Performance Calculation Blog

It should be a simple enough question: what is the performance of your investment fund? However, as the saying goes, the devil is in the details. For Investors it is imperative to earn a competitive return. This is best illustrated through a fund’s performance, which can seem simple to display but can be especially challenging for different types of funds and also very challenging when comparing different fund strategies with varying fund attributes. This can also become a persistent obstacle for portfolio managers when trying to determine what metrics they should display for their fund, aside from the core performance metrics.

Fund performance varies by different fund strategies as well as by the different metrics used to illustrate it, even within each of those strategies. In this post we wanted to highlight some different methodologies fund managers use to capture performance for their funds. Fund performance can diverge significantly depending on the data provided and also how that data represents the performance of fund strategies. Therein lies one of the key challenges, in that performance when measured across different types of investment funds is not always consistently applied, which can make finding a relevant comparison quite a difficult exercise to accomplish at times.

When capturing the performance of a fund it is important to illustrate cumulative (also called effective or absolute) and/or annualized performance. At first blush capturing historical performance seems rudimentary, however, many managers, especially those managing institutional funds, address specific performance considerations that can be challenging to adopt. Looking at the performance time horizon should be measured by capturing an investment track record over a sufficiently long period of time. This may be since fund inception or, if the fund has been in existence for many, many years, common practice may be to capture performance over the most recent 10 year time period. Performance time horizon is best captured in full annual periods, while also capturing at least quarterly (and often monthly and daily) performance for the most recent calendar year. And because the goal in the end is to compare like funds, YTD and 1Y should always be part of the performance displayed.

The second key dimension of fund performance criteria should be to evaluate if cumulative (effective or absolute) vs. annualized performance should be displayed. Usually both of these metrics are displayed, but for time periods that may not be comparable, such as since inception, annualized returns are very important so that investors can compare a common view of performance. This is one of the most commonly used metrics in the mutual fund world. On the other hand, cumulative performance can be a very important consideration for private investors. Annualization for time frames under one year is frowned upon by the institutional investor community, as highlighted in the CFAI Global Investment Performance Standards (GIPS).

An important and more detailed consideration for this second key dimension when using annualization is the need to define day count conventions. For this you need to define two parameters: First, how do you count the days which fall in the observed time period. Second, how many days make up a full time period (when annualizing a complete year). Usually ACT/ACT is common – so the actual amount of days in the observed time period is looked at and the year is made up of 365 or 366 days (depending on if it is a leap year or not). Other conventions like 30/360, ACT/360, ACT/365 or ACT/365.25 are also used. The cumulative impact on performance for day count conventions might be minimal, but should be considered nonetheless.

The third dimension of performance we wanted to look at today is from a financial math perspective, viewing linear vs. discrete vs. continuous performance. Usually discrete performance is shown (which is intuitive for a broad audience). Only for stats based on performance figures does it sometimes make sense to use continuous compounding.
So, depending on what audience you are marketing to, performance figures should vary also to show your fund’s strength, strategies and expertise.

And lastly, investment performance cannot be captured in isolation – instead it should be considered in the context of the riskiness of the investment fund. How these returns perform relative to risk can give good perspective on the best overall performance, taking into account both risk and return. We will be highlighting some of these risk considerations next time, so keep an eye peeled for our next blog post, which will cover volatility and its calculation, the most wide-spread risk measurement in finance. Also, if you’d like to read more about this topic and have future articles delivered conveniently to your inbox, please sign up for our newsletter.



For those of you looking to dig deeper into how the performance calculations in this post were calculated, the following footnotes contain the formulas we utilize to calculate and annualize linear, discrete, continuous returns:

As we know from interest calculation, given an annual linear interest rate R_l(t,T) at time t for time period t until T, with (T-t) the time difference in years, 100 units increase to

    \[100 \cdot [ 1 + R_l(t,T) \cdot (T-t) ].\]

Given a discrete interest rate R_d(t,T), 100 units increase to

    \[100 \cdot [1 + R_d(t,T)] ^ {(T-t)}.\]

Given a continuous interest rate R_c(t,T), 100 units increase to

    \[100 \cdot e ^{R_c(t,T) \cdot (T-t)}.\]


Thus, for effective performance the following holds:

Given prices P_t at time t and P_T at time T , the effective linear performance is calculated in a one-period setting using

    \[PE_l(0,1) =\frac{P_1-P_0}{P_0} = \frac{P_1 }{ P_0 } - 1 .\]

When we look at a multi-period setting we need to account for the fact that earned interest is not allowed to be reinvested for linear returns. This makes this concept a pure theoretical one. So instead of looking at one complete time period with interest earned at the end you might interpret a linear multi-period setting as the following: Earning interest in each period and putting it aside, so not reinvesting it. In the end of the last period you sum up all the saved interest earned until then.

For the multi-period setting, the effective linear performance can be calculated using the following formula. The part -(P_t-P_0) reflects that all interest earned so far is put aside and is not reinvested:

    \begin{flalign*} PE_l(t,T) &=\frac{P_{T}-(P_t-P_0)}{P_0}-1 \\ &=\frac{P_{T}-P_t+P_0}{P_0}-1 \\ &=\frac{P_{T}-P_t}{P_0}+\frac{P_0}{P_0}-1 \\ &=\frac{P_{T}-P_t}{P_0} \end{flalign*}

The effective discrete performance PE_d(t,T) is in one-period as well as multi-period settings calculated using

    \[PE_d(t,T) = \frac{P_T-P_t}{P_t} = \frac{P_T }{ P_t } - 1 .\]

It assumes that in the end of every period interest is paid and then reinvested. For the effective continuous performance interest is paid continuously and not only at discrete points in time, e.g. at the end of each period. The effective continuous performance is calculated using

    \[PE_c(t,T) = ln \left( \frac{P_T }{ P_t} \right) .\]


Annualizing those returns means extra- or interpolating the effective performance calculated based on the time frame t until T to a fictive time frame of exactly one year.

Given the effective linear, discrete and continuous performances PE_l(t,T), PE_d(t,T) and PE_c(t,T) for the time period t until T, the annualized linear, discrete and continuous performances are given by

    \[PA_l(t,T) = PE_l(t,T) \cdot K,\]

    \[PA_d(t,T) = (1 + PE_d(t,T)) ^{K} - 1,\]

    \[PA_c(t,T) = PE_c(t,T) \cdot K\]

with K the annualization factor. For calculating this annualization factor the time difference in years (T-t) is converted into the time difference in actual days, for simplicity let’s denote this as \Delta_d .

Then it holds for day count convention ACT/365:

    \[K = \frac{365 }{\Delta_d }\]

For  ACT/ACT we have

    \[K = \frac{365}{\Delta_d} \text{   or   } K = \frac{366}{\Delta_d}\]

depending if T is part of a leap year or not.

For 30/360 it is more difficult to calculate K, as the 31st day of the respective months are ignored and February is extended to a month of 30 days. So it depends on where T and t lie to be able to calculate this exactly.



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