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- 1. King’s College ACS 310a Midterm Test 2 1.5 hours Calculator Allowed Instruction: Make your answer as brief and to-the-point as possible. Draw graphs and give formulas whenever applicable. A short-hand form answer is acceptable. Broad and vague answers without any reference to theories will not get a score. Part A(30%) : Evaluate the following statement: First, tell whether the given statement is True or False. And then, explain why the statement is true or false by using the formal theories along with relevant formula(s), graph(s) or words. Your marks depend on the rigorousness and quality of your explanation: Ether draw a graph or give a formula. And then give a few lines of explanation: at the maximum, no more than a half page should be written for each answer. 1. “According to the Dividend-Discount Model for the stock (price) valuation, when interest rates go up, the price of stock should go up because in general the rate of return of equities should also go up as well.” False: P = D1/(rcs-g). Thus, when the interest or ‘rcs’ goes up, and g is constant, the (rcs –g) goes up, so, the price of a stock or P falls. 2. “If you have the investment portfolio with a certain level of return and risk, you should add an asset only with a higher rate of expected return, and a lower level of risk. It does not make a sense to newly add (to your existing portfolio) an asset with a lower rate of expected return and a higher value of risk or standard deviation.” False or Uncertain: It depends on the correlation between your present portfolio and the new asset that you are about to add: If the two assets are perfectly positively correlated, then it does not make a sense as adding the new asset to the portfolio will lower the expected rate of return and raise the risk or the standard deviation.
- 2. Page 2 of 6 0100090000037c00000002001c00000000000400000003010800050000000b0200000000050 000000c023e07b00d040000002e0118001c000000fb021000070000000000bc020000008601 02022253797374656d0007b00d000080c8110072edc630605916000c020000b00d00000400 00002d0100001c000000fb02a8ff0000000000009001000000000440001254696d6573204e6 57720526f6d616e0000000000000000000000000000000000040000002d0101000400000002 010100040000002d010100050000000902000000020d000000320a60000000010004000000 0000ac0d3a0720244b00040000002d010000030000000000 However, if the two assets are not perfectly positively correlated or imperfectly correlated, then the resultant portfolios would have the locus of a hooked curve. The upper part of the hooked curve is Efficient Frontier. As you are moving from your current portfolio to the mixture with an increasing amount of the new asset, the rate of return may fall and so does the risk. Thus the portfolios on the upper part of the hooked curve are all comparable. This is an expanded opportunity set. 0100090000037c00000002001c00000000000400000003010800050000000b0200000000050 000000c023e07b00d040000002e0118001c000000fb021000070000000000bc020000008601 02022253797374656d0007b00d000080c8110072edc630605916000c020000b00d000004000 0002d0100001c000000fb02a8ff0000000000009001000000000440001254696d6573204e657 720526f6d616e0000000000000000000000000000000000040000002d010100040000000201 0100040000002d010100050000000902000000020d000000320a60000000010004000000000 0ac0d3a0720244b00040000002d010000030000000000 3. “I am extremely risk-averse (hating any risk) when it comes to financial investment. Thus, I should not invest on anything other than so-called `risk free assets’ such as bank deposits or government T-bills. I should not invest any amount on risky assets or their combinations.” False. Theoretically, you can mix two risky assets to approximate a risk free asset as long as the two assets have the correlation coefficient equal to -1. The resultant portfolio might have zero standard deviation and still a higher rate of return than the risk free asset has. 0100090000037c00000002001c00000000000400000003010800050000000b0200000000050000 000c023e07b00d040000002e0118001c000000fb021000070000000000bc020000008601020222 53797374656d0007b00d000080c8110072edc630605916000c020000b00d0000040000002d0100 001c000000fb02a8ff0000000000009001000000000440001254696d6573204e657720526f6d616 e0000000000000000000000000000000000040000002d0101000400000002010100040000002d0 10100050000000902000000020d000000320a600000000100040000000000ac0d3a0720244b000 40000002d010000030000000000
