Investment Analysis

38. You buy 300 shares of Qualitycorp for $30 per share and deposit initial margin of 50%. The next day Qualitycorp's price drops to $25 per share. What is your actual margin? A) 50% B) 40% C) 33% D) 60% E) 25% Answer: B Difficulty: Moderate Rationale: AM = [300 ($25) - .5 (300) ($30)] / [300 ($25)] = .40

30. Assume that you purchased 200 shares of Super Performing mutual fund at a net asset value of $21 per share. During the year you received dividend income distributions of $1.50 per share and capital gains distributions of $2.85 per share. At the end of the year the shares had a net asset value of $23 per share. What was your rate of return on this investment? A) 30.24% B) 25.37% C) 27.19% D) 22.44% E) 29.18% Answer: A Difficulty: Moderate Rationale: R = ($23-21+1.5+2.85)/$21 = 30.238% 31. Assume that you purchased shares of High Flying mutual fund at a net asset value of $12.50 per share. During the year you received dividend income distributions of $0.78 per share and capital gains distributions of $1.67 per share. At the end of the year the shares had a net asset value of $13.87 per share. What was your rate of return on this investment? A) 29.43% B) 30.56% C) 31.19% D) 32.44% E) 29.18% Answer: B Difficulty: Moderate Rationale: R = ($13.87-12.50+0.78+1.67)/$12.50 = 30.56% 32. Assume that you purchased shares of a mutual fund at a net asset value of $14.50 per share. During the year you received dividend income distributions of $0.27 per share and capital gains distributions of $0.65 per share. At the end of the year the shares had a net asset value of $13.74 per share. What was your rate of return on this investment? A) 2.91% B) 3.07% C) 1.10% D) 1.78% E) -1.18% Answer: C Difficulty: Moderate Rationale: R = ($13.74-14.50+0.27+0.65)/$14.50 = 1.103%

15. You want to evaluate three mutual funds using the Sharpe measure for performance evaluation. The risk-free return during the sample period is 6%. The average returns, standard deviations and betas for the three funds are given below, as is the data for the S&P 500 index.

The fund with the highest Sharpe measure is __________.
A) Fund A
B) Fund B
C) Fund C
D) Funds A and B are tied for highest
E) Funds A and C are tied for highest

Answer: C Difficulty: Moderate
Rationale: A: (24% - 6%)/30% = 0.60; B: (12% - 6%)/10% = 0.60; C: (22% - 6%)/20% = 0.80; S&P 500: (18% - 6%)/16% = 0.75.

SHARPE, TREYNOR AND JENSEN'S RATIOS
SHARPE RATIO
This ratio measures the return earned in excess of the risk free rate (normally Treasury instruments) on a portfolio to the portfolio's total risk as measured by the standard deviation in its returns over the measurement period. Or how much better did you do for the risk assumed.

S = Return portfolio- Return of Risk free investment
Standard Deviation of Portfolio

Example: Let's assume that we look at a one year period of time where an index fund returned 11%
Treasury bills earned 6%
The standard deviation of the index fund was 20%

Therefore S = 11-6/.20 = 25

The Sharpe ratio is an appropriate measure of performance for an overall portfolio particularly when it is compared to another portfolio, or another index such as the S&P 500, Small Cap index, etc.

That said however, it is not often provided in most rating services.

TREYNOR RATIO
This ratio is similar to the above except it uses beta instead of standard deviation. It's also known as the Reward to Volatility Ratio, it is the ratio of a fund's average excess return to the fund's beta. It measures the returns earned in excess of those that could have been earned on a riskless investment per unit of market risk assumed.

T = Return of Portfolio - Return of Risk Free Investment
Beta of Portfolio

The absolute risk adjusted return is the Treynor plus the risk free rate.

Assume two portfolios A B

Return 12 14
Beta .7 1.2
Risk Free Rate= 9%
Ta=
.12 - .09 = .043 Risk adjusted rate of return of Portfolio A = .043+ .09 = .12 = 13.3%
.07
Tb=
.14 - .09 = 0.04 Risk adjusted rate of return of Portfolio B = 0.04 + .09 = .13 = 13%
1.2

For many investors, without any analysis of risk, if you ask them what is the better number (12% or 14%) almost universally they state 14%. (I did this with about 1,000 HP employees who owned considerable sums of mutual funds in 401(k) plans). However, when you point out the risk adjusted rate of return, many adjust their thinking.
The example I used was for 1990 - 1993 (roughly) where Fidelity Magellan had earned about 18%. Many bond funds had earned 13 %. Which is better? In absolute numbers, 18% beats 13%. But if I then state that the bond funds had about half the market risk, now which is better? You don't even need to do the formula for that analysis. But that is missing in almost all reviews by all brokers. For clarification- I do not suggest they put all the money into either one- just that they need to be aware of the implications. It is information, not advice per se. But if you give really good information, the advice is implied.

