Wednesday, December 4, 2013
Saturday, October 19, 2013
After the game, legendary Dodgers broadcaster Vin Scully was quoted as saying something to the effect of, "Losing feels worse than winning feels good."
When I heard this quote, it struck me that this applies to portfolio returns as well. Which is one of the points of one of the readings at the Expert Level of the CIPM curriculum: Reading 18 - How Sharp is the Sharpe Ratio.
One of the ideas that modern portfolio theory emphasizes is that we expect returns to fall in a symmetrical, normal distribution. In a way, the idea of efficient portfolios at least implies that this symmetry is desirable. One of the ideas of post modern portfolio theory, however, is that investors do not want symmetry in returns... investors definitely have a preference for the return history to have asymmetry to it.
Certain return statistics are referred to as "moments" in a distribution (history) of returns. These moments describe the shape of that history. In a return distribution:
- mean return is the first moment, describing the center of the distribution
- variance (or, alternatively, standard deviation) is the second moment, describing the range of the distribution
- skewness is the third moment, describing the tendency to have returns in the tails of the distribution
- kurtosis is the fourth moment, describing the flatness or peakedness of the distribution
- high mean returns
- low standard deviation (or variance)
- positive skewness (i.e., more returns in the right tail than a normal distribution would typically have)
- low kurtosis (a flatter distribution, which means more extreme returns (again, preferably on the right)
This reading makes the point that investors typically feel losses more painfully than they enjoy gains. Some of the risk statistics in this reading are designed specifically to focus on the number of extreme returns (painful losses or extreme wins).
The Adjusted Sharpe Ratio adjusts for skewness and kurtosis by using a penalty factor for negative skewness and excess kurtosis.
Downside deviation and downside potential focus on losing returns, while upside risk and upside potential focus on winning returns.
Omega ratio is upside potential divided by downside potential.
Conditional Value at Risk considers the size and shape of the tails of a return distribution, while Modified Value at Risk adjusts standard VaR for kurtosis and skewness. The Conditional Sharpe Ratio and Modified Sharpe Ratio are, then, modifications of the standard return to VaR formula that substitute either Conditional VaR or Modified VaR, respectively - in an attempt to focus on extreme losing returns.
This is just a sampling from your reading. Your focus for this reading is not the mathematics of these calculations, per se; rather, you should know the concepts behind why we want to measure extreme returns.
To summarize some differences between modern portfolio theory (MPT) thinking and post modern portfolio theory (post MPT) thinking:
- assumes normal distribution of returns
- tracking error (the standard deviation of excess returns) measures active risk
- returns above the mean and below the mean observation (of either absolute or relative return) are treated the same
- recognizes that investors prefer upside risk rather than downside risk
- hedge funds (and other post MPT portfolios) are designed to be asymmetric with (emphasis on) variability on the upside but not on the downside
Wednesday, October 16, 2013
(Note: I wrote this post as a guest blogger for the STP Investment Services blog, but it appears that blog is offline and we received a request for the information, so I decided to post the information directly here.)
The What, Why and Who of the GIPS Standards
In this blog post, I give you some quick and simple answers to the “what,” “why” and “who” questions that many people ask with respect to the GIPS Standards.
What are the GIPS Standards?
The GIPS Standards are voluntary global standards for the presentation of investment performance results to prospective clients. That the standards are “voluntary” and deal with presentation to “prospective clients” are two of the most important aspects of the GIPS Standards.
By voluntary, we mean that investment firms may choose to comply with the GIPS Standards – or they may choose not to. There is no governing body that forces firms to comply with GIPS. Different countries may have various laws that govern the presentation of performance results, separate from the GIPS Standards. In fact, the GIPS Standards require compliant firms to follow the law in situations where the law differs from the GIPS Standards (in such a situation firms must disclose in their presentations the manner in which the regulations differ from the GIPS Standards). It should be noted that locally, regulators may cite firms that have a false claim of compliance with the GIPS Standards. Otherwise, with compliance being voluntary, the GIPS Standards represent a form of self-regulation that the investment industry has adopted on a global basis.
