## Friday, September 20, 2013

### 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:

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:

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:

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:

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:

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:

### Generalizing the Equation

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

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

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!