Efficient Frontier and the Famous Portfolio Theory of Markowitz - Assignment Example

Ans Efficient Frontier: Efficient Frontier was derived originally from the famous portfolio theory of Markowitz. An efficient frontier represents the curve which provides a maximum rate of return along a particular level of risk. Similarly, it provides the lowest level of risk for a particular level of expected rate of return. The theory behind this is that investors are risk averse and they can diversify their risk by holding portfolio which have different correlation, but to mention not equal to 1. Any two securities which have a different correlation structure but are less than 1 can diversify the risk of the portfolio. Figure 1: Efficient frontier for risky securities If we assume that riskless borrowing and lending were allowed than the efficient portfolio would be a line passing from Risk free rate and tangent to the efficient frontier as evident from the figure provided below Figure 2: Efficient Frontier with risk free security The efficient frontier in the above figure starts from the point where the portfolio return is minimal. The efficient theory is derived from mean variance theory which holds the following assumptions: Investors are risk averse; and they prefer more for less for the same level of risk Expected returns for the assets in the portfolio are known The variances and covariance structure between different class of assets are also known The returns follow the normal distribution (Elton & Gruber, 1995) Only the first assumption seems to be viable while all the other assumptions are not practically valid. It is very difficult to forecast the expected rate of returns therefore forecasting comes into play which tries to provide a reliable value but not the actual that will be in the marketplace. Errors in estimating of mean returns can have a dramatic impact on the portfolio returns and risk and thus can lead to ineffective decision making. Furthermore, the quantity of estimates required for calculating portfolio risk can be huge depending on the number of assets. Regardless of the quantity, the quality of even some of the factors can deteriorate the calculation of the whole portfolio and makes the mean variance theory inefficient and impractical. Minimum Variance Portfolio: Minimum Variance Portfolio is relevant with the concept of efficient frontier, as in order to find the efficient portfolio, we must identify the portfolios that have minimum variance for each level of expected rate of return. All such portfolios are called minimum variance portfolio. The portfolio variance for two assets is computed by XA and XB in the above formula represent the weightage of each stock. The minimum variance is computed by assigning values to XA and XB in such a way that it gives us the minimum variance. For a two asset portfolio the weightage of assets in the minimum variance portfolio can be computed by If borrowing or lending were not allowed: The efficient frontier curve would have changed if the assumption that borrowing or lending is not allowed at the same rate. Since in the practical world, borrowing has implied credit risk therefore financial institution charge a premium on the risk free rate to cover for the credit risk. Thus borrowing would have becomes more costly. Figure 3: Efficient frontier with borrowing and lending not allowed at the same rate. If there have been differences in borrowing or lending rates than the efficient frontier would have been the line passing from RF_____G___H___I. Ans 2 a) The cost of equity or the discount rate can be calculated using the Capital Asset Pricing Model The capital asset pricing model requires information regarding the following = Risk free rate = Beta of the asset = Average Market rate of return Cost of equity = k = 2 + 1.1 (6 – 2) Cost of equity = 2 + 1.1 (4) Cost of equity = 2 + 4.4 Cost of equity = k = 6.4% Using the Dividend Discount Model we can calculate the price of the share using the following assumptions The price of the stock will be the present value of all the future dividends Price = D1 / (1 + k) + D2 / (1 + k)2 + D3 / (1 + k)3 + (D4 / k – g) / / (1 + k)4 We are assuming that dividends of the price technology firm will grow at a stable growth rate of 6% after the third year till perpetuity. D0 = £0.8 During the first three years dividends will grow at a rate of 10% b) Price to Earning Ratio = Price per Share / Earning per Share Price to Earning Ratio = 236.82 / 10 Price to Earning Ratio = 23.68 This ratio might be different from the PE ratio based on the actual price of the share since this price which we have calculated through is a particular model. Not all the investors will be using the same assumptions and the same model to calculate the price of the share which makes the actual price different from the price we have calculated. Thus a different price implies a different PE ratio. For instance if the market price is £175, than the PE ratio would be Price to Earning Ratio = 175 / 10 Price to Earning Ratio = 17.5 Ans 3 Years remaining to Maturity Coupon Principal 10 5 120 8 0 100 5 10 200 7 20 100 Assuming a Yield to Maturity of 8% and a face value of the bond £100 We will assume a 10 basis change in points Bond 1: Bond 2: Bond 3: Bond 4: Where w is the proportion of the market value of each bond in the portfolio. Market Value of Bonds: Ans 4 Uses of Duration as a tool: Duration is the most common measure of evaluating the risks of bond. It is defined as the approximate change in price of the bond for a 100 basis change in interest rates. It indicates how price sensitive is the bond to sensitive in changes in interest rates (Stanton, 2009). The duration of bond is of real interest to the investors since it gives him an idea of the bond instrument to changes in interest rate. If the interest rates are expected to fall, than the duration is extended and the investor can infer that it will take a greater time to recover the funds. For instance, fund managers use duration to construct a portfolio for an investor based on his/her priorities. Low duration portfolios are often used as an alternative for traditional cash instruments such as money markets as they are assumed to show the liquidity. Advantages of Duration: It provides a credible measure of risk for a bond to the investor. Disadvantages of Duration: Duration requires some sort of subjectivity in estimating its parameters which can lead to erroneous conclusions. Duration does not provide accurate results for large yield changes Ans 5 Probability of default means that the chances that the bond issuer will default on its payment. Generally, these probabilities are calculated using the historical default rates of certainly rated instruments published by Rating Agencies. For instance, Standard’s & Poors issues an opinion of the future ability of the issuer to make full and timely payments due to investors. As we know that the bonds are priced based on the future value of all the coupon and principal payments and discounting them with the market determined yield. Lets assume that the price of the bond is captured by the equation below Here we have only a single payment due at the end of the year which will be discounted back by the market determined yield. The payoff at the end of the maturity will be either the bond will default or it will not. Figure 4: Probability of default and bond Price Here p represents the probability of default and f represents the amount that will be recovered in the case of default. In this simple case we can calculate the price of bond by incorporating the probability of default as follows. This approach assumes risk neutrality. However, there is another approach does not uses the risk neutrality approach and adjusts the probability of default through additional risk premium. Credit Spread = (Probability of default) x (1 – Recovery Rate) This spread can be used to determine the required yield and will determine the price of the bond. References Elton, E. J. and Gruber, M. J. 1995, Modern Portfolio Theory and Investment Analysis. 5th ed. New York: John Wiley & Sons Stanton, E. 2009. Practical Uses of Duration and Convexity for Bond Buyers. [online] Available at: < http://www.thestreet.com/story/751721/practical-uses-of-duration-and-convexity- for-bond-buyers.html> Read More