It is also important to consider what the standards are not. The standards are not calculation standards or reporting standards. When we say that the standards are not calculation standards, we mean that they are not an “A-to-Z” reference manual as to how the calculation of performance must be done. Yes, it is true that there are certain basic requirements for the calculation of performance that must be met (covered in other chapters in this guide). At the same time, it is up to the firm claiming compliance with GIPS to determine (and document) its policies and procedures for establishing and maintaining compliance with GIPS – including the calculation of returns, dispersion and other performance data. As long as the requirements of GIPS are met, firms have a lot of leeway in defining how they perform the calculations. One of the most important aspects of the GIPS Standards is that firms define (and document) their policies as clearly and objectively as possible, and that they apply their policies consistently. This supports two main objectives of the GIPS Standards – fair representation and full disclosure.
When we say that the GIPS Standards are not reporting standards, we mean that the standards do not dictate the format of the firm’s compliant presentation. The standards do prescribe requirements and recommendations for the content (i.e., the presentation elements) that must go into the composite presentation, and the accompanying disclosures. It is up to the firm to format this information. So again, the firm has a lot of flexibility in creating compliant composite presentations. Again, the ideals of fair representation and full disclosure should be met.
What items must a firm show in a composite presentation?
This list is not exhaustive, but a quick summary of what compliant firms must show includes:
- Generally, time-weighted returns (TWR) that separate client contribution from manager results. For closed end real estate and private equity funds, since inception internal rate of return is shown.
- Annual composite returns (a composite is the aggregation of accounts managed to the strategy).
- A measure of the internal dispersion (i.e., range) of returns of portfolios within the composite.
- As a measure of risk, the variability (standard deviation)of the composite’s historical returns.
- The amount of assets in the composite each year and the number of portfolios in the composite
- The amount of firm assets each year.
- Disclosures about the firm and the given composite designed to help the reader of the presentation understand the firm, the composite, and the performance history being shown.
The GIPS Standards promote the comparability of manager performance across firms and across borders. By requiring firms to show the same information, the prospect is better equipped to compare managers and make an informed decision as to which manager it should hire.
Why are there standards for performance presentation to prospective clients?
The GIPS Standards are a direct “descendent” of other predecessor standards that were created in various local areas dealing with presentation of performance results to prospective clients. The GIPS Standards, introduced in 1999, are global standards that incorporate the best practices from the participating local country sponsors.
In the late 1970s and early 1980s, there were several abuses that were becoming commonplace as far as how firms presented their performance to prospective clients. Some of the typical problems were:
- Presentations that only showed the firm’s best performing accounts
- Returns calculated based on unsubstantiated pricing
- Annualization of partial annual periods
- Reporting/presentation of best performing periods, omitting poor performing periods
- Comparisons of performance with either low-return, or inappropriate benchmarks
- Calculations that did not segregate manager returns from the client contribution
- Presentations created by marketing departments that underplayed unfavorable data and highlighted persuasive elements
Because of these problems, prospective clients had difficulty making informed, sound decisions regarding what investment manager they should hire.
Key events in the history of performance standards development:
- 1966: Peter Dietz’s seminal work, “Pension Funds: Measuring Investment Performance,” was published, introducing what came to be known as the time-weighted return.
- Late 1960s: the Bank Administration Institute published return calculation guidelines based on Dietz’s work.
- 1987: Financial Analysts Federation created the Committee for Performance Presentation Standards (CPPS). Key recommendations from their report:
- The use of time-weighted return was recommended.
- Presentation of performance gross-of-fees was recommended.
- The report recommended the inclusion of cash in portfolio returns.
- Construction and presentation of asset-weighted composites was recommended.
- 1990: The Association for Investment Management and Research (AIMR) board of governors endorse the AIMR-PPS.
- 1993: The AIMR-PPS is published.
- 1997: 2nd edition of AIMR-PPS published
- 1999: The first edition of the GIPS Standards was published.
- 2005: The second edition of the GIPS Standards was published.
- 2010: The third and current edition of the GIPS Standards was published, going into effect on January 1, 2011.
