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Performance measurement is an important part of the knowledge base for anyone involved in investing. The dynamic nature of the investment field and the creativity of its participants ensure a constant stream of innovation in performance measurement.
The difference between what we put in and what we got back is the return; we invest to provide this return. The higher the degree of return uncertainty, or risk, in a given investment, the more return we require.
THE INVESTMENT PROCESS
With most investments, we cannot initially be sure of the value of the income and gains we will receive. This includes analyzing how well alternative scenarios for diversifying investments across asset classes and funds are expected to meet our objectives, and then selecting the scenario with the most potential.
WHY MEASURE PERFORMANCE?
Once funded, managers will decide to implement these strategies through the selection of securities and optimal diversification among them to achieve the highest expected return given the level of risk. It is important to differentiate the two because as investors we are interested in forward and backward looking measures of risk and return.
PERFORMANCE MEASUREMENT AND THE INVESTOR
Most investors outsource all or part of their fund management to professional investment managers. Constant analysis of historical risk and return is the only way to verify that the manager's investment process is delivering what we are paying for.
PERFORMANCE MEASUREMENT AND THE INVESTMENT MANAGER
The sum of such decisions is used to determine the strengths and weaknesses of the manager's strategy. For institutional investors, measures of return and risk facilitate a dialogue about the manager's philosophy, investment process and future expectations.
THE PERFORMANCE MEASUREMENT PROCESS
The data helps clients monitor whether the manager is performing as expected given the capital market environment over the period. Compute statistics that represent managerial skill by studying the patterns of returns produced by the manager and by relating returns to the risks taken.
RETURN MEASUREMENT
RISK MEASUREMENT
We will also examine the evaluation of absolute risk in a mean-variance framework, where risk is represented by the standard deviation of returns. Finally, we discuss the methods used to determine the incremental risk the manager takes over the risk implied by the benchmark, or the benchmark's relative risk.
EFFICIENCY AND SKILL MEASUREMENT
Then we look at measures where risk is defined as the potential to lose money or downside risk.
PERFORMANCE ATTRIBUTION
PERFORMANCE PRESENTATION
In this section we summarize industry standards for presenting performance measurement results to investors. Given the degree to which the standards affect not only performance presentation, but also other aspects of performance measurement, attention is paid to the calculation and interpretation of the required statistics, including compound returns, and compound statistics of equal distribution. and weighted with assets. .
SUMMARY
RETURN
In subsequent periods, it is equal to the final market value of the previous period. The market value at the end of the investment period and the income earned during that period equals the accumulated value for the period.
MARKET VALUE
Revenue is included in the numerator because it is part of the profit over the period. We include income in the denominator because it is part of the capital at risk at the beginning of the period.
TRANSACTIONS WITHIN THE FUND
Because the accrual would be part of the proceeds if the security were sold on the valuation date, it is included in the market value calculation. Instead, we include cash received at the sale in the total fund market value calculation.
CASH FLOWS
Cash flow is a general term for various types of transactions with the economic effect of adding or removing investment in the fund. So, the term cash flow refers to a transfer of assets into or out of the portfolio, valued at the market value of these assets at the time of the transfer, regardless of whether the transaction is actually made in cash.
RETURN ON INVESTMENT
If the additional contribution was made at the beginning of the period, the investor did not need the money for the entire period. If the investment was made sometime during the period, the investor actually had use of the capital for part of the period.
COMPOUNDING
The first expression (EMV + NOF) is used instead of the final market value used in the ROR calculation. Returns can be equated to interest rates used in calculating the future value of a fixed income investment.
IMPACT OF CASH FLOW TIMING ON RETURN
Given the same profits, returns are higher when the investment is made at the end of the period. In Exercise 2.9, the advertised return for the period would be the 10% return measured from the beginning of the period to the end.
TIMING OF INVESTMENT MANAGER DECISIONS
SEGREGATING INVESTOR AND MANAGER TIMING DECISIONS
The decisions the manager makes to allocate assets and select securities within the portfolio. The actual return experienced by the investor is influenced by a combination of the two effects.
PRECISION OF RETURN CALCULATIONS
The second can also be considered to be attributed to the investor because he decided to hire a manager. In contrast, an ideal statistic to measure the return generated by investment managers would only consider their asset allocation and security selection decisions, since they typically have no control over the timing of external cash flows.
MONEY-WEIGHTED RETURN
To reflect these transactions, MWR considers not only the amount of the flows, but also the timing of the cash flows. If there is a cash flow, we need to consider the amount and timing of the flow.
