Question

To illustrate the risk management approaches, create a sample portfolio of investment vehicles, with varying levels of risk. Be sure to work out the numbers, using appropriate formulas based on your understanding of the position.

Answer 1

The diversification plays a very important role in the modern portfolio theory. Markowitz approach is viewed as a single period approach: at the beginning of the period the investor must make a decision in what particular securities to invest and hold these securities until the end of the period. Because a portfolio is a collection of securities, this decision is equivalent to selecting an optimal portfolio from a set of possible portfolios.

A good investment portfolio will spread your risk. It is an almost universally accepted concept that most portfolios should include a mix of investments, such as stocks, bonds, mutual funds, and other investment vehicles. A portfolio should also be balanced. That is, the portfolio should contain investments with varying levels and types of risk to help minimize the overall impact if one of the portfolio holdings declines significantly. Spreading your investment over multiple asset classes should help reduce your risk of losing your entire investment. However, remember that there is no guarantee that any investment strategy will be successful and that all investing involves risk, including the possible loss of principal. There are few key points to take away while creating an investment portfolio:

• Asset Allocation: It means deciding how your investment dollars should be allocated among broad investment classes, such as stocks, bonds, and cash alternatives. Rather than focusing on individual investments (such as which company's stock to buy), asset allocation approaches diversification from a more general viewpoint. For example, what percentage of your portfolio should be in stocks? The underlying principle is that different classes of investments have shown different rates of return and levels of price volatility over time. Also, since different asset classes often respond differently to the same news, your stocks may go down while your bonds go up, or vice versa. Though neither diversification nor asset allocation can guarantee a profit or ensure against a potential loss, diversifying your investments over various asset classes can help you try to minimize volatility and maximize potential return.

• Diversification: Diversification isn't limited to asset allocation, either. Even within an investment class, different investments may offer different levels of volatility and potential return. For example, with the stock portion of your portfolio, you might choose to balance higher-volatility stocks with those that have historically been more stable (though past performance is no guarantee of future results).

The expected rate of return of the portfolio:

The expected rate of return of a portfolio should depend on the expected rates of return of each security included in the portfolio. This method for calculating the expected rate of return on the portfolio (E(r)p) is the weighted average of the expected returns on its component securities:

n E(r)p = Σ wi * Ei (r) = E1(r) + w2 * E2(r) +…+ wn * En(r),

Here, i=1

                wi - the proportion of the portfolio’s initial value invested in security i;

Ei(r) - the expected rate of return of security I;

n - the number of securities in the portfolio.

Risk of the portfolio:

The relationship between the assets can be estimated using the covariance and coefficient of correlation. As covariance can range from “–” to “+” infinity, it is more useful for identification of the direction of relationship (positive or negative), coefficients of correlation always Investment Analysis and Portfolio Management 56 lies between -1 and +1 and is the convenient measure of intensity and direction of the relationship between the assets.

Risk of the portfolio, which consists of 2 securities (A ir B):

Here:

wA ir wB - the proportion of the portfolio’s initial value invested in security A and B ( wA + wB = 1);

δA ir δB - standard deviation of security A and B;

kAB - coefficient of coreliation between the returns of security A and B.

Standard deviation of the portfolio consisting n securities:

Here:

wi ir wj - the proportion of the portfolio’s initial value invested in security i and j ( wi + wj = 1);

δi ir δj - standard deviation of security i and j;

kij - coefficient of coreliation between the returns of security i and j

Capital Asset Pricing Model (CAPM) :

CAPM predicts what an expected rate of return for the investor should be, given other statistics about the expected rate of return in the market and market risk (systematic risk).

Measuring Risk in CAPM is based on the identification of two key components of total risk (as measured by variance or standard deviation of return):

• Systematic risk
• Unsystematic risk

Systematic risk is that associated with the market (purchasing power risk, interest rate risk, liquidity risk, etc.)

Unsystematic risk is unique to an individual asset (business risk, financial risk, other risks, related to investment into particular asset).

In CAPM investors are compensated for taking only systematic risk.

Portfolio risk and the level of diversification

E(r j) = Rf + β(j) * ( E(rM) - Rf ),

here: E(r j) - expected return on stock j;

Rf - risk free rate of return;

E(rM) - expected rate of return on the market

β(j) - coefficient Beta, measuring undiversified risk of security j

Equation in formula represents the straight line having an intercept of Rf and slope of β(j) * ( E(rM) - Rf ).