- 2012: Guidance Statement on Alternative Investment Strategies and Structures issued
- 2013: Exposure draft for the Guidance Statement on the Application of the GIPS Standards to Pension Funds, Endowments, Foundations and Other Similar Entities reeased
To whom do the GIPS Standards apply?
The GIPS Standards are voluntary standards that may be adopted and complied with by any investment firm with discretion over assets. This includes all of the traditional asset classes (cash, equities, fixed income) and alternative asset classes (commodities, private equity, real estate, hedge funds). The GIPS Executive Committee has recently introduced a document clarifying that pension funds, and other managers of managers, can claim compliance with the GIPS Standards. While the GIPS Standards are commonly followed by managers seeking institutional clients, there are also a large number of retail investment managers that claim compliance with GIPS.
How does compliance with GIPS benefit managers, beyond being hired?
Compliance with GIPS can benefit investment firms in many ways, including:
- providing track record transparency and the ability to have a full/fair review of performance results for internal purposes
- creating a framework within which the manager’s firm can document the decisions made and the justification behind them when a new scenario occurs in the performance processing and/or investment operations of the firm
- allowing a better process for ensuring the marketing literature reflects the underlying information
- promoting an improved reputation due to the recognition of GIPS compliance
How does the existence of GIPS benefit investors?
Some of the benefits of GIPS for investors include:
· An enhanced ability to compare the performance of strategies among managers
· The improved likelihood that investors can understand the information in managers’ presentations and the data behind it; thus, they are able to ask relevant questions to enhance their understanding of the strategy
What is Verification and How Does It Add Value for Investment Firms?
Verification is the use of an independent third party to test an investment firm’s claim of compliance with the GIPS Standards. Verification tests two things, specifically. First, it tests whether the firm has complied with all of the composite construction requirements of GIPS on a firm-wide basis. Secondly, it tests whether the firm’s policies and procedures are designed to calculate and present performance in compliance with the GIPS Standards. Firms that claim compliance with GIPS are not required to be verified, but for investment managers seeking institutional business, it has become a de facto requirement. Most institutional investors will inquire as to whether managers they are considering a) claim compliance with GIPS, and, b) have been verified by an independent third party. Managers are often eliminated from consideration if they have to answer no to these questions. Thus, establishing and maintaining compliance with GIPS, accompanied by verification by an independent third party are essential to investment managers seeking to grow their business.
I hope this post helps give you an introduction to the GIPS Standards and the concepts of compliance and verification. If you have any questions on these topics, please feel free to contact me at firstname.lastname@example.org.
Tuesday, October 15, 2013
OK, I'm just kidding... there is no grand battle here between economists (and Nobel prize winners) Eugene Fama and William Sharpe.
That being said, CIPM candidates at the Principles level *are* expected to deal with the following Learning Outcome Statement regarding the work of these economists:
- Compare Sharpe's return-based style analysis and Fama and French's three-factor model
First, it might be helpful to review what style analysis is. Style analysis is a process that is used to try to classify either a portfolio or an individual stock, based on traits (or characteristics) of the given portfolio or stock. It is a process that evolved over some time, and major advances in this area came from separate contributions by William Sharpe and Eugene Fama (along with Kenneth French).
Equity style itself is based on the idea that stocks that share certain traits tend to have similar returns. Style attempts, then, to aggegate stocks at an intermediate level between the broad market (on the macro end of the spectrum) and by industry or other small groupings (at the micro end of the spectrum). If we think of the linear market equation as being a predecessor to style analysis, that equation attempted to describe stock or portfolio performance in terms of a single factor, the market factor (or beta):
Work by researchers in the 1970s found that two factors explained a large portion of stock returns: capitalization (i.e., size) and valuation. The capitalization factor is based on the idea that large cap stocks perform differently than small cap stocks, generally speaking. The valuation factor is based on the idea that stocks that sell for low multiples of earnings or book value based variables tend to perform differently than stocks selling for high multiples of earnings or book value based variables.