INTERNAL RATE OF RETURN
The period weight used for each of the cash flows is a percentage of the total period considered. If the cash flows are outside the portfolio, the cash flow adjustment is negative.
MODIFIED DIETZ RETURN
Similar to the IRR, in the modified Dietz calculation, cash flows are adjusted by weighting the flows by the number of days they were available for investment during the period. MWR results are affected by the time frame and volume of cash flows during the period.
TIME-WEIGHTED RETURN
The next step is to note the value of the portfolio immediately before the cash flow. The denominator is the initial market value adjusted for the cash flow into or out of the portfolio.
ESTIMATING THE TIME-WEIGHTED RETURN
This linked MWR estimate of TWR provides a reliable approximation of the TWR in situations where the cash flows are small relative to the portfolio size and there is not a large amount of return volatility in the period. If the cash flow is large and/or there is large volatility within the period, the MWR estimate of TWR is inaccurate.
ELIMINATING DISTORTION CAUSED BY LARGE CASH FLOWS
The stopwatch version of the Modified Dietz return of 35.50% is a better approximation of TWR than the 36.52% unmodified Modified Dietz calculation. If the valuations were actually available before each of the cash flows, we would use the TWR formula and not an estimation method.
COMPARING THE TWR AND THE MWR
One period return can be calculated by valuation dates by summing the returns of one period into cumulative returns. In addition, returns calculated over multiple periods are often presented as an average of the returns of each period within the period, usually on an annual average basis.
CUMULATIVE RETURN
Growth rates compounded over several periods are called cumulative growth factors. These growth factors are useful for calculating cumulative returns over multiple periods without needing to know interim returns or growth rates. Figure 5.2 shows the calculation of returns in each of the possible holding periods according to the growth rates from Figure 5.1.
COMPRESSING PERIODS
We calculate cumulative returns when we are interested in the performance of investments over long term periods. The increase in value at the end of each period, measured by returns, is treated as if it were income reinvested in the portfolio in the following period.
ARITHMETIC MEAN RETURN
Using the arithmetic mean return to reconcile the beginning and ending investment value will overestimate the ending value. The average return we need to use for this application must be lower than the arithmetic mean return to account for the compounding process between periods.
GEOMETRIC MEAN RETURN
Geometric mean return is the nth root of the compound return, where n = the number of periods used to calculate the cumulative return. Multiplying et to the average of the continuously compounded growth rates yields the geometric mean return.
ANNUALIZING RETURNS
If the holding period's return is greater than one year, the rate is usually adjusted to an annual basis using the inverse of the compounding formula used above. Note that we calculate the annualized return by first taking the nth root of the cumulative growth rate, or 1.191 in our example, as opposed to taking the nth root of the cumulative return.
COMPOUND ANNUAL INTERNAL RATE OF RETURN
MULTIPLE PERIOD RETURN ANALYSIS
This is because the MWR is the only interest rate that reconciles the initial and final values of an investment made, given the pattern of cash flows. Because there are no cash flows, the MWR for Clients A and C is equal to the TWR.
MANAGEMENT FEES AND EXPENSES
Sometimes the management fee is not paid by the fund itself, but is paid by the investor separately. We can extend this methodology to calculate net returns from other investment expenses.
TAXES
The taxes generated by these transactions are attributable to the Trustee; we should subtract these capital gains taxes from the numerator of the return calculated to measure the manager. AIMR suggests a method where we adjust the numerator of the return calculation upwards for the impact of these gains. 3To calculate an after-tax return that removes the effect of the non-discretionary capital gains, we must.
MUTUAL FUND RETURN CALCULATION
Shareholders receive a proportionate share of the capital gains and income earned on the fund's holdings. The NAV of the portfolio will fall on the day the fund goes ex-dividend by the amount of the dividend (excluding the change in market value for that day).
RETURN BEFORE EXPENSES AND TAXES
Income distributions are made up of the income on the underlying investments in the portfolio, including dividends and interest attribution. In relation to our example, the cumulative return is 38.64% assuming reinvestment of the dividend on each ex-dividend date.
RETURN AFTER SALES CHARGES
As with the no-load example, this equals the assumed investment of $1,000 divided by the NAV at the end of the day the contribution is made. We then adjust the final market value by the load amount and use the resulting number in the return calculation.
RETURN AFTER EXPENSES
Treat the accrual of the costs as a non-taxable distribution and calculate reinvestment shares for the accrual. 12B-1 fees have the effect of shifting charges to the fund's expense structure as they are accrued and offset in the daily NAV calculation in the same manner as management fees and other expenses.
