| January 30, 2017

FINA312 / FINA517 – Semester B 2015



To be carried out based on closing prices of stocks on Friday, October 02, 2015

My heartiest congratulations for reaching the final stage of the FMSP. We have learnt various aspects of fund management in the real world perspective. It ranges from the setting-up of the funds, determining asset allocation, stock selection, and portfolio rebalancing. We also learnt how to deal with redemption and inflow of new fresh funds, which of course, are the day-to-day challenges of fund managers in the real world.

In this stage, the FMSP concludes with portfolio performance evaluation process. Any investment or investment-related textbook will deemed as incomplete if it does not have a chapter on portfolio performance evaluation, and most textbooks will save this chapter as the last concluding chapter. It is easy to understand given the merit and importance of learning the right techniques and ways of evaluating the performance of investment portfolios.

As such, for the passive fund, you are required to do the followings:-

To measure the tracking error of the passive fund with the selected index as benchmark.

Your passive fund has an objective to track a selected index. At the setting up of the fund, you have chosen a stock market index to act as the benchmark (for example NZX50) of your passive fund. Tracking error is a measurement of how much the return on a portfolio deviates from the return on its benchmark index. Tracking erroris a measure of how closely a portfolio follows the index to which it is benchmarked.Tracking error represents a divergence between the price behavior of a position or a portfolio and the price behavior of a benchmark. This is often in the context of a hedge or mutual fund that did not work as effectively as intended, creating an unexpected profit or loss instead.

Many portfolios are managed to a benchmark, typically an index. Some portfolios are expected to replicate, before trading and other costs, the returns of an index exactly (e.g., anindex fund), while others are expected to ‘actively manage’ the portfolio by deviating slightly from the index in order to generateactive returns. Tracking error (also calledactive risk) is a measure of the deviation from the benchmark; the aforementioned index fund would have a tracking error close to zero, while an actively managed portfolio would normally have a higher tracking error. Dividing portfolio active return by portfolio tracking error gives theinformation ratio, which is a risk adjusted performance measure.

If tracking error is measured historically, it is called ‘realized’ or ‘ex post’ tracking error. If a model is used to predict tracking error, it is called ‘ex ante’ tracking error. Ex-post tracking error is more useful for reporting performance, whereas ex-ante tracking error is generally used by portfolio managers to control risk. Various types of ex-ante tracking error models exist, from simple equity models which use beta as a primary determinant to more complicated multi-factor fixed income models. In a factor model of a portfolio, the non-systematic risk (i.e., the standard deviation of the residuals) is called “tracking error” in the investment field.

Tracking error is the difference between the return on a portfolio or fund, and the benchmark it is expected to mirror (or track). There are two methods to calculate the tracking error. The first is the easiest. Simply subtract the fund’s return from the return of the index it is supposed to track. For instance, a mutual fund that is pegged to the S&P 500 had a 7% return for the year, whereas the S&P had an 8% return. The tracking error is 1%.

The second method to calculate the tracking error is more complicated, but more informative. This calculation involves taking the standard deviation of the difference in the fund’s and index’s returns over time. The formula is:

Standard deviation of tracking error = 1/(n – 1) ?(xi – yi)2

Where n is equal to the number of periods, x equals the fund’s return for each given period and y equals the benchmark’s return for each period.

By using the standard deviation calculation, investors get a better idea of how the fund will perform compared to the benchmark over time. A low standard deviation means the fund tracks the benchmark fairly closely. A higher standard deviation means the fund does not track its benchmark very well. These figures indicate how well a fund is managed. Investors seeking a fund that accurately tracks their preferred index should look for funds with low tracking errors. For FMSP Report 05, please use the second method to calculate the tracking error of your passive fund.

For the active fund, you are required to do the followings:-

To evaluate the performance of the active fund using the four (4) known techniques as below:

Sharpe Ratio
The Sharpe ratio or Sharpe index or Sharpe measure or reward-to-variability ratio is a measure of the excess return (or Risk Premium) per unit of risk in an investment asset or a trading strategy. Since its revision made by the original author in 1966, it is defined as:

.gif” alt=”S = \frac{r-r_f}{\sigma} =”>

whereR is the asset return,Rf is the return on a benchmark asset, such as the risk free rate of return,E[R ? Rf] is the expected value of the excess of the asset return over the benchmark return, and? is the standard deviation of the asset excess return.

The Sharpe ratio is used to characterize how well the return of an asset compensates the investor for the risk taken. When comparing two assets each with the expected returnE[R] against the same benchmark with returnRf, the asset with the higher Sharpe ratio gives more return for the same risk. Investors are often advised to pick investments with high Sharpe ratios.