In the American stock markets, we typically categorize styles based on size on one dimension (large cap, mid cap, small cap) and value on the other dimension (value, core/neutral, and growth). Combinations of these are also considered styles or sub-styles (e.g., large-cap growth, small cap value, etc). This is a system that was popularized by Morningstar in the 1990s, when they began to classify mutual funds in this fashion.
William Sharpe, in 1988, published work based on his methodology for using regression analysis to explain the performance of any portfolio or stock by doing linear regression to find the appropriate mix (or, equivalently, the allocation) of style indexes. The major contributions of Sharpe with respect to style analysis, as cited in your reading, are:
- All portfolios except style index funds are a mix of styles, and that style is a continuum
- His research allowed plan sponsors to separate manager style bets from the manager’s pure alpha
- His work focused on price/book variables while that of his predecessors' focused on price/earnings variables; this influenced future construction of style benchmarks
where SMB is the size effect (small minus big) and HML is the value effect (high multiples minus low multiples).
So, to summarize, Sharpe and Fama/French have the following similarities:
- both are methods of returns-based style analysis; and , thus, use style as a means of understanding performance
- both used book value based variables rather than earnings values to explain the fundamental value of companies
A couple of points of contrast:
- Sharpe and Fama/French used different factors in their returns-based style analysis approaches. Sharpe explained performance as a weighted scheme of style indexes. Fama/French explained performance in terms of the three factors (beta, size, value) and coefficients assigned to those factors.
- Your reading tells you that Fama/French used book value based variables, whereas it points out specifically that Sharpe used price/book (aka P/B), which was a major change from predecessors and shaped the development of style indexes into the future
P.S.: Yesterday I mentioned that Eugene Fama's work that recently won the Nobel prize in economics can be found here. If you want to read William Shape's Nobel prize winning work, you can find that here.
Monday, October 14, 2013
Congratulations to Professor Eugene Fama of the University of Chicago! The Royal Swedish Academy of Sciences just announced that Mr. Fama (along with Robert Schiller and Lars Peter Hansen) has won this year's Nobel prize in Economics, for their work on developing methods to study trends in stock, bond and housing markets. For more information on the announcement, see here.
Of course, Professor Fama should be familiar to CIPM candidates and certificants for his work developing the Fama-French three factor model. In honor of Mr. Fama's accomplishment, I will spend some time discussing the Fama-French model in a post later today. But first, I wanted to separately acknowledge this accomplishment by an individual that has contributed to the world of performance measurement. Congratulations!
P.S.: Their paper can be found here.
Friday, October 4, 2013
At the CIPM Expert Level, the topic of "performance triangles" is covered, and there is one specific Learning Outcome Statement on the subject:
Demonstrate the use of performance triangles vs. benchmarks in assessing a manager's track record
Having said that, the curriculum does not give much background, and it's not uncommon that students in our prep classes have questions on this subject. Most have never seen a performance triangle (sample pictured above) prior to enrolling in the CIPM Program. Thus, I thought I'd devote a post to this subject.
Performance triangles are covered in the performance appraisal part of the curriculum. Remember that performance appraisal is the process of trying to evaluate manager skill. Thus, performance triangles are designed to be a report tool to assess manager returns. Recall also that with performance appraisal, we define skill as returns that exhibit magnitude and consistency over time.
Performance reports tend to focus on single period results (e.g., historical annual returns, or monthly returns or quarterly returns, etc.). Occasionally, reports may show cumulative periods in addition to the single period returns (e.g., latest year, latest 3 years, latest 5 years, inception to date), but the end date of those periods is the same. Thus if one wants to view single period returns and cumulative returns through a different date, a different report must be run.
A simplistic approach to evaluating a manager is to look at his/her returns (or excess returns) and conclude the manager has skill if those returns are generally positive.
A more comprehensive approach is to analyze the pattern of returns over time... the performance triangle is a tool for this purpose.
The performance triangle is a multi-period performance report that gives a view into how single period returns affect cumulative (i.e., compound period) returns over time.
The construction of the triangle is as follows:
- On the horizontal axis are a set of period end dates, in ascending order from left to right. These are “from dates.”
