3.1.1 The Compounding Assumption

Compounding is the recognition of interest on interest or, more generally, earnings on earnings. Simple interest is an interest rate computation approach that does not incorporate compounding. But returns are often compounded. For example, earning 10% over one year is equivalent to earning 9.64% per year compounded quarterly: [1 + (.0964/4)]⁴ = 1.10.

Continuous compounding assumes that earnings can be instantaneously reinvested to generate additional earnings. Discrete compounding includes any compounding interval other than continuous compounding such as daily, monthly, or annual.

3.1.2 Logarithmic Returns

Denote R as a total (non-annualized) return or rate with no compounding. Adding 1 to R forms a wealth ratio. A log return is a continuously compounded return that can be formed by taking the natural logarithm of a wealth ratio:

(3.1)

where ln( ) is the natural logarithm function, R^{m = ∞} is the log return, or continuously compounded return, and m is the number of compounding intervals per year.

For example, the rate or return that discounts a value of $110 to be received in the future to a present value of $100 expressed as a total (non-annualized) rate is 0.10. Since R = .10, then the log return (R^{m = ∞}) is 0.0953. With continuous compounding at 9.53% for one year, $100 grows to $110.

For very small returns, we can roughly think of R^{m = ∞}and ln(1 + R) as being equal to R: as R → 0 then R → R^{m = ∞} and R → ln (1 + R).

But for larger returns, simple returns (R) and log returns can differ substantially. Generally, the use of continuous compounding and log returns provides mathematical ease and generates straightforward modeling. For example, the advantages of using log returns rather than returns based on simple interest or discrete compounding are demonstrated in the next section and involve aggregation of returns over shorter periods of time into returns over longer periods of time.

3.1.3 The Return Computation Interval and Aggregation

The return computation interval for a particular analysis is the smallest time interval for which returns are calculated, such as daily, monthly, or even annually. Sometimes the length of the smallest time interval for which a return is calculated is referred to as the granularity, the time resolution, or the frequency of the return measurement. While some financial studies regarding microstructure or other very short-term trading issues compute returns as often as from tick to tick (i.e., trade to trade), most studies regarding alternative investments use daily returns or returns computed over longer time intervals, such as months, quarters, or even years.

Two common tasks in return analysis involve (1) aggregating a number of returns from smaller sub-periods (e.g., days) into one larger time period (e.g., months), and (2) determining an average return (e.g., finding an average daily return based on a monthly return). Different compounding assumptions typically require different formulas for these two tasks and can introduce substantial complexities. One way to simplify many analyses is to express all rates and returns using continuous compounding (i.e., using log returns).

Let's look at an example of aggregating short-term returns into a longer-term return. The challenge is calculating multiperiod returns from single-period returns in a way that reflects compounding and therefore the true long-term growth rate. Our example begins by using simple interest for the sub-periods. We refer to the total return of an asset over the T periods from time t = 0 to t = T as R_0,T, which can be expressed as being equal to the following product in terms of the returns of the asset over the sub-periods (R_t):

(3.2)

In most cases, this equation is not as easy to work with as the analogous equation using continuously compounded returns (i.e., log returns), which involves simple addition:

(3.3)

Equations 3.2 and 3.3 demonstrate that whereas simple periodic returns require multiplication for aggregation, log returns require only addition when they are aggregated.

For example, an asset earns a return of 10% in the first time period and 20% in the second time period. What is the total return over both time periods assuming discrete compounding and continuous compounding? Using discrete compounding, the total return is 32%, found as [(1.1 × 1.2) – 1]. If the returns had been expressed with continuously compounded returns (log returns), the process would be simplified to addition as 30%, found as (10% + 20%). Thus, an asset growing with continuous compounding for one period at 10% and a second period at 20% grows at a total rate of 30% compounded continuously.

The advantage of this additivity is useful in a variety of modeling contexts, including the computation of averages. The mean of a series of log returns has special importance:

(3.4)

When the arithmetic mean log return is converted into an equivalent simple rate, that rate is referred to as the geometric mean return. Alternatively, geometric mean returns are computed from the total (non-annualized) return over an interval as:

(3.5)

The geometric mean return should be used with care in interpreting long-term performance realizations.

3.2.1 The Challenge of Returns on Positions with Zero Value

Subsequent chapters provide an extensive discussion of forward contract prices and returns. For the purposes of this discussion, a forward contract can be simply defined as an agreement to make an exchange at some date in the future, known as the delivery date. For example, a hedge fund with an undesired exposure to receiving a payment in Japanese yen in three months and with a preference to receive that payment in euros might enter into a forward contract with a major bank. The forward contract might require the hedge fund to deliver 100 million yen in exchange for 1 million euros at a particular date, such as in three months. The hedge fund has effectively transformed its receipt of yen into a receipt of euros.

