of past performance and expected future trends. For example, you can invest around 70% of your portfolio in renewable energies and the rest in other industries to reduce the risk. This enhances your return prospects from the concentration while reasonably mitigating risk.

# Portfolio Variance

To gauge the risk level in a portfolio beforehand, investors assess the portfolio’s variance. In simple terms, the variance of the portfolio is a measure of its volatility. It shows how much the value of the assets in the portfolio can fluctuate. To calculate this variance, the investor needs to calculate the standard deviation of each asset in the portfolio, and correlations between each pair of two assets in the portfolio.

High correlations between assets increase the variance of the portfolio, since correlated assets tend to move in the same direction. Low correlations, on the other hand, decrease the portfolio variance, which is the case in a diversified portfolio.

The equation for calculating variance of a portfolio that consists of two assets (for simplicity) goes as follows:

$$ \begin{aligned} \text{Portfolio Variance} = (W_1^{\space2} \times SD_1^{\space2}) + (W_2^{\space2} \times SD_2^{\space2}) + \\ (2 W_1\times SD_1 \times W_2 \times SD_2 \times Cov_{1,2}) \end{aligned} $$*W _{1 }* → the weight of the asset in the portfolio (the percentage)

*W _{2}* → the weight of the asset in the portfolio (the percentage)

*SD _{1 }* → standard deviation of asset 1

*SD _{2 }* → standard deviation of asset 2

*Cov _{1,2}* → correlation coefficient between the two assets. This coefficient describes the extent with which two assets move in tandem.

The portfolio variance gives you an overall picture of the possible volatility in the portfolio (to the upside or to the downside). This portfolio stems from the volatility of each asset, measures by standard deviation, and the correlation between different assets, which can either increase or decrease portfolio volatility.

# Using Futures to Reduce Portfolio Risk

Another way to reduce the risk of the portfolio is by holding futures such as options or forward contracts. Those contracts give the investor certainty in that regardless of market fluctuations, the investors will still be able to sell or buy the assets he wants at a predetermined price, although this comes at a cost (that is, the cost of the futures contract). For example, an investor can buy soy beans at $10 expecting the price to rise to$15. The investor also buys a future contract to sell the soybeans at $9 for$1.5 at a later date. This way, even if the price drops below $9, the investor can sell his soybeans safely.

Future contracts reduce the variance of the portfolio as they limit risk for a certain cost.

# Modern Portfolio Theory

Modern portfolio theory (MPT) entails that any investor can choose the optimal selection of assets based on his risk tolerance level, from a possible set of portfolios that lie on the efficient market frontier.

The efficient market frontier is a curve on a two axis graph. The two axis are the expected return for the portfolio and the portfolio risk (variance). The investor should choose a portfolio at any point on the efficient market frontier since it is the most efficient investment. In other words, selecting a portfolio based on this curve guarantees you are compensated adequately for the level of risk you are taking. Otherwise, their portfolio is either too risky or has very little return.

Source: Investopedia

The main premise of the theory is that investors must take risk to receive returns. However, the two factors mush be proportionate to one another. For example, investors who have low tolerance for risk should expect to make low returns. On the other hand, investors who are willing to take on high risk are more likely to receive high returns. For this reason, risk-averse investors can choose portfolios with more bonds in them since bonds generate fixed returns, where risk-tolerant investors can include more stocks in their portfolio since their returns fluctuate.

## Capital Asset Pricing Model

One key tenet of the modern portfolio theory is the capital asset pricing model, which entails that the two key factors affecting the price of the asset are the volatility of the stock in relation to the volatility of the market, and the volatility of the market itself.

$$ E(r) = R_f + \beta(R_m-R_f) $$*E(r)* → the expected return on the asset

*R _{f}* → the risk free rate (the rate of return that investors expect for an investment that has zero risk, such as the interest rate on sovereign bonds)

β → asset volatility as compared to the volatility of the broader market

*R _{m}* → the return of the market.

The difference between the return of the market and the risk free rate is called the risk premium (the extra reward to compensate investors for taking on additional risk).

The equation captures the volatility of the market times the risk premium in addition to the risk free rate.

**The Arbitrage Pricing Model:**

The arbitrage pricing model, on the other hand, takes into account many factors that affect the price of the asset. Those can include macroeconomic, microeconomic, and industry specific factors.

This model can sometimes be called the factor-based pricing model, since it takes into account risks generated by various risks.

$$ E(r) = R_f + B_1 \times RP_1 + B_2 \times RP_2 + \mathellipsis + B_n \times RP_n $$*R _{f}* = the risk-free interest rate

*B _{i}* = the sensitivity of the asset to the factor

*i*

*RP _{i}* = the risk premium associated with factor

*i*

# Conclusion

Whether you are a graduate student with a small capital, or a hard worker who managed to save a decent amount, or someone who inherited an amount of money and would like to invest it, it is strongly advised to keep your investment process simple.

In very simple terms, two key factors influence the return you can get on your investments. The first factor is the level of risk, and the second is the expected return. Those two factors can be calculated for assets as well as for portfolios, and they should guide your decision making during investment.