What Is a Performance Triangle?

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

 I hope this background helps explain the report a bit... happy studying!

 

Fast Calculation of Internal Rate of Return (in Multiple Choice Situations...), Part II

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

Given the above, is the monthly internal rate of return closest to:
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

 In fact, Modified Dietz gives a first-order approximation to the internal rate of return.   (One missing element is the time value of money.)  Plus, if you are like most people, you probably think calculating Modified Dietz returns is easier than calculating internal rate of return!

Recall the formula for Modified Dietz is:

Plugging our data into this formula, we get the following:

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

P.S. #2:  The picture above is Michael Phelps, considered to be the fastest person in water. I figured I'd use him today, since we used the fastest person on land (Usain Bolt) yesterday.

Happy studying!

Fast Calculation of Internal Rate of Return (in Multiple Choice Situations...), Part I

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

Given the above, is the monthly internal rate of return closest to:
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:

If you are given a question along the lines of the above exercise, where you are given three choices for a valid answer, rather than doing all of the steps to solve IRR that I outlined 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%:

 In the above, 69.6 represents the ending value of 69.6 million, 56.3 is the beginning value and 9.8 is the external cash flow, which occurs on the 11th day of a 30 day month.  Once we take the future value of all cash flows (i.e., the beginning value and the one contribution), and sum those future values, it is not equal to the ending value.  Thus, option a) is not the correct answer.  So, we try the return of 2.7%, which is option b):

Again, the sum of the future values does not equal the ending value, so option b) is not the correct answer.

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:

Chances are that if you apply this "process of elimination" method, you will arrive at the correct answer faster than if you executed the 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.

Happy studying!

P.S.:  the athlete in the picture above is Jamaica's Usain Bolt, the world's fastest person!

Performance with Leverage: Part II

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.

In this example, the cash basis return is 8%, and the leveraged return is 8.75%.  If you would like to review how these returns are determined, please see the previous post here.

 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: 

In this portfolio, there are two positions:

• The real estate investment, which earns a return of 8%
• The cash borrowed, which has a cost of 5%

The 500 million euro real estate investment constitutes a weight of 125% of the total portfolio value of 400 million euro at the start of the period,  Thus, the return contribution of this position is 10%, which is the weight of 125% multiplied by the return of 8%.

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.

Happy studying!

Performance with Leverage, Part I

Leverage can be a confusing topic, so I figured it is worth covering in a few blog posts.  In this first post, we'll deal with return calculations for portfolios that employ leverage.

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

 For example, let's say an investor has 400 million euro to invest but wants to invest in a 500 million euro real estate property.  If the investor limits herself to her cash at hand, she can't puchase the property.  But, if she uses leverage (i.e., borrows 100 million euro) she can acquire the property.  The investor will have to pay a cost of borrowing (we will assume that is 5% interest per year).  In this scenario:

•  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

Assume over the investment period of one year, the property has appreciated to 540 million euro.  We can calculate the cash basis return as follows:

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.

Happy studying!

Common Themes: "Dietz-Style Equations"

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 Flows

In 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:

1.1

In this equation, R is our period return, V "sub" E is our ending value and V "sub" B is our beginning value.  The numerator of this equation is the amount earned by the portfolio manager and the denominator is the amount of money available to earn the return; i.e., the basis.  In the absence of cash flows, this equation gives us a precise return.

The Problem of External Cash Flows

External 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:

1. Exactly at the start of the period
2. Exactly at the end of the period
3. Sometime during the period

 Adjusting the Equation When Cash Flows Occur at the Start of the Period

When 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:

1.2

This adjustment gives us a precise return formula for this situation.

Adjusting the Equation When Cash Flows Occur at the End of the Period

When 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:

1.3

This adjustment gives us a precise return formula for this situation.

Comparing the Equations:  Flow at Start vs. Flow at End

If 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:

1.4

This adjustment gives us a precise return formula for this situation.

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:

1.5

If we have multiple cash flows all occurring either at the start of the period or all occurring at the end of the period, we simply sum the cash flows:

1.6

Generalizing the Equation

We can generalize the two equations in (1.6) to come up with a single equation to cover both circumstances:

1.7

In this equation, we apply (i.e., multiply) the cash flow sum by a weight:

• 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.

Thus, we now have precise formulae for calculating return for two of our three scenarios:  when flows all occur at the start of the period and also when flows occur at the end of the period.

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):

Note that the weight of 1/2 is between 0 and 1.

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

1.8

 Thus, we now have equations to cover all three scenarios:

1. Exactly at the start of the period (equation 1.6 with a weight of 1)
   
2. Exactly at the end of the period (equation 17. with a weight of 0)
   
3. 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.

Happy studying!

Free webinar on Ethics from CFA Institute!

As you know, Ethics is a component of the CIPM curriculum - and one that candidates should not take lightly. 

CFA Institute announced today that they will be holding a free webinar on Ethical Decision Making.  I recommend that both Principles Level and Expert Level candidates consider attending one of the sessions.  I plan to attend one of the sessions to brush up on my own understanding of ethics.

I also recommend the session to all investment professionals, but especially holders of the CIPM or CAIA designations

(picture above of Plato and Aristotle...)