## Sunday, September 22, 2013

### 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!

## Saturday, September 21, 2013

### 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!

## 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!