- On the vertical axis are the same set of period end dates, in descending order from top to bottom. These are “to dates.”
- In the “report grid” are the cumulative returns for each period (based on the from and to dates). The data in the grid naturally form a triangle based on which periods have returns (grid will be the "upper left triangle of the grid; the bottom right is naturally blank.
- The hypotenuse of the triangle has all of the single period returns. The rest of the grid is cumulative performance from the "from date" (on the horizontal axis) through the "to date" (on the vertical axis).
Thus, with a single report, a reader can glean things such as
- how the manager's performance is in single periods and in various cumulative periods through different ending dates
- trends in the manager's results
- whether the manager's positive or negative performance cumulatively is due to consistent good (or bad) results as opposed to extreme good (or bad) returns
- recovery time after losing returns, as well as the amount of time it took to lose the value gained by positive returns
Thursday, October 3, 2013
In my last post, I covered a "fast" way to solve multiple choice internal rate of return exercises. In today's post, I look at a second quick method.
Recall the details from the last post:
- Account market value on 3/31 is $56.3 million
- Account market value on 4/11 is $58.2 million (prior to contribution on same day)
- Contribution of $9.8 million is made on 4/11
- Account market value of $69.6 million on 4/30
a) 9.3%, or
b) 2.7%, or
c) 5.6 ?
The second fast method involves use of the Modified Dietz formula. Modified Dietz is, in fact, a money-weighted return. It exhibits the following traits of money-weighted returns:
- the portfolio is valued only at the start and end of the period
- interim external cash flows are day-weighted within the evaluation period
Recall the formula for Modified Dietz is:
From this it is clear that the answer is option c), the return of 5.6%. That wasn't painful at all! This method may be even faster than the last method!
Modified Dietz as an approximation to IRR should work fine for fairly short evaluation periods and non-extreme cash flows.
Hope this helps!
P.S. #1: Modified Dietz can, of course, be used to calculate sub-period returns, which we then geometrically link to get the time-weighted return. Recall the following traits of the time-weighted return:
- geometrically linked sub-period returns
- revaluation on a frequent basis (rather than just simply at the start and end of the period)
- if valuations are done on external cash flow dates, the geometrically linked return is the so-called "true time-weighted return"
- if valuations are done frequently but not on the cash flow dates, then the geometrically linked return is an estimate of the time-weighted return
Tuesday, October 1, 2013
I have covered the steps to calculating internal rate of return in a few different posts on the blog, including here, here and here. I have also covered the steps to use the cash flow worksheets of the TI BA II Plus and the HP 12 C.
You may have read the curriculum and the steps in the blog posts cited above, and you may have thought to yourself, "Wow, that's a lot of steps!"
It is possible that a question could be phrased in such a way (given that you are taking a multiple choice exam) that allows you to use a strategy to quickly calculate the internal rate of return.
For example, assume the question is as follows:
- Account market value on 3/31 is $56.3 million
- Account market value on 4/11 is $58.2 million (prior to contribution on same day)
- Contribution of $9.8 million is made on 4/11
- Account market value of $69.6 million on 4/30
a) 9.3%, or
b) 2.7%, or
c) 5.6 ?
Recall that internal rate of return is the rate R that equates the ending market value for a period with the sum of:
- the future value of the beginning market value growing at the rate R for the entire period
- each contribution and withdrawal growing at the rate R for the fraction of the period that remains at the time of the given cash flow.
Thus, internal rate of return is the rate R that satisfies the following equation:
here, you could simply plug each of the possible answers into the above equation, and see if you get equality. If you do, that's the correct answer. If you don't get equality, you should try a different multiple choice option.
To illustrate, using the first possible answer, option a), of 9.3%:
At this point, if you trust your calculations and wanted to save time, you should be able to conclude that option c), the return of 5.6%, is the correct answer. But, if you want to confirm this, you can take a few more minutes to prove that to yourself:
longer set of steps to calculate IRR. Now keep in mind that you may encounter an exam question that is not structured in a way that you can do these "fast" steps, but if it is, and if you can recognize that, you have a good process to use.
