#### Parameters to define performance – risk and return-

When measuring and evaluating investment performance, it’s important to consider both the return earned and the associated risk. However, many people tend to focus solely on the return without taking into account the level of risk required to achieve that return. Therefore, a comprehensive performance measurement should include both return and risk assessment.

#### Rate of return measures-

**Holding period return** (HPR)-

Holding period return (HPR) is a simple and popular rate of return used to measure the performance of an investment. It calculates the income generated by an investment and the change in its value over the period it was held, expressed as a percentage per annum, relative to the beginning value of the investment.

For example, if an investor’s portfolio was valued at Rs.1,00,000 on 1 April, 2018, and on 31 March, 2019, the value was Rs.1,20,000, the HPR would be 20%, as calculated using Equation 1: HPR = (E-B)/B = ((120000-100000))/100000 = 20%.

If the investor also received Rs. 5000 by way of dividend and interest income during the same period, the HPR would be calculated using Equation 2: HPR = (I + (E -B)) / B = (5000 + (120000-100000))/100000 = 25%.

While this measure assumes that all income distributions are made at the end of the year, it is widely used and accepted as a starting point for performance measurement.

Time-Weighted Rate of Return (TWRR) and Money-Weighted Rate of Return (MWRR) are two ways to measure the rate of return over multiple holding periods. For instance, consider a portfolio that generated the following annual holding period return from 2015 through 2019:

-2015: -5.00%

-2016: -15.20%

-2017: 8.10%

-2018: 30.75%

-2019: 17.65%

**Time Weighted Rate of Return (TWRR) versus Money Weighted Rate of Return (MWRR)-**

Suppose further that an investor has invested Rs. 75,000 in this portfolio by making contributions at the beginning of the year. The portfolio value at the end of 2019 is Rs. 105,920.31.

To calculate the rate of return generated during the five-year period, we can use the internal rate of return (IRR) or MWRR. The IRR is calculated by discounting the terminal value of the investment to the cashflow contributions made. In this case, the IRR is 15.15%.

Alternatively, to calculate the underlying investment performance without being influenced by the timing of cash flows, we can use TWRR. TWRR is the compound rate of growth over the stated period. When there is an external cash flow, TWRR requires computing a set of returns for each period, which are then linked together to compute TWRR for the evaluation period.

In this example, there are five sub-periods, and the first step towards calculating the TWRR is to calculate each period return. The next step is to link these sub-period returns together using wealth relatives. Wealth relative is the ending value of one unit of money. We create a cumulative wealth relative for the entire evaluation period by multiplying each period wealth relative. The compound annual return can then be calculated as TWRR, which is 6.021%. Note that unless the sub-periods constitute exactly one year, TWRR will not be expressed as an annual rate, and we have to annualize the return.

**Geometric Mean Return (GMR) vs. Arithmetic Mean Return (AMR)**

Geometric Mean Return (GMR) and Arithmetic Mean Return (AMR) are two methods for calculating investment returns. To illustrate their differences, let’s consider two portfolio choices available for investment of Rs.100,000. The first choice produces a -50% return in the first year and a 100% return in the second year, while the second choice produces a 10% return in each of the two years.

At the end of the second year, the value of the portfolio for the first choice remains unchanged at Rs.100,000, resulting in an average annual return of 25%. Meanwhile, the value of the portfolio for the second choice increases to Rs.121,000, with an average annual return of 10%. Although the second choice has a lower AMR, it is still the better option.

AMR is the simple average of individual yearly returns, while GMR is the rate at which the sum invested at the beginning of the period will accumulate to a given sum at the end of the period by compounding. The GMR depends only on the initial and final values of the portfolio, not on the path by which that value was realized.

To calculate the GMR, we need to calculate wealth relatives for each holding-period rate of return, link these sub-period returns, and raise the cumulative wealth ratio to the power of 1/n, where n is the number of years. The GMR is always less than the AMR except when all the individual yearly returns are exactly the same.

The GMR is important for analyzing long-term returns on assets, while the AMR is a better estimate for a future year’s return based on the past year’s returns. While a fund manager can increase the average annual return by increasing risk, the GMR is the only way to compare long-term accumulations and understand what has really happened to investments. To estimate the probability distribution of terminal wealth, the GMR is the better option.

**Gross versus net return**

The gross return is the total return on an investment before any fees or expenses are deducted, and is usually stated for a specific period of time. The net return, on the other hand, is the return on an investment after all fees and expenses have been taken into account. This is the return that the investor actually receives.

It is important to consider both gross and net return when evaluating the performance of an investment. While gross return can provide a broad overview of how well an investment is performing, it can be misleading because it does not take into account any fees or expenses. Net return, on the other hand, provides a more accurate picture of the return an investor is actually earning.

