ECON366 - KONSTANTINOS KANELLOPOULOS
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INSTRUCTOR: Mr. Konstantinos Kanellopoulos, MSc (L.S.E.), M.B.A.
COURSE: MBA-680-50-SUII12 Corporate Financial Theory
SEMESTER: Summer Session II
Risk and Return
10th August 2012
CASE STUDY I ON RISK AND RETURN
Percival Hygiene has $10 million invested in long-term corporate bonds. This bond portfolio’s expected annual rate of return is 9%, and the annual standard deviation is 10%. Amanda Reckonwith, Percival’s Financial adviser, recommends that Percival consider investing in an index fund that closely tracks the Standard and Poor’s 500 Index. The Index has an expected return of 14%, and its standard deviation is 16%.
a. Suppose Percival puts all his money in a combination of the index fund and Treasury bills. Can he thereby improve his expected rate of return without changing the risk of his portfolio? The Treasury bill yield is 6%.
b. Could Percival do even better by investing equal amounts in the corporate bond portfolio and the index fund? The correlation between the bond portfolio and the index fund is +.1.
a. Percival’s current portfolio provides an expected return of 9% with an annual standard deviation of 10%. First we find the portfolio weights for a combination of Treasury bills (security 1: standard deviation = 0%) and the index fund (security 2: standard deviation = 16%) such that portfolio standard deviation is 10%. In general, for a two security portfolio:
(P2 = x12(12 + 2x1x2(1(2(12 + x22(22
(0.10)2 = 0 + 0 + x22(0.16)2
x2 = 0.625 ( x1 = 0.375
rp = x1r1 + x2r2
rp = (0.375 ( 0.06) + (0.625 ( 0.14) = 0.11 = 11.0%
Therefore, he can improve his expected rate of return without changing the risk of his portfolio.
b. With equal amounts in the corporate bond portfolio (security 1) and the index fund (security 2), the expected return is:
rp = x1r1 + x2r2
rp = (0.5 ( 0.09) + (0.5 ( 0.14) = 0.115 = 11.5%
(P2 = x12(12 + 2x1x2(1(2(12 + x22(22
(P2 = (0.5)2(0.10)2 + 2(0.5)(0.5)(0.10)(0.16)(0.10) + (0.5)2(0.16)2
(P2 = 0.0097
(P = 0.985 = 9.85%
Therefore, he can do even better by investing equal amounts in the corporate bond portfolio and the index fund. His expected return increases to 11.5% and the standard deviation of his portfolio decreases to 9.85%.
CASE STUDY II ON RISK AND RETURN
John and Marsha hold hands in a cozy French restaurant in downtown Manhattan. Marsha is a future-market trader. John manages a $125 million common-stock portfolio for a large pension fund. They have just ordered tournedos financiere for the main course and flan financiere for desert. John reads the financial pages of the Wall Street Journal by candlelight.
John: Wow! Potato futures hit their daily limit. Let’s add an order of gratin Dauphinoise. Did you manage to hedge the forward interest rate on that euro loan?
Marsha: John, please fold up that paper (He does so reluctantly). John, I love you. Will you marry me?
John: Oh, Marsha, I love you too, but … there’s something you must know about me-something I’ve never told anyone.
Marsha (concerned): John, what is it?
John: I think I’m a closet indexer.
Marsha: What? Why?
John: My portfolio returns always seem to track the S&P 500 market index. Sometimes I do a little better, occasionally a little worse. But the correlation between my returns and the market returns is over 90%.
Marsha: What’s wrong with that? Your client wants a diversified portfolio of large-cap stocks. Of course your portfolio will follow the market.
John: Why doesn’t my client just buy an index fund? Why are they paying me? Am I really adding value by active management? I try, but I guess I’m just an … indexer.
Marsha: Oh, John, I know you’re adding value. You were a star security analyst.
John: It’s not easy to find stocks that are truly over- or undervalued. I have firm opinions about a few, of course
Marsha: You were explaining why Pioneer Gypsum is a good buy. And you’re bullish on Global Mining.
John: Right, Pioneer. (Pulls handwritten notes from his coat pocket.). Stock price $87.50. I estimate the expected return as 11% with an annual standard deviation of 32%.
Marsha: Only 11? You’re forecasting a market return of 12.5%
John: Yes, I’m using a market risk premium of 7.5% and the risk-free interest rate is about 5%. That gives 12.5%. But Pioneer’s beta is only 0.65. I was going to buy 30,000 shares this morning, but I lost my nerve. I’ve got to stay diversified.
Marsha: Have you tried modern portfolio theory?
John: MPT? Not practical. Looks great in textbooks, where they show efficient frontiers with 5 or 10 stocks. But I choose from hundreds, maybe thousands, of stocks. Where do I get the inputs for 1,000 stocks? That’s a million variances and co variances!