Treynor Ratio
The Treynor ratio is a measurement of the returns earned in excess of that which could have been earned on a riskless investment (i.e. Treasury Bill) (per each unit of market risk assumed). The Treynor ratio (sometimes called reward-to-volatility ratio) relates excess return over the risk-free rate to the additional risk taken; however systematic risk instead of total risk is used. The higher the Treynor ratio, the better the performance under analysis.

.gif” alt=”T = \frac{r_i – r_f}{\beta_i} “>


.gif” alt=”T \equiv”>Treynor ratio,

.gif” alt=”r_i \equiv “>portfolio i’s return,

.gif” alt=”r_f \equiv “>risk free rate

.gif” alt=”\beta_i \equiv “>portfolio i’s beta

Like the Sharpe ratio, the Treynor ratio (T) does not quantify the value added, if any, of active portfolio management. It is a ranking criterion only. A ranking of portfolios based on the Treynor Ratio is only useful if the portfolios under consideration are sub-portfolios of a broader, fully diversified portfolio. If this is not the case, portfolios with identical systematic risk, but different total risk, will be rated the same. But the portfolio with a higher total risk is less diversified and therefore has a higher unsystematic risk which is not priced in the market.

Jensen’s Alpha
An alternative method of ranking portfolio management is Jensen’s alpha, which quantifies the added return as the excess return above the security market line in the capital asset pricing model. As the Jensen and Treynor determine rankings based on systematic risk alone, they will both rank portfolios identically. In finance, Jensen’s alpha (or Jensen’s Performance Index, ex-post alpha) is used to determine the excess return of a security or portfolio of securities over the security’s theoretical expected return.

The security could be any asset, such as stocks, bonds, or derivatives. The theoretical return is predicted by a market model, most commonly the Capital Asset Pricing Model (CAPM) model. The market model uses statistical methods to predict the appropriate risk-adjusted return of an asset. The CAPM for instance uses beta as a multiplier.

Jensen’s alpha was first used as a measure in the evaluation of mutual fund managers by Michael Jensen in the 1970s. The CAPM return is supposed to be ‘risk adjusted’, which means it takes account of the relative riskiness of the asset. After all, riskier assets will have higher expected returns than less risky assets. If an asset’s return is even higher than the risk adjusted return, that asset is said to have “positive alpha” or “excess returns”. Investors are constantly seeking investments that have higher alpha.

In the context of CAPM, calculating alpha requires the following inputs:

· the realized return (on the portfolio),

· the market return,

· the risk-free rate of return, and

· the beta of the portfolio.

Jensen’s alpha = (Portfolio Return – Risk Free Rate) – (Portfolio Beta * (Market Return – Risk Free Rate))

.gif” alt=”\alpha_J = (r_i – r_f) – (\beta_{im} \cdot (r_m – r_f)) “>

Since” title=”Eugene Fama”>Eugene Fama, many academics believe financial markets are too efficient to allow for repeatedly earning positive Alpha, unless by chance. To the contrary, empirical studies of mutual funds usually confirm managers’ stock-picking talent, finding positive Alpha. However, they also show that after fees and expenses are deducted, the effective Alpha for investors is negative. Nevertheless, Alpha is still widely used to evaluate mutual fund and portfolio manager performance, often in conjunction with the Sharpe ratio and the Treynor ratio.

Risk Adjusted Performance (RAP)
• Adjust the risk of the portfolio to equalize the risk of the market or benchmark portfolio;

• Compare the returns after risk adjustment to the benchmark portfolio returns; and

• Resulting values larger than the market return (or other benchmark used) would indicate superior performance.

The formula for RAP is given by:

RAP =RFR + (SDm/SDp)(Rp – RFR)

Upon performing all of the above, you will have a clear understanding on the performance of both your passive and active fund. For the passive fund, the smaller the tracking error, the better it is and in general, it is commendable if you are able to get a tracking error of less than 5%. As for the active fund, the higher the ratios (for Sharpe and Treynor) and the higher the Alphas (for Jensen and RAP), the better it is.

Lastly, your report should also include the followings:

NAV/unit of each of the funds (passive and active) on October 02, 2015;
Realized gains (if any) of each of the funds (passive and active) on October 02, 2015
The NAV/unit shall give an indication of the overall performance of each of the funds while the realized gain shall provide an assessment of the ability of the fund to distribute its income. You should submit your report using the Web Submission. Thank you.

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