Performance metrics

The calculations for performance metrics will be explained in three sub-sections.

The basis for our analysis in this section will be the cumulative return and the risk-adjusted performance measures. We follow the GIPS standard used by the GPFG.

5.4.1 Arithmetic average and standard deviation:

We calculate the arithmetic average return and the standard deviation of the portfolios. The calculations are reported on an annualized basis. This provides a general impression of how the portfolios have performed during the sample period, and provide input for the risk-adjusted return calculation.

5.4.2 Cumulative return

To calculate the cumulative return of the companies, we adopt the methodology from GIPS used by the GPFG. GIPS requires that the portfolios use

time-weighted rate of return (CFA Institute, 2012). The time-time-weighted return (TWR) is used to eliminate the effect of external flows of capital. The portfolio gets an inflow of capital at the time of each exclusion wave for the GPFG. The weights and composition of stocks in the portfolio changes with every inclusion of new companies. We use the weighted average of asset returns to calculate the return of the portfolio for each period. We also calculate the monthly holding period return of assets by controlling for cash flows to ensure that the portfolio returns are accurate. We use the same methodology to calculate the time-weighted return for benchmark. Time-weighted return, as the name implies, put appropriate weights according to the duration of the sub-period. In our case, all returns will receive

equal weights to reflect the monthly frequency of our data. The time-weighted return assumes that all cash distribution is reinvested into the portfolio. The returns are geometrically linked to calculate the cumulative return, using the following equation:

𝑇𝑊𝑅 = [∏(1 + 𝑟_𝑖)

𝑛

𝑖=1

] − 1

The value calculated will show how much the cumulated return of the portfolio during the sample period. The annualized absolute return (AR) is calculated using the annualized geometrical mean return. Since we have over 12 monthly

observations, we use the following equation to calculate the annualized geometric mean return.

𝐴𝑅 = [(1 + 𝑇𝑊𝑅)^12𝑛] − 1

n is the number of monthly returns in our sample. In the portfolio of exclusions, we use 141 monthly observations to calculate the cumulative return. To calculate the annualized return, we take (1+ TWR) to the power of (12/141). The

annualized geometric mean return often differs from the arithmetic average because it considers the effect of compounding. Larger fluctuations result in higher discrepancy between the arithmetic and geometric mean (Bodie, Z., Kane, A., & Marcus, A. J. 2014). Although the arithmetic average is considered a good estimate for future predictions, the geometric mean presents a more realistic measure of historical portfolio performance

5.4.3 Risk-adjusted return measures

We use the Sharpe ratio, 𝑀² and Jensen’s alpha as our risk-adjusted performance measures. The Sharpe ratio is among the most common performance measures for risk-adjusted return, and was developed by William F. Sharpe in 1966. We use Sharpe ratio to measure the risk-reward tradeoff of the portfolios. The Sharpe ratio is calculated by dividing the excess return over to total volatility of the portfolio.

We follow the same formula used by the GPFG in their reporting (NBIM, 2010).

The excess return is the arithmetic average return of the portfolio subtracted by the risk-free rate. We use the annualized rate of the 3-month US treasury bills as a proxy for the risk-free rate. The total volatility is the standard deviation of the portfolio. The calculated Sharpe ratio is the excess return per unit of risk.

𝑆𝑅 =𝑟̅_𝑝− 𝑟̅_𝑓 𝜎_𝑝

The portfolio with the highest Sharpe ratio has the higher risk-adjusted return.

However, the numerical value of the difference in ratio is difficult to interpret.

This issue is solved by using 𝑀². The 𝑀² measure is derived from the Sharpe ratio, as it measures the return over the total risk of the portfolio. It is set up by creating an adjusted portfolio with positions in the managed portfolio and T-bills to match the volatility of the benchmark. We use a slightly altered version of the M² measure by using the GPFG fixed income portfolio as the quasi risk-free asset.

The adjusted portfolio will be a composite of the portfolio and GPFG fixed income portfolio. We create an adjusted portfolio to match the volatility of GPFG equity and benchmark, with the purpose of finding the difference in return for the same unit of total risk.

We use the solver function in excel to find the appropriate weights in the adjusted portfolio that yields the same standard deviation as the benchmark and GPFG equity. The return of the adjusted portfolio is calculated as the weighted average of the returns from the portfolio and GPFG fixed income portfolio. The advantage of the 𝑀² measure is that the numerical value of difference is more intuitive. The 𝑀² is calculated by subtracting the annualized mean return of the adjusted portfolio from the benchmark, using the following equation:

𝑀_𝑝² = 𝑟_𝑝^∗− 𝑟_𝑏

Where the 𝑟_𝑝^∗ is the total return of the adjusted portfolio, and 𝑟_𝑏 is the return of benchmark or GPFG Equity. The 𝑀² will provide a numerical value of how much the portfolio outperforms/underperforms the benchmark given the same unit of risk.

Jensen’s alpha measures the performance of the portfolio relative to a market index (Brooks, 2014). It is a risk-adjusted performance measure that represents the average return on a portfolio above or below that predicted by the capital asset pricing model (CAPM). The equation of Jensen’s alpha is:

𝛼_𝑝 = 𝑟_𝑝− [𝑟_𝑓+ 𝛽_𝑝(𝑟_𝑚− 𝑟_𝑓)]

where 𝑟_𝑝 is realized annual return of the portfolio, 𝑟_𝑚 is the realized annual return of the benchmark, 𝑟_𝑓 is the annual risk-free rate of return, 𝛽_𝑝 is the beta

coefficient of the portfolio with respect to constructed market index. The equation in the brackets on the right-hand side is the CAPM. To estimate the alpha, we use the constructed benchmark as a proxy for the market index and the equation above is rearranged to following equation:

𝑅_𝑝,𝑡 − 𝑅_𝑓,𝑡 = 𝛼_𝑝^𝑗 + 𝛽(𝑅_𝑚,𝑡− 𝑅_𝑓,𝑡) + 𝑢_𝑝,𝑡

Where 𝑅_𝑝,𝑡 is return on portfolio over time, 𝑅_𝑚,𝑡 is return on benchmark portfolio over time, 𝑅_𝑓,𝑡 is the risk-free rate, 𝛽 is the OLS estimate of the slope coefficient in the CAPM and 𝑢_𝑝,𝑡 is the disturbance term. Jensen’s alpha measured on a monthly level is the OLS estimate of the alpha in the regression above, that being 𝛼_𝑝^𝑗.

A positive alpha indicates that the portfolio has outperformed the benchmark, and negative alpha indicates underperformance. A 95 percent confidence interval around the alpha will be constructed using the OLS standard error of the intercept in the regression.