JENSEN'S ALPHA
This is the difference between a fund's actual return and those that could have been made on a benchmark portfolio with the same risk- i.e. beta. It measures the ability of active management to increase returns above those that are purely a reward for bearing market risk. Caveats apply however since it will only produce meaningful results if it is used to compare two portfolios which have similar betas.
Assume Two Portfolios | A | B | Market Return | Return | 12 | 14 | 12 | Beta | .7 | 1.2 | 1.0 |
Risk Free Rate +9%
The return expected= Risk Free Return + Beta portfolio (Return of Market - Risk Free Return)
Using Portfolio A, the expected return = .09 + .7 (.12 - .09) = .09 + .02 = .11
Alpha = Return of Portfolio- Expected Return= .12 - .11 = .01 = 1%
As long as "apples are compare to apples"- in other words a computer sector fund A to computer sector fund b- I think it is a viable number. But if taken out of context, it loses meaning. Alphas are found in many rating services but are not always developed the same way- so you can't compare an alpha from one service to another. However I have usually found that their relative position in the particular rating service to be viable. Short term alphas are not valid. Minimum time frames are one year- three year are more preferable.

Measure Your Portfolio's Performance
December 16 2012| Filed Under » Growth Investing, Portfolio Management, Risk Management
Many investors mistakenly base the success of their portfolios on returns alone. Few consider the risk that they took to achieve those returns. Since the 1960s, investors have known how to quantify and measure risk with the variability of returns, but no single measure actually looked at both risk and return together. Today, we have three sets of performance measurement tools to assist us with our portfolio evaluations. The Treynor, Sharpe and Jensen ratios combine risk and return performance into a single value, but each is slightly different. Which one is best for you? Why should you care? Let's find out.

Treynor Measure
Jack L. Treynor was the first to provide investors with a composite measure of portfolio performance that also included risk. Treynor's objective was to find a performance measure that could apply to all investors, regardless of their personal risk preferences. He suggested that there were really two components of risk: the risk produced by fluctuations in the market and the risk arising from the fluctuations of individual securities.

Treynor introduced the concept of the security market line, which defines the relationship between portfolio returns and market rates of returns, whereby the slope of the line measures the relative volatility between the portfolio and the market (as represented by beta). The beta coefficient is simply the volatility measure of a stock portfolio to the market itself. The greater the line's slope, the better the risk-return tradeoff.

The Treynor measure, also known as the reward to volatility ratio, can be easily defined as: (Portfolio Return – Risk-Free Rate) / Beta |

The numerator identifies the risk premium and the denominator corresponds with the risk of the portfolio. The resulting value represents the portfolio's return per unit risk.

To better understand how this works, suppose that the 10-year annual return for the S&P 500 (market portfolio) is 10%, while the average annual return on Treasury bills (a good proxy for the risk-free rate) is 5%. Then assume you are evaluating three distinct portfolio managers with the following 10-year results: Managers | Average Annual Return | Beta | Manager A | 10% | 0.90 | Manager B | 14% | 1.03 | Manager C | 15% | 1.20 |

Now, you can compute the Treynor value for each:

T(market) = (.10-.05)/1 = .05

T(manager A) = (.10-.05)/0.90 = .056

T(manager B) = (.14-.05)/1.03 = .087

T(manager C) = (.15-.05)/1.20 = .083

The higher the Treynor measure, the better the portfolio. If you had been evaluating the portfolio manager (or portfolio) on performance alone, you may have inadvertently identified manager C as having yielded the best results. However, when considering the risks that each manager took to attain their respective returns, Manager B demonstrated the better outcome. In this case, all three managers performed better than the aggregate market.

Because this measure only uses systematic risk, it assumes that the investor already has an adequately diversified portfolio and, therefore, unsystematic risk (also known as diversifiable risk) is not considered. As a result, this performance measure should really only be used by investors who hold diversified portfolios.