To illustrate the difference between gross and net return, let’s look at an example of a portfolio. In this example, the size of the portfolio is Rs. 100 lacs, the investment period is one year, and the profit made during the year is 20% on the capital contribution. Other expenses such as brokerages and DP charges are charged on the gross value of the portfolio at 0.50%.

After taking into account all fees and expenses, the net return on this investment is 13.88%. This is lower than the gross return of 20%, but it provides a more accurate picture of the return an investor is actually earning.

**Compounded Annual Growth Rate-**

Compounded Annual Growth Rate (CAGR) is a measure of an investment’s annual growth rate over time, taking into account the effect of compounding. The calculation assumes that any dividends, income, or rent declared by the investment are reinvested in the same investment at the day’s market price.

The formula for calculating CAGR between the opening and closing wealth is (A/P)^(1/t)-1, where A is the closing wealth, P is the opening wealth, and t is the time period in years.

For example, if an investment of Rs. 100,000 is made in a certain investment for five years and the year-on-year returns are as follows: -5.00% in 2015, -15.20% in 2016, 8.10% in 2017, 30.75% in 2018, and 17.65% in 2019, the terminal wealth or the end-of-period value of the investment can be calculated.

Using the investment values at the end of each year as shown in the table, the CAGR can be calculated using the compound interest formula. In this case, the CAGR is 6.02%.

CAGR is a constant rate of growth required to reach the same terminal value over a given period of time. In the table, the investment value at the end of each year is shown along with the investment value at the end of the year growing at a rate of 6.02%.

**Pre-tax versus post-tax return-**

The pre-tax rate of return refers to the return on an investment before any taxes are paid, while the post-tax rate of return is the return after taxes have been paid on investment income and realized capital gains. Investors may belong to different tax brackets, so the performance of investments is communicated as the pre-tax rate of return. However, investors or their financial advisors need to calculate the post-tax return by adjusting the pre-tax return to the applicable tax rates.

The pre-tax return enables comparisons across different investments and strategies because different investors may be subject to different levels of taxation. However, what ultimately matters to the investor is the post-tax return. Therefore, investors make investment decisions based on post-tax performance.

An example of post-tax return calculation is provided below: Suppose an individual achieves a 5% pre-tax rate of return for stock XYZ and is subject to a capital gains tax of 15%. The post-tax rate of return would be calculated as follows:

Post-tax return = Pre-tax return x (1-tax rate)

Post-tax return = 5% x (1-15%) = 4.25%

**Annualizing return-**

Annualizing return is a method of calculating the compound average annual return of an investment in order to facilitate comparisons. To annualize a return, the chain linking method is used, and the product of the linking is raised to the reciprocal of the number of years in the evaluation period. For example, if a portfolio generated returns of 15.5% in 2017, 9.50% in 2018, and -6.90% in 2019, the first step in annualizing the return is to calculate the wealth relative. Then, the product of the wealth relative is taken and raised to the reciprocal of the number of years, and finally, 1 is subtracted from the result to obtain the annualized return.

It is important to note that annualizing a return for a period of less than a year is not advisable, as it involves extrapolating the return to a full year and may not accurately reflect the performance of the investment. This is particularly relevant for investments where returns can fluctuate significantly during the remaining period.

**Cash drag adjusted return-**

Cash drag adjusted return is a method of computing investment returns that takes into consideration the impact of uninvested cash on the overall portfolio return. When portfolio managers hold cash or other assets that are not invested in the portfolio, it can reduce the overall return of the portfolio. The cash drag adjusted return method adjusts for this by including the impact of the uninvested cash in the calculation of the portfolio return.

For example, if an investor has invested Rs. 100 lacs in an equity investment portfolio, but the manager has invested only Rs. 75 lacs in equities and Rs. 25 lacs are in liquid funds, the return on investment (ignoring the cash drag) would be calculated as 10% based on the Rs. 7.5 lacs return on the equities. However, adjusting for the cash drag, the return on investment would be calculated as (10% x Rs. 75 lacs) + (4% x Rs. 25 lacs) = Rs. 85,000 or 8.5% on Rs. 100 lacs.

By using the cash drag adjusted return method, investors can get a more accurate picture of the actual return on their investments, and portfolio managers can better evaluate the effectiveness of their investment strategies.

**Alpha and beta-**

Metric | Definition | Calculation | Interpretation |
---|---|---|---|

Alpha | Return generated by a portfolio over the required rate of return as per CAPM | Actual Return – Required Rate of Return | Reward for bearing non-market risk |

Beta | Measure of a portfolio’s sensitivity to market risk | Covariance between portfolio and market returns divided by variance of market returns | Reward for bearing market risk |

In simple terms, alpha is a measure of a portfolio’s performance in excess of the expected return based on its beta, while beta is a measure of a portfolio’s sensitivity to market risk.