Marsha: Actually only about 500,000, dear. The co variances above the diagonal are the same as the co variances below. But you are right, most of the estimates would be out-of-date or just garbage.
John: To say nothing about the expected returns: Garbage in, garbage out.
Marsha: But John, you don’t need to solve for 1,000 portfolio weights. You only need a handful. Here’s the trick: Take your benchmark, the S&P 500, as security 1. That’s what you would end up with as an indexer. Then consider a few securities you really know something about. Pioneer could be security 2, for example. Global, security 3. And so on. Then you could put your wonderful financial mind to work.
John: I get it. Active management means selling off some of the benchmark portfolio and investing the proceeds in specific stocks like Pioneer. But how do I decide whether Pioneer really improves the portfolio? Even if it does, how mush should I buy?
Marsha: Just maximize the Sharpe ratio, dear.
John: I’ve got it! The answer is yes!
Marsha: What’s the question?
John: You asked me to marry you. The answer is yes. Where should we go on our honeymoon?
Marsha: How about Australia? I’d love to visit the Melbourne Stock Exchange.
• The following table reproduces John’s notes on Pioneer Gypsum and Global Mining. Calculate the expected return, risk premium, and standard deviation of a portfolio invested partly in the market and partly in Pioneer. (You can calculate the necessary inputs from the beats and standard deviations given in the table). Does adding Pioneer to the market benchmark improve the Sharpe ratio? How mush should Jon invest in Pioneer and how much in the market?
| |Pioneer Gypsum |Global Mining |
|Expected return |11.0% |12.9% |
|Standard deviation |32% |20% |
|Beta |0.65 |1.22 |
|Stock price |$87.50 |$105.00 |
• Repeat the analysis for Global Mining. What should John do in this case? Assume that Global accounts for .75 of the S&P index.
John neglected to mention the standard deviation of the S&P 500. We will assume 16%. Recall that stock i’s beta is just the ratio of its covariance with the market (σim) to the market variance σm2, where σm2 = .162 = .0256. For Pioneer Gypsum, β = .65 = σim/.0256, which gives a covariance of σim = .01664. The covariance also equals the
correlation coefficient ρ times the product of the stock’s and market’s standard deviations σi and σm. For Pioneer, σim = ρσiσm = .01664 = ρ×.32×.16, which implies ρ = .325.
Here is the 2×2 covariance matrix for the market and Pioneer.
Now calculate the portfolio return rP, portfolio standard deviation σP and the Sharpe ratio for different fractions invested in the market and Pioneer. For example, suppose that the market gets 99% of investment and Pioneer 1%.
rP = .99×.125 + .01×.11 = .12485
σP2 = .992×.0256 + 2×.99×.01×.01664 + .012×.1024 = .0254
σP = √.0254 = .1595
Sharpe ratio = (rP – rf)/σP = (.12485 - .05)/.1595 = .4694
It turns out that the Sharpe ratio is maximized by putting about 95% in the market and 5% in Pioneer.
|S&P 500 |Pioneer |Sharpe ratio |
|1.0 |0 |.4688 |
|.99 |.01 |.4694 |
|.98 |.02 |.4698 |
|.97 |.03 |.4701 |
|.96 |.04 |.4702 |
|.95 |.05 |.4702 |
|.94 |.06 |.4699 |
We can follow the same procedures for Global Mining. Global’s covariance is .03123 and its correlation with the market is extremely high at .976. (Perhaps John has underestimated Global’s standard deviation and thus overestimated its correlation with the market.) The 2×2 covariance matrix is:
Global’s return is not attractive: with a beta of 1.22, it should offer an expected rate of return of .05 + 1.22×.075 = .1415, over 14%. But John’s estimate is only 12.9%. Therefore he should sell Global. In fact he should eliminate it from his portfolio.
Suppose that John’s starting portfolio matches the market and includes 0.75% in Global. Then he should sell all the Global shares and put the proceeds back into the overall market. The resulting portfolio weights are 100.75% in the market and -0.75% in Global. That is, the portfolio should “underweight” Global by -0.75% in order to reduce holdings of Global to zero. The underweight increases the Sharpe ratio from .4688 to .4693:
|S&P 500 |Global |Sharpe ratio |
|.99 |.01 |.4680 |
|1.0 | 0 |.4688 |
|1.005 |-.005 |.4691 |
| 1.0075 |-.0075 |.4693 |
The Sharpe ratio gets still better if the portfolio weight for Global is reduced below -.75%. A weight below -.75% means selling short. In order to sell short, John would have to borrow Global shares, sell them (with an obligation to repurchase and return the shares later) and invest the sale proceeds in the market. But we doubt that John is allowed to sell short from the portfolio he manages.
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