- 3. Page 3 of 6 Part B: 4. (20%) Consider two stocks, A and B. Stock A is currently price at $10 per share, and stock B at $25 per share. For each, there are four possibilities that may take place next year, depending on how the economy performs along business cycles: Economic State Probability Price of A Price of B Boom 25% $13 $27.50 Above average 25% 11.5 27.50 Below average 25% 10.50 27 Bust(Recession) 25% 9 25 1) What is the expected annual return and the risk for asset A and B resepctively? Percentage: Expected Return(%)= (Ending Saling price – Purchasing price) / purchasing price * 100% Economic State Probability Price of A E(RA) (%) Price of B E(RB) (%) Boom 25% $13 30 $27.50 10 Above average 25% 11.5 15 27.50 10 Below average 25% 10.50 5 27 8 Bust(Recession) 25% 9 -10 25 0 E(R) = ΣRipri E(RA) = 0.3*0.25 + 0.15*0.25+0.05*0.25+ (-0.1)*0.25 =0.1=10% E(RB) = 0.1*0.25 + 0.1*0.25 +0.08*0.25 + 0*0.25 =0.07=7% σ= {Σ[Ri – E(R)]2pri }1/2 σA = {[(0.3-0.1)2*0.25]+ [(0.15-0.1)2*0.25] + [(0.05-0.1)2*0.25] + [(-0.1-0.1)2*0.25]}1/2 = (0.01 + 0.000625 +0.000625 +0.01)1/2 =(0.02125) 1/2 = 0.1458 σB = {[(0.1-0.07)2*0.25] + [(0.1-0.07)2*0.25] + [(0.08-0.07)2*0.25] + [(0-0.07)2*0.25]} 1/2 = (0.000225 + 0.000225 + 0.000025 + 0.001225) 1/2 =(0.0017 ) 1/2= 0.041231 2) If an investor mixes A and B at the ratio of 6 to 4, what would be the expected return and the risk of the resultant portfolio?
- 4. Page 4 of 6 E( RP) =ΣwiE(Ri) = [6 /(6+4)] *0.1 + [4 /(6+4)]*0.07 = 0.088=8.8% σAB = {Σ[RA,i – E(RA)] [RB,i – E(RB)] pri } = [(0.3-0.1)(0.1-0.07)*0.25 ]+ [(0.15-0.1) (0.1-0.07)*0.25]+ [(0.05 – 0.1)(0.08-0.07)*0.25] + [(-0.1-0.1)(0-0.07)*0.25] = 0.0015 + 0.000375 - 0.000125 + 0.0035 =0.00525 σP = [(wAσA)2+ (wBσB)2+2wAwBσAB]1/2 = [(0.6*0.1458)2 + (0.4*0.041231)2 +2*0.4*0.6*0.00525]1/2 = (0.00765+ 0.000272+0.00252) 1/2 =0.10219 5. (20%) Suppose that the correlation coefficient between the stock of company called ‘Inert Technologies Ltd’ and the overall stock market (portfolio) index is 0.30. The rate of return on a risk free asset , such as 30-day T-Bill, is 8%. Overall, the average rate of return on entire stocks is 9% higher than the expected rate of return on risk free asset. The standard deviation of the stock market index is 0.25, and the standard deviation of the returns on the Inert Technologies Ltd stock is 0.35. From the statement, we can get: ρ AB = 0.30, RF =8%, σ Μ = 0.25, σ I =0.35, (Rm -RF) =9% 1) Give the formula for the beta, which shows the beta as a function of the relevant covaraince and variance. Calcualte the numerical value for this stock. β = coverariance of Inert and the stock market index / (stock market index’s variance)= σ Ι,Μ / σ Μ = (ρ AB *σ i *σ M )/ σ Μ 2 = (ρ AB *σ A )/ σ Μ β Ι = (0.35*0.3) / 0.25 =0.42 2) What should be the expected rate of return for this “Inert Techonologies Ltc” stock according to the Capital Asset Pricing Model? Give the formula, and then calculate the ‘fair’ expected rate of return including the risk premium. E(Ri ) = RF + β Ι ( Rm -RF ) Required Rate = 0.08 + 0.42 (0.09) = 0.1178 3) What is the advantage of the beta in general (from the Capital Asset Pricing Model), compared with the standard deviation (from the Mean-Variance Approach), in terms of the accuracy of the risk premium for the financial asset? Explain it briefly by using the numbers given here in this question. A few setences should be enough. By the Mean-Variance Approach to analyse: The Mean of risk in the market is σ Μ =0.25, and σ I= 0.35, so Variance between them is 40% (= (0.35-0.25)/ 0.25 * 100%). However, the mean of risk premium is 9% in the market, but the risk premium of the company is 0.0378 (0.42*0.09=0.0378), so, the Variance of risk premium between them is – 58% (( 0.0378 -0.09) /0.09 *100% = -58%).