Hopefully this gives you a good test taking tip, and also reinforces your understanding of the internal rate of return. Later this week, I will show you a second "fast" method.
P.S.: the athlete in the picture above is Jamaica's Usain Bolt, the world's fastest person!
Labels: CIPM, CIPM expert, CIPM Principles, internal rate of return, IRR, performance measurement, private equity, rate of return, real estate, return measurement, SI-IRR, The Spaulding Group
Sunday, September 22, 2013
Yesterday, I covered return calculation for a portfolio with leverage. To review, the background information is:
- The investor wants to acquire a 500 million euro property but only has 400 million in cash
- The investor borrows 100 million euro in order to acquire the property; cost of borrowing is 5% per year.
- Over a one year period, the property appreciates in value by 40 million.
So to summarize, by using leverage, the investor has amplified the return of 8% (the cash basis return the investor would realize if they acquired a 400 million euro investment in the 500 million euro property) to realize a levered return of 8.75%.
In this post, I look at contribution to see the relationship between the investment and the leverage, with respect to return impact.
Recall that the return of a portfolio is the sum of the contribution from all of the positions in the portfolio:
- The real estate investment, which earns a return of 8%
- The cash borrowed, which has a cost of 5%
The cash obligation (the borrowed cash of 100 million euro) has a weight of -25% of the total portfolio. The return on this position is the interest cost of 5%. Thus, the contribution of the leverage is -25% multiplied by 5% which equals -1.25%.
The portfolio return is, therefore, the sum of the contribution from the positions: 10% plus -1.25% is the same 8.75% that we calculated using portfolio values in the previous blog post.
The data for the return contributions are shown here, to summarize:
Hopefully this second view into the calculation of portfolio return helps you to understand how leverage can amplify returns. From a contribution standpoint, the use of leverage has been effective because:
- the underlying assets constitute more than 100% of the portfolio value, which increases the contribution from 8% to 10%
- the cash borrowed is a short position, so the interest cost will erode the contribution amplification from the underlying assets. But, because the interest cost of 5% is less than the 8% return of the underlying assets, there is still a benefit to the use of leverage. The contribution of the leverage is -1.25%, eroding the 10% contribution from the underlying assets, resulting in an overall contribution of 8.75%. Thus, there was a 75 basis points benefit in this example due to the use of leverage.
Saturday, September 21, 2013
Leverage is the use of borrowing, typically with an intent to amplify investment gains (and thus, returns). The use of leverage is also sometimes referred to as margin borrowing.
When a portfolio uses leverage, we can refer to two different returns:
- the leveraged return is the actual return based on the portfolio's total invested capital
- the cash return is the unleveraged return; i.e., the return on the underlying assets, ignoring the use of leverage
- the cash return is the return on the 500 million euro property she acquires
- the levered return is the return on her entire portfolio; i.e., her 500 million euro property and her -100 million cash borrowed
Note that this cash basis return is the same return that the investor would have if she was somehow able to purchase 400 million worth of the 500 million euro property.
The levered return, however, is higher:
The investor has successfully amplified returns. The levered return of 8.75% is higher than the 8% cash basis return. This is true because the return on the underlying asset (i.e., 8%) is higher than the cost of borrowing (the interest cost of 5%).
Hope this example helps you understand the impact leverage can have on returns. I'll give a different view on this in the next post.
Friday, September 20, 2013
For today's post, I'd like to review some of the "Dietz-style" formulae we use to calculate true time-weighted return and estimated time-weighted return. I've never actually seen the formulae presented this way, but hopefully doing it in this fashion will help candidate see that we are using essentially the same basic formula in all of the following cases... just applying them in different ways.
Note: the term "Dietz-style" is my own term... used to reference the basic equation style we see with the Original Dietz and Modified Dietz formulae.
Return Calculations in the Absence of Cash FlowsIn the absence of cash flows, return calculation is simple. We measure the change in value of the assets from the beginning of the period to the end of the period, and compare (i.e., divide by) the beginning value:
The Problem of External Cash FlowsExternal cash flows cause the previous equation to not be completely accurate, because the cash flows change the amount of money available to the manager to earn return. Specifically, contributions increase the amount available to earn return, and withdrawals decrease that amount.