This relationship between the expected return and Beta is known as Security Market Line (SML). Each security can be described by its specific security market line, they differ because their Betas are different and reflect different levels of market risk for these securities.

Security market Line (SML)

Coefficient Beta (β). Each security has it’s individual systematic - undiversified risk, measured using coefficient Beta. Coefficient Beta (β) indicates how the price of security/ return on security depends upon the market forces.

Thus, coefficient Beta for any security can be calculated using formula :

Portfolio Beta can be calculated using formula:

Here:

wi - the proportion of the portfolio’s initial value invested in security i;

βi - coefficient Beta for security i

Arbitrage Pricing Theory (APT) :

APT is the rational statement that the market return is determined by a number of different factors. These factors can be fundamental factors or statistical. If these factors are essential, there to be no arbitrage opportunities there must be restrictions on the investment process.

Here arbitrage can be understood as the earning of riskless profit by taking advantage of differential pricing for the same assets or security. Arbitrage is widely applied investment tactic. APT states, that the expected rate of return of security J is the linear function from the complex economic factors common to all securities and can be estimated using formula:

here: E(rJ) - expected return on stock J;

E(ŕJ) - expected rate of return for security J, if the influence of all factors is 0;

IiJ - the change in the rate of return for security J, influenced by economic factor i (i = 1, ..., n);

βiJ - coefficient Beta, showing sensitivity of security’s J rate of return upon the factor i (this influence could be both positive or negative); εJ - error of rounding for the security J (expected value – 0).

The Concept of Total Return

The total return on an investment tells the investor the percentage gain or loss on an asset based on its purchase price. It is calculated by dividing the selling value of the position plus any dividends received by its total cost. In essence, this works out to capital gains plus dividends as a percentage of the money you laid out to buy the investment.

Calculating Total Return :

Say that an investor had a cost basis of $15,100 in Tesco stock (she purchased $15,000 worth of Tesco stock and paid $100 total commissions on the buy and sell orders). She received $300 cash dividends during the time she held the stock. Later, she sold the position for $35,000. What was her total return?

We can plug the variables into the total return formula to find our answer:

• Start with the $35,000 received upon the sale of the stock
• Add the $300 cash dividends received to get $35,300
• Divide this by the cost basis of $15,100

The result is 2.3377 percent or 133.77 percent total return on invested principal (remember that 1.0 of the total return is the principal so you must subtract it out to express the gain or loss as a percentage;

(2.3377 - 1.0) = 1.3377, or 133.77 percent expressed as a percentage.

Had the result been 1.5, the total return expressed as a percentage would have been 50 percent (1.5 - 1.0) = .5, or 50 percent.

To answer if this was a good return:

If the investor earned 133.77 percent in five years, it's cause for celebration. However, if it took the investor twenty years to produce such a return, this would have been a terrible investment

Compound Annual Growth Rate (CAGR):

A stock position might be up 40 percent one year and down 5 percent the next. CAGR provides the annual return for such an investment as if it had grown at a steady, even pace. In other words, it tells you how much you would have to earn each year, compounded on your principal, to arrive at the final selling value.

Practically all of the best stock investments in history have experienced declines of 50 percent or more, peak-to-trough, all while making their owners fabulously wealthy.

Calculating Compound Annual Growth Rate (CAGR):

In order to calculate CAGR, you must begin with the total return and the number of years in which the investment was held. In the above example, the total return was 2.3377 (133.77 percent). You also know the investment was held for ten years.

You can find the CAGR for this scenario by plugging the information into the following formula:

CAGR = Ending value / Starting value)^(1 / # years) -1

Multiply the total return (2.3377) by the X root (X being the number of years the investment was held).

This can be simplified by taking the inverse of the root and using it as an exponent as shown in the above formula.

In this example, 1/10, or .10 (had the number of years been 2, you could have taken 1/2 or .5 as the exponent, 3 years would be 1/3 or .33 as the exponent, four years would be 1/4, or .25, and so on and so forth.)

In the above example, CAGR would be calculated as follows:

CAGR = 2.377(.10) = 1.09

This answer equates to a 9-percent compound annual growth rate (again, recall the 1.0 represents the principal value which must be subtracted; ergo, 1.09 - 1.0 = .09, or 9 percent CAGR expressed as a percentage).

In other words, if the gains on the Tesco investment were smoothed out, the investment grew at 9 percent compounded annually. To check the result, use the future value of a single amount. In essence, this means that if the investor had taken the roughly $15,000 to a bank for ten years and earned 9 percent on her money, she would have ended up with the same balance of $35,300 at the end of the period.