Alpha is a way to assess the value that a portfolio manager adds through their stock picking and market timing skills, above and beyond the overall market’s return. A positive alpha indicates that the portfolio has outperformed the market, while a negative alpha indicates underperformance.

Beta, on the other hand, measures how much the portfolio’s returns move in relation to the market as a whole. A beta of 1 indicates that the portfolio moves in lockstep with the market, while a beta greater than 1 indicates greater volatility than the market and a beta less than 1 indicates less volatility. A beta of 0 indicates no correlation with the market.

**Portfolio Return-**

The portfolio return is calculated as the weighted average return of individual securities in the portfolio. For example, if a portfolio consists of four securities with different weights and returns, the portfolio return can be calculated as follows:

Security A has a return of 15% and a weight of 30% in the portfolio.

Security B has a return of 10% and a weight of 20% in the portfolio.

Security C has a return of 12% and a weight of 20% in the portfolio.

Security D has a return of 18% and a weight of 30% in the portfolio.

The weighted average return of the securities in the portfolio is calculated by multiplying each security’s weight by its return and adding up the results. In this example, the portfolio return would be:

Portfolio Return = (15% x 30%) + (10% x 20%) + (12% x 20%) + (18% x 30%) = 14.30%

Therefore, the portfolio return for this example is 14.30%.

#### Risk Measures-

Risk is the possibility of loss, damage, or harm, and it is a crucial aspect of performance measurement and portfolio management. There are different definitions of risk for investments, but it is commonly defined as variability in the expected return or limited to losses or worse than expected outcomes only. Two measures of risk that have gained support in theory are the variance and the standard deviation of the estimated distribution of expected returns. However, downside risk, which includes concepts such as semi-variance/standard deviation and target semi-variance/standard deviations, is also relevant.

Risk Type | Description |
---|---|

Total Risk & Downside Risk | Total risk is the overall risk associated with an investment, while downside risk is the risk of losing money. Total risk can be mitigated through diversification, while downside risk cannot be eliminated completely. Investors should be aware of both types of risk and develop strategies to manage them. |

Portfolio Risk vs Individual Risk | Individual risk is the risk associated with a single stock or investment, while portfolio risk takes into account the diversification of an investor’s portfolio. A well-diversified portfolio can reduce the overall portfolio risk. Investors should consider both types of risk when building their investment portfolios. |

Market Risk | Market risk is the risk associated with the overall market. It’s affected by economic, political, and social factors, and can’t be eliminated through diversification. It’s important to note that market risk affects all investments in the market. Investors should be aware of the market risk associated with any investment before investing. |

Volatility | Volatility is a measure of the amount of price movement an investment experiences over time. High volatility can lead to higher returns but also higher risk. Investors should be aware of the risks associated with volatility before investing in any particular stock or investment. |

Tracking Error | Tracking error is the difference between the performance of an investment and its benchmark. It’s important to keep track of tracking error, as it can indicate whether an investment is underperforming or outperforming its benchmark. Investors should monitor tracking error to ensure their investments are performing as expected. |

Systematic Risk & Unsystematic Risk | Systematic risk is the risk associated with the overall market or a specific sector. It can’t be eliminated through diversification, as it affects all investments in the market. Unsystematic risk, on the other hand, is the risk associated with a specific company or industry. It can be reduced through diversification. Investors should consider both types of risk when building their investment portfolios. |

Beta | Beta is a measure of the volatility of a stock or portfolio in relation to the overall market. It’s an important metric to consider when investing, as it can help investors understand the risk associated with a particular investment. A beta of 1 indicates that the stock or portfolio is as volatile as the market, while a beta greater than 1 indicates higher volatility and a beta less than 1 indicates lower volatility. |

Liquidity Risk | Liquidity risk is the risk associated with the ability to buy or sell an investment at a fair price and in a timely manner. Some investments may be illiquid, meaning they can’t be easily sold or bought. Investors should be aware of the liquidity risk associated with any investment before investing. |

Credit Risk | Credit risk is the risk associated with the ability of a borrower to repay a debt. It’s important to consider the credit risk associated with any investment that involves lending money, such as bonds or loans. Investors should evaluate the creditworthiness of the borrower before investing and consider diversification to reduce the overall credit risk of their portfolio. |

#### Risk-adjusted return measures:-

Ratio | Description | Formula | Example |
---|---|---|---|

Sharpe Ratio | The Sharpe Ratio measures the risk-adjusted performance of an investment by comparing its return to the level of risk taken on. A higher Sharpe Ratio indicates better risk-adjusted performance. | (Rp – Rf) / σp | If an investment has an annual return of 10%, the risk-free rate is 2%, and its standard deviation is 15%, its Sharpe Ratio would be (10% – 2%) / 15% = 0.53. |