RETURN AFTER TAXES ON INCOME AND CAPITAL GAINS
After-tax pre-redemption proceeds deduct the tax due on distributions from the fund to the investor. After-tax returns after redemption also deduct the tax when the investor sells shares back to the fund company.
RETURN AFTER TAXES ON REDEMPTION GAINS
If the net amount is negative, the long-term gain was less than the short-term loss. If the amount is negative, the long-term loss was greater than the short-term gain.
ADJUSTING RETURNS TO THE INVESTOR’S BASE CURRENCY
We can isolate the currency loss by calculating a currency return, equal to the change in the exchange rate divided by the initial rate. We also discuss at a summary level the methodology behind the third option - creating and calculating returns for market indices and other benchmarks.
PEER GROUPS
For measuring manager performance, however, it is relative performance that is of interest. The chapter closes with a discussion of the methodology for determining the relative value added by the manager over time.
PERFORMANCE UNIVERSE
We can rank and compare the performance of the funds in the narrowed list of results. If the total number of funds is an even number, then we can take the median as the average of the middle two observations.
PEER GROUP ANALYSIS
A percentage return P% for a data set is the value that is greater than or equal to (1 − P%) of returns but is less than P% of returns (keeping our convention that the best quantity is the highest return).
PEER COMPARISON CONSIDERATIONS
The goals, constraints and strategies of peer group portfolios should be aligned as closely as possible to ensure good analysis. Since the number of funds in a peer group decreases with increasing returns, the statistical significance of peer group comparisons decreases with increasing time period.
MARKET INDICES
Balanced fund mandates are compared using an index calculated as the sum of the weighted returns of several cash, fixed income and equity indices. For a particular institutional or private client account, the benchmark is part of the investment policy statement, along with the fund's liquidity requirements, income, permitted investments and other information that provides guidance to the manager.
EQUITY INDEX RETURN CALCULATION
At the start of the measurement period, we initialize the index level to an arbitrary base value, or level, e.g. 1000. We can calculate cumulative index returns over multiple periods by taking the ratio of the index levels between any two dates.
EQUITY INDEX MAINTENANCE
We must account for these new shares in calculating the index return Day 3. The relative market cap of the individual securities within the index determine the securities' weights.
CUSTOMIZED BENCHMARKS
One of the main considerations in managing these portfolios is that the growth in liabilities matches the growth in assets. The value of the assets will vary inversely with the change in liabilities as interest rates fluctuate.
VALUE ADDED
Arithmetic value added is the difference between the fund's return and the standard return for the period. In summary, the geometric value added is the value that causes the benchmark's return to increase in the fund's return over a period.
DEFINING RISK
Here, the client communicates their risk tolerance through the benchmark setting and will judge the manager based on the effective use of the relative risk of the benchmark to achieve added value. The added value can be compared to the excess risk to measure whether the manager has planned his active risk well, where active risk is defined as the risk associated with positioning the portfolio in a different way from the reference.
MEASURING RISK
So since we can only expect higher returns in exchange for taking on risk, if volatility were a good measure of risk, we would expect volatility to be correlated with returns. We can see that while small company stocks had the highest average return, at 17.3%, the range of annual returns experienced by investors in this asset class ranged from 200.9%, from low to high annual returns .
CLASSIFYING RISK MEASURES
Forward-looking risk estimation (fore-risk) is the process of predicting the expected risk of the current portfolio. Retrospective (ex-post risk) risk is the measurement of the historical risk experienced by the investor in the portfolio or the measure of relative risk produced by the manager.
RANGE OF RETURNS
The chapter concludes with a discussion of the properties of the normal distribution and statistics that indicate a departure from normality. Range = Highest Return − Lowest Return (9.1) The range is the difference between the highest and lowest observed return.
HISTOGRAM
The series comparison is an easily calculated and understandable representation of the relative volatility of the two funds. Although in a real world situation we would hesitate to draw conclusions based on 13 months of returns, we can see a similar distribution of returns for both entities.
MEAN RETURN
The arithmetic average return does not account for the reinvestment of periodic profits and income. However, the arithmetic mean return is useful in analyzing the distribution of periodic returns that implies investment risk because it provides the center around which the periodic observed returns are distributed.
RETURN DEVIATIONS
The sum of the distances of the returns above the average return is equal to the sum of the distances of the returns that fall below the average return. The average return is the point at which the sum of the return deviations equals zero.
MEAN ABSOLUTE DEVIATION
9.4) The average absolute deviation is the arithmetic mean of the absolute value of the difference between each of the observed returns and the arithmetic mean of the returns. Exhibit 9.5 illustrates the calculation of the mean absolute deviation for our sample fund and benchmark, equal to 3.54% and 3.35%, respectively.