Cash flows can occur in one of three ways:
- Exactly at the start of the period
- Exactly at the end of the period
- Sometime during the period
Adjusting the Equation When Cash Flows Occur at the Start of the PeriodWhen a cash flow occurs exactly at the start of the period, this logically means that the beginning value has been adjusted to include the cash flow. Thus, the corresponding adjustment we can make to our initial equation is to add the cash flow to the beginning value in all instances that it appears in the formula:
Adjusting the Equation When Cash Flows Occur at the End of the PeriodWhen a cash flow occurs exactly at the end of the period, this logically means that the ending value implicitly includes the cash flow's impact. Thus, the corresponding adjustment we can make to our initial equation is to subtract the cash flow from the ending value in all instances that it appears in the formula:
Comparing the Equations: Flow at Start vs. Flow at EndIf we compare the numerators of the two equations (1.2 and 1.3), after evaluating the parentheses in both cases, we see that the numerators are equal:
The denominator of the equations are different, however. Basically, the cash flow is part of the denominator if it occurs at the start of the period, and it isn't part of the denominator if the flow occurs at the end of the period. Thus, we can rewrite our equations 1.3 and 1.4:
Generalizing the EquationWe can generalize the two equations in (1.6) to come up with a single equation to cover both circumstances:
- If the flows all occur at the start of the period, the weight is 1
- If the flows all occur at the end of the period, the weight is 0.
What If Flows Occur During the Period?The Original Dietz and Modified Dietz equations are extensions of the formula (1.7) above to handle the case where cash flows occur during the period. In both of these cases, the formula gives us an estimate of the manager's return.
In the case of Original Dietz, we assume all cash flows occur in the middle of the period; thus, a weight of 1/2 is applied to all cash flows (through multiplication):
In the case of Modified Dietz, rather than assuming that all cash flows occur at a single point in time (start, middle or end of the period), we will consider the timing of each individual cash flow, and apply (through multiplication) a weight that corresponds to the fraction of the period that remains at the time of the cash flow. Thus, a weight (W "sub" i) is calculated for each cash flow F "sub" i using the following equation:
(CD - D) / CD
where CD is the number of calendar days in the period and D is the day of the cash flow within the period. For example, if the period is January and the flow occurs on the 10th of January, then CD = 31 and D = 10. Note that this assumes that the given cash flow occurs at the end of the day. Some prefer to assume the cash flow occurs at the end of the day, in which case the weight is calculated as:
(CD - D + 1) / CD
Note that these weights will be between 0 and 1 in all cases. Rather than the entire period remaining at the time of the cash flow (i.e., a flow at the start of the period which implies a weight of 1) and rather than none of the period remaining at the time of the cash flow (i.e., a flow at the end of the period which implies a weight of 0), the cash flow occurs sometime during the period, so a fraction of the full period remains (i.e., a weight between 0 and 1).
Thus the formula for Modified Dietz is
- Exactly at the start of the period (equation 1.6 with a weight of 1)
- Exactly at the end of the period (equation 17. with a weight of 0)
- Sometime during the period (either equation 1.7, which is Original Dietz, or equation 1.8, which is Modified Dietz). Both of these equations are estimates of the return. Each cash flow's weight is a fraction somewhere between 0 and 1.
Why Is The "Case 3" Return Only an Estimate?If we want to improve our estimate and make it precise, we must break the single period into sub-periods, and revalue the portfolio at the time of the cash flow. We then calculate the return for each sub-period, and the formula for the sub-periods will either be a Case 1 (start of period) or Case 2 (end of period) situation. Geometric linking of the sub-period returns gives us the cumulative time-weighted return for the entire period; i.e., across all of the sub-periods.
Hopefully this post helps to explain the relationship between all of the above formulae. At some later date I will come back and add some numeric examples, but for now I think you can get the picture, without burdening an already long post with some math.