Sharpe Ratio
The Sharpe ratio is almost identical to the Treynor measure, except that the risk measure is the standard deviation of the portfolio instead of considering only the systematic risk, as represented by beta. Conceived by Bill Sharpe, this measure closely follows his work on the capital asset pricing model (CAPM) and by extension uses total risk to compare portfolios to the capital market line.

The Sharpe ratio can be easily defined as: (Portfolio Return – Risk-Free Rate) / Standard Deviation |

Using the Treynor example from above, and assuming that the S&P 500 had a standard deviation of 18% over a 10-year period, let's determine the Sharpe ratios for the following portfolio managers: Manager | Annual Return | Portfolio Standard Deviation | Manager X | 14% | 0.11 | Manager Y | 17% | 0.20 | Manager Z | 19% | 0.27 |

S(market) = (.10-.05)/.18 = .278
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S(manager X) = (.14-.05)/.11 = .818

S(manager Y) = (.17-.05)/.20 = .600

S(manager Z) = (.19-.05)/.27 = .519

Once again, we find that the best portfolio is not necessarily the one with the highest return. Instead, it's the one with the most superior risk-adjusted return, or in this case the fund headed by manager X.

Unlike the Treynor measure, the Sharpe ratio evaluates the portfolio manager on the basis of both rate of return and diversification (as it considers total portfolio risk as measured by standard deviation in its denominator). Therefore, the Sharpe ratio is more appropriate for well diversified portfolios, because it more accurately takes into account the risks of the portfolio.

Jensen Measure
Like the previous performance measures discussed, the Jensen measure is also based on CAPM. Named after its creator, Michael C. Jensen, the Jensen measure calculates the excess return that a portfolio generates over its expected return. This measure of return is also known as alpha.

The Jensen ratio measures how much of the portfolio's rate of return is attributable to the manager's ability to deliver above-average returns, adjusted for market risk. The higher the ratio, the better the risk-adjusted returns. A portfolio with a consistently positive excess return will have a positive alpha, while a portfolio with a consistently negative excess return will have a negative alpha

The formula is broken down as follows: Jensen\'s Alpha = Portfolio Return – Benchmark Portfolio Return | Where: Benchmark Return (CAPM) = Risk-Free Rate of Return + Beta (Return of Market – Risk-Free Rate of Return) |

So, if we once again assume a risk-free rate of 5% and a market return of 10%, what is the alpha for the following funds? Manager | Average Annual Return | Beta | Manager D | 11% | 0.90 | Manager E | 15% | 1.10 | Manager F | 15% | 1.20 |

First, we calculate the portfolio's expected return:

ER(D)= .05 + 0.90 (.10-.05) = .0950 or 9.5% return

ER(E)= .05 + 1.10 (.10-.05) = .1050 or 10.50% return

ER(F)= .05 + 1.20 (.10-.05) = .1100 or 11% return

Then, we calculate the portfolio's alpha by subtracting the expected return of the portfolio from the actual return:

Alpha D = 11%- 9.5% = 1.5%

Alpha E = 15%- 10.5% = 4.5%

Alpha F = 15%- 11% = 4.0%

Which manager did best? Manager E did best because, although manager F had the same annual return, it was expected that manager E would yield a lower return because the portfolio's beta was significantly lower than that of portfolio F.

Of course, both rate of return and risk for securities (or portfolios) will vary by time period. The Jensen measure requires the use of a different risk-free rate of return for each time interval considered. So, let's say you wanted to evaluate the performance of a fund manager for a five-year period using annual intervals; you would have to also examine the fund's annual returns minus the risk-free return for each year and relate it to the annual return on the market portfolio, minus the same risk-free rate. Conversely, the Treynor and Sharpe ratios examine average returns for the total period under consideration for all variables in the formula (the portfolio, market and risk-free asset). Like the Treynor measure, however, Jensen's alpha calculates risk premiums in terms of beta (systematic, undiversifiable risk) and therefore assumes the portfolio is already adequately diversified. As a result, this ratio is best applied with diversified portfolios, like mutual funds.

The Bottom Line
Portfolio performance measures should be a key aspect of the investment decision process. These tools provide the necessary information for investors to assess how effectively their money has been invested (or may be invested). Remember, portfolio returns are only part of the story. Without evaluating risk-adjusted returns, an investor cannot possibly see the whole investment picture, which may inadvertently lead to clouded investment decisions.