STANDARD DEVIATION
So far we have calculated the standard deviation of the periodic returns around the arithmetic mean return. We could instead have calculated the standard deviation of the returns around the geometric mean return.
ANNUALIZED STANDARD DEVIATION
We multiply the standard deviation by the square root of the periodicity to convert the periodic statistic into a statistic comparable to the annual arithmetic return. The standard deviation of annualized returns for the same period was actually slightly higher at 15.84%.2 In the second.
NORMAL DISTRIBUTION
The normal distribution is an accurate description of the spread of values around the mean for many things, such as the height and weight of the population. About 68% of observed returns will be within a range of one standard deviation above and below the mean return.
HISTORICAL VALUE AT RISK
We calculate the return at 1.96 standard deviations below the mean by multiplying the standard deviation by 1.96 and subtracting the result from the mean return, which is at zero standard deviations. We derive the VaR in dollars by multiplying the result by the original investment of 10,000.
OTHER DISTRIBUTIONS
SKEWNESS
The skewness statistic is a measure of the degree of asymmetry in the dispersion of returns around the average return. The skewness statistic can only be interpreted as a measure of the shape of the return distribution.
KURTOSIS
Like skewness, kurtosis has no meaning in terms of returns; it is merely a measure of the shape of the return distribution. The return distribution from our example has an excess kurtosis of −1.04 for the fund and −1.59 for the benchmark.
TESTING FOR A NONNORMAL DISTRIBUTION
Standard deviation is not a good measure of return variability if the distribution of returns is skewed or otherwise non-normal. If the distribution of returns were not normally distributed, we would expect to experience a different number of returns at a particular point in the distribution than the normal distribution would indicate.
PROBLEMS WITH STANDARD DEVIATION
However, given the variation in returns around the average return, this may not be how we all would conceptualize the risk of investing. First, return deviations are calculated, which make up the standard deviation, with the mean return as the reference return.
ABSOLUTE RISK VERSUS DOWNSIDE RISK
The semideviation is the square root of the semivariance, and the mean semivariance is the variance of the returns below the mean return. Figure 10.3 shows that it is the part of the return distribution to the left of the mean that is considered risky, using half a deviation as a measure.
RETURN TARGETS
SHORTFALL RISK
Shortfall risk indicates that 38.46% of periodic monthly returns fell below our target return of 1.20% per month. We can compare the default risk of different funds to see how well they meet the investor's needs for target return.
EXPECTED DOWNSIDE VALUE
We can determine the asset allocation for a multimanager fund where the combined risk of expected annual shortfall is equal to 5%. Absence risk is a simply calculated statistic with a powerful application: it can indicate risky strategies that are not normally considered risky.
DOWNSIDE DEVIATION
In light of the relationship between risk and return, we are also interested in whether the portfolio's risk level differs from the benchmark. The volatility of the periodic return differences to the benchmark measures the risk specific to the strategy being evaluated.
COVARIANCE
Return of the fund and the average fund return, and the ■ Return of the benchmark and the average benchmark return. Covariance is difficult to interpret as anything other than the average product of the differences between the fund and benchmark return variances.
CORRELATION
Periods when the fund and benchmark returns move in opposite directions relative to their averages contribute negatively to the covariance. To calculate the correlation, we first calculate the covariance of the fund and benchmark returns.
COEFFICIENT OF DETERMINATION
-squared is the proportion of variability in fund returns that we can relate to the variability of the benchmark returns. Covariance The direction and degree of association of the fund and benchmark returns, as well as the magnitude of the variation in the fund and benchmark returns.
REGRESSION ANALYSIS
We can use the line of best fit to interpolate the fund's return if we were given the benchmark return. We can see in Appendix 11.9 that the line passes through the fund's rate of return at a point between 0% and 1%.
REGRESSION BETA
11.4) The linear regression equation assumes that the value of one of the variables is at least partially dependent on the other. Beta measures how an asset fluctuates with the market itself and is equal to the covariance of the fund and benchmark return divided by the variance (square of the standard deviation) of the benchmark returns.
REGRESSION ALPHA
The beta equal to 1 indicates that the amount of covariance in the fund and benchmark returns is equal to the amount of variance in the benchmark returns. It does indicate that 22% of fund returns are not explained by variation in the benchmark return.
TRACKING RISK
11.8) WhereThis is the periodic differences between the return of the fund and the benchmark and is the average of the return differences. Tracking risk is a function of the standard deviation of the fund's returns and the correlation between the fund's returns and the benchmark.
COEFFICIENT OF VARIATION