Treynor Ratio | The Treynor Ratio is similar to the Sharpe Ratio but instead compares the excess return of an investment to the beta of a market index. A higher Treynor Ratio indicates better performance relative to market risk. | (Rp – Rf) / βp | If an investment has an annual return of 12%, the risk-free rate is 2%, and its beta is 1.5, its Treynor Ratio would be (12% – 2%) / 1.5 = 6.67%. |

Sharpe vs. Treynor | The Sharpe vs. Treynor measures compare the risk-adjusted performance of two investments, allowing investors to choose between them based on their relative risk and return. | (Rp1 – Rp2) / √(σ12 + σ22 – 2ρσ1σ2) | If Investment A has an annual return of 8%, a standard deviation of 20%, and Investment B has an annual return of 12%, a standard deviation of 25%, and the correlation between them is 0.6, their Sharpe vs. Treynor measure would be (12% – 8%) / √(20%^2 + 25%^2 – 2(0.6)(20%)(25%)) = 0.99. |

Sortino Ratio | The Sortino Ratio measures the risk-adjusted performance of an investment by focusing on downside risk, using only the standard deviation of negative returns. A higher Sortino Ratio indicates better performance in managing downside risk. | (Rp – Rf) / σd | If an investment has an annual return of 9%, the risk-free rate is 2%, and its downside standard deviation is 12%, its Sortino Ratio would be (9% – 2%) / 12% = 0.58. |

Modigliani and Modigliani Ratio (M2) Measure | The M2 Ratio measures the risk-adjusted performance of a portfolio or investment by comparing its return to the return of a benchmark index with the same level of risk. A higher M2 Ratio indicates better performance relative to the benchmark. | (Rp – Rb) / σp | If a portfolio has an annual return of 15%, the benchmark has a return of 10%, and the standard deviation of the portfolio is 18%, its M2 Ratio would be (15% – 10%) / 18% = 0.28. |

#### Performance Evaluation: Benchmarking and peer group analysis-

Performance evaluation is a crucial step in investment management, and it involves benchmarking and peer group analysis. A benchmark is a point of reference that serves as an objective test of the effective implementation of investment strategy. Popular choices for benchmarks include indices, and a good benchmark should meet several criteria such as being clearly defined, investable, consistent with the portfolio’s investment approach, having the same risk-return profile as the portfolio, and being measurable.

Sometimes market-based indices may not meet the criteria of a good benchmark, which calls for a customized benchmark that reflects the portfolio manager’s investment strategies and style. However, constructing and maintaining such benchmarks can be costly. Choosing the wrong benchmark or making benchmarking errors can lead to problems in evaluating portfolio performance accurately.

Peer group analysis is another popular way of evaluating portfolio performance. It involves comparing the performance of a portfolio to that of its peers with similar investment approaches and strategies. Portfolio tracking firms have developed databases to facilitate peer group analysis, ranking portfolios based on some risk-adjusted return measure.

#### Performance attribution analysis-

Analysis Category | Description |
---|---|

Assets and Sector Allocation | Determines the impact of the allocation of assets among various sectors on portfolio returns. This is measured against a benchmark. |

Selection | Measures the performance of the portfolio manager in selecting individual securities that outperform the benchmark. |

Market Timing versus Selectivity | Examines whether the performance of the portfolio was due to the timing of market movements or the ability to select profitable investments. |

Net Selectivity | This is an absolute measure of performance, calculated as the annualized return of the fund less the yield of a risk-free investment. The result is then reduced by the standardized expected market premium times the total risk of the portfolio under review. |

Local Currency versus Foreign Currency Denominated Investment Return | Calculates the return of foreign currency-denominated investments adjusted for fluctuations in the local currency against those foreign currencies. This determines whether currency fluctuations enhanced or reduced the return of the portfolio. |

Description:

**Assets and Sector Allocation:**This analysis determines the impact of the allocation of assets among various sectors on portfolio returns. It is measured against a benchmark to identify if over-investing in a specific economic sector resulted in differential returns.**Selection:**This analysis measures the performance of the portfolio manager in selecting individual securities that outperform the benchmark. It examines the ability of the portfolio manager to identify securities that perform well relative to the benchmark.**Market Timing versus Selectivity:**This analysis examines whether the performance of the portfolio was due to the timing of market movements or the ability to select profitable investments. It compares the portfolio manager’s ability to anticipate market movements against their ability to identify profitable investments.**Net Selectivity:**This is an absolute measure of performance that takes into account the total risk of the portfolio. It calculates the annualized return of the fund, reduces it by the yield of a risk-free investment, and further reduces it by the standardized expected market premium times the total risk of the portfolio under review.**Local Currency versus Foreign Currency-Denominated Investment Return:**This analysis calculates the return of foreign currency-denominated investments adjusted for fluctuations in the local currency against those foreign currencies. It determines whether currency fluctuations enhanced or reduced the return of the portfolio. This analysis is particularly relevant to investors who buy and sell securities denominated in currencies other than their local currency.