Performance measurement is an important task for both investors and investment managers. Whether it is for a mutual fund, leveraged fund, derivative or fund of funds, performance must be calculated. Returns are most commonly quoted in absolute terms, but in reality they should always be compared to the strategy, and ultimately to the benchmark, they are designed to beat. Goals for each investor, whether they are individual or institutional, come in all shapes and sizes:

- risk-adjusted return
- maximum income
- preservation of capital
- liability matching

These varied goals mean that performance must be calculated frequently and accurately. Although institutions use industry standards to report performance, individual investors typically use less-sophisticated methods. Let's take a look at some time-tested ways to calculate investment returns and determine how you can use them to judge your portfolio's profitability. (Read *Measure Your Portfolio's Performance* for more information.)

**Return Calculation**To calculate rates of return accurately, you must account for the various transactions that occur during the period being evaluated. Buys, sells, income, distributions and contributions can all be included in the calculation, but the beginning market value (BMV) and ending market value (EMV) are the most important figures. All else being constant, a rough estimation can be made of the return.

To obtain these data points, asset values must be accurate:

- Stock prices tend to be quoted based on the last trade of the day from the exchange on which they are traded. This value is rarely disputed, because it represents the price of the stock at that point.
- Bonds, on the other hand, carry some variances in pricing. Because most bonds do not trade on an exchange, their prices are usually fed off a tape in a matrix pricing model.
- Total bond market values can also include accrued income, which might or might not be included in the market value (MV), depending on the style of reporting. (Read more in
*Bond Market Pricing Conventions*.)

In addition to variation in prices, there is also the debate over using trade date or settlement date as the method for evaluating MV. The difference between the two is that securities traded within settlement date of the month-end can "hang" out because they have not been delivered. Because the trade date is a better representative of what actually happened, the transactions can be backed into the portfolio's transaction for the month as if they had settled. This provides a clean accounting of the portfolio's actual value.

**Time-Weighted Rate Of Return (TWRR)**Prior to the 1960s, returns were presented in various formats without much consideration as to when the cash flows were accounted for. Time-weighted rate of return takes cash flows into account for each period and standardizes them so they can't hinder or help comparative performance. The TWRRs for each period are linked together to create the geometric return, instead of the arithmetic return, which is just the average. (Learn more about why the method of calculating returns matters in

*All Returns Are Not Created Equal*.)

The calculation for TWRR is as follows:

TWRR=(EMV-BMV-CF)/(BMV+.5CF) |

Where:

BMV = beginning market value

EMV = ending market value

CF = net cash flows

This calculation is also known as the Dietz algorithm.

The simplicity of the formula moves the net contribution to the middle of the month, thus standardizing the cash flow to prevent disruption of the true return. A modified version of the formula weights the cash flow on the actual day it way received or distributed. The modified Dietz method pegs the net cash flows to the exact days they occurred. For example, if a cash flow was received on day 23 of a 30-day month, a weighting proportional to the remaining days in the month would replace the .5 in the formula for the mid-month.

In this case:

(30-23)/30 = .23 |

This process would not be necessary if the portfolio was valued daily and performance could be calculated.

Because dividends and interest income are not considered cash flows, they are not accounted for in the formula, but are automatically reflected in the EMV. That is why accrual methods for valuing a portfolio are important if any payments are pending. (Read more about accrual accounting in *Operating Cash Flow: Better Than Net Income?*)

**Geometrically Linking TWRR**After the intervals are computed (preferably monthly), they can be linked together geometrically to form quarterly and annual rates of return. Linking returns requires the positive and negative values to be relative, so they are added to 1.

LR = ((1+r1)x(1+r2)x(1+r3) |

Where:

LR = linked return

r = period TWRR

This compounds the TWRR for an accurate and comparable rate of return.

Here is an example of linking returns:

PERIOD |
TWRR |
TWRR+1 |

R1 | 6.5 | 1.065 |

R2 | 5.5 | 1.055 |

R3 | 2.2 | 1.022 |

The arithmetic quarterly return for these three months would be 14.2%, while linking the returns produces a slightly larger number:

(1.065)*(1.055)*(1.022) = 14.8% |

The highest return for the quarter counted more because it occurred early in the data stream. The reverse would happen if a large negative return occurred early in the stream. For either outcome, using TWRR and linking the returns geometrically is a much more accurate method for calculating performance.

Until now, we have been discussing the performance of the entire fund or portfolio. In reality, to be able to compare asset class returns, they must be segmented out. For example, when comparing the performance of a large-cap equity manager to a benchmark like the S&P 500, the portfolio would be at a disadvantage because it would most likely have some cash in the portfolio. The index, on the other hand, is calculated on a fully invested basis. The process for segmentation is simple: just process the market values of each asset class in the portfolio separately - stocks, bonds and other asset classes.

**Verification Of Returns**For the most part, published returns are audited and verified. Mutual funds are marked to market daily, so their valuation and performance are somewhat transparent. For the rest of the world, investors kind of have to take the asset manager's word for it. One comforting footnote along with posted performance numbers is that the assetmanager's calculations were compliant with the CFA Institute's Global Investment Performance Standards (GIPS). (Read more on this topic in

*A Guide To Global Investment Performance Standards*.)

Although this does not provide any sort of guarantee or legal verification, it does let you know that the investment company representing the numbers has taken the time to apply the standards relating to valuation time-period minimums, the treatment of cash flows, compiling and segmenting returns by asset classes and the handling of performance numbers calculated by third parties.

**Conclusion**Accurately calculating performance is required to reflect how one portfolio or fund ranks against an absolute target and a relative target, such as an index. Although the process is not that complex, it does require some time to process; an entire industry is devoted to just that business. Because cash flows can affect how one portfolio performs, the TWRR method properly weights the cash flows when they occur so as not to upset the real return. After computing the TWRR for each period, the numbers can be linked geometrically to produce quarterly and annual data. These methods are industry-standard in the institutional realm; as individual investors become more sophisticated and aware of the nuances of calculating performance, they will upgrade their methods.