- 5. Page 5 of 6 However, the beta could give the systematic risk to us: because β Ι =0.42 is 58% smaller than β Μ , so , the risk premium of the company is 58% smaller than the risk premium of the overall market. In other words, the risk premium is not proportional to the total market risk given by standard deviation, but to the systematic risk given by the beta value. 4) Suppose that one year later, the actual rate of return of the above stock turns out to be a lower than the expected rate of return from the above question # 2). What does this mean? Does this mean that the Capital Asset Pricing Model is wrong? Two to three setences should be enough. No, CAPM is not wrong. The CAPM is a formula for the expected rate of return. Here we are looking at the actual or realized rate of return. In a word, actual ROR = expected ROR + random factor(unsystematic error). It simply means that there was a negative unexpected (random) factor, which pulled down the rate of return below the expected or fair rate of return. 6. (10%) Answer the following questions: If you invest on stock of company ABC, you will start getting the annual dividend payment one year later. The first (annual) dividend will be $1, and is expected to increase at 3% per annum thereafter. The interest rate in the financial market is 5%. The actual stock price today is $55 in the financial market. 1) Calculate the ‘fair price’ of the sock according to the dividend discount model. P = D/(r-g) 1/(0.05-0.03) = 50 dollars 2) What would be the recommended investment strategy by the `market fundamentalist’ for this stock, given the actual market price of the stock? This stock is over-valued: the actual price is higher than what the price should be according to the dividend discount model. Thus, you must (short) sell the stock as the price of 55 dollars will fall to 50 dollars in the future. What would be the evaluation by the economist who is in line with ‘rational expectations’ theory? A few sentences for each group should be enough. There is no mispriced asset in the financial market. The difference 5 dollars reflect some information that the above dividend discount model does not capture/include. 7. (10%) Answer the following questions: 1) What does Tobin’s Separation Theorem mean for the selection of risky assets in portfolio? A few sentences should be enough. Whatever diverse risk preferences investors might have, the optimal combination of risky assets is the same for all the investors.
- 6. Page 6 of 6 2) What is the (bond) rating for a Junk Bond? (for example, the highest grade bonds have either Aaa, or AAA ratings) The junk bonds are the bond rated below BBB by Standard and Poor's and Baa by Moody's. 8. (15%) Suppose that as a financial planner, you are coming up with the best portfolio from the combination of three assets, two of which are risky and one of which is risk-free, for your client who has a certain risk preference. How would you do it? Describe the procedure of coming up with the best possible variety of investment portfolios by mixing the given assets, and making a choice of the best portfolio for your client. Give appropriate graphs (efficient frontier(s) and indifference curve must be drawn), and short and to-the-point verbal explanation. About 2 pages of answer, including graphs, should be enough. Make sure that a student has the following points in graphs or in words: 1) Portfolio Diversification of Two Risky Assets, and their Efficient Frontier (Envelope Curve); 2) Portfolio Diversification of Risk Free Asset and the result of above 1), and their Efficient Frontier (Straight line); 3) Indifference Curve; 4) The Optimal Choice of Portfolio

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