Forward contracts can usually be viewed as starting with a value of zero because the initial value of the item to be delivered is usually equal to the value of the item to be received. However, as soon as time begins to pass, it would be expected that the value of the contract would become positive to one side of the contract and negative to the other side of the contract. For example, if the value of the yen rose substantially relative to the value of the euro after the forward agreement was established, the hedge fund would perceive the commitment that it made through the forward contract as having a negative value.

Assuming the hedge fund reports its performance in euros and that the change in the yen–euro exchange rate caused a loss to the fund of 1,000 euros, the rate of return on the forward contract would need to be computed. The traditional formula for the return without any interim cash flows is:

The forward contract, however, has a starting value of zero, which would lead to division by zero. The next two sections discuss solutions to this challenge.

3.2.2 Notional Principal and Full Collateralization

One solution to the problem of computing return for derivatives is to base the return on notional principal. The return on notional principal divides economic gain or loss by the notional principal of the contract. Notional principal or notional value of a contract is the value of the asset underlying, or used as a reference to, the contract or derivative position. In the case of a forward contract on currency, it would be 100 million yen, 1 million euros, or even the value of either in terms of a third currency. Selecting 1 million euros as the notional principal, the change in value in the previous example could be expressed as:

However, the figure of –0.10% has little economic importance for the hedge fund, since it has not invested any capital into the contract. Usually a percentage loss is interpreted as being based on the amount of capital invested, so it has an intuitive meaning. The problem of calculating the rate of return when there is no initial investment is identical to the problem of calculating the rate of return on a fully leveraged position, such as when a position in a risky asset, like a common stock, is fully financed through borrowing.

To provide greater economic meaning, the return is often expressed on a fully collateralized basis. Fully collateralized means that a position (such as a forward contract) is assumed to be paired with a quantity of capital equal in value to the notional principal of the contract. Thus, the hedge fund computes the return on the combination of the forward contract and a hypothetical investment of full collateral, meaning collateral equal to the notional principal. Often a fully collateralized position has equivalent risk and return to a long position in the underlying asset using the cash or spot market.

A fully collateralized position has two components of return: (1) the change in the value of the derivative, and (2) any return on the collateral. Specifically, it is usually assumed that the investor is able to receive a short-term interest rate, such as the riskless rate on the collateral.

Defining R as the percentage change in the value of the derivative based on notional value and using continuous compounding (i.e., log returns), as discussed earlier in this chapter, the return on a fully collateralized position, R_fcoll, can be expressed as

(3.6)

where R is the change in the derivatives price divided by its previous price or notional value.

The first term on the right-hand side of Equation 3.6 is the continuously compounded percentage change in the fully collateralized position due to changes in the value of the derivative. The second term is the percentage change in the fully collateralized position from interest on the collateral. The sum represents the total return on the fully collateralized position. All three are expressed as continuously compounded rates (log returns) and are based on one period, such as a year.

3.2.3 Partially Collateralized Rates of Return

The previous section detailed the computation of return for a fully collateralized position on a derivative contract. The concept of full collateralization is typically hypothetical; the party to the derivative has usually not actually set aside the full collateral amount in a dedicated account. However, in practice, parties to a derivative position are often required to deposit specified levels of funds to partially collateralize the position. A partially collateralized position has collateral lower in value than the notional value.

Suppose that the notional principal of a derivative contract is l times the quantity of collateral required (i.e., the amount of collateral required is 1/l times the notional principal). For example, with l = 10, there would be a requirement of posting one unit of cash collateral for every 10 units of notional principal (i.e., $10,000 would be the required or other collateral for a derivative position of $100,000). The formula for the log return of a partially collateralized position, R_pcoll, reflects the same change in the derivative contract, R, but must be adjusted to reflect the reduced denominator (starting value) due to reduced required collateral (i.e., use of leverage). The amount of interest received on the collateral declines but remains constant as a percentage of the collateral:

(3.7)

The use of leverage magnifies the effect of changes in the derivative as a percentage of the money invested. This is expressed in Equation 3.7 by the use of leverage, l, to multiply the derivative's notional return, R.

3.3.1 Defining the IRR

The internal rate of return (IRR) can be defined as the discount rate that equates the present value of the costs (cash outflows) of an investment with the present value of the benefits (cash inflows) from the investment. Using the terminology and methods of finance, the IRR is the discount rate that makes the net present value (NPV) of an investment equal to zero.

Let CF₀ be the cash flow or a valuation related to the start of an investment (i.e., at time 0). CF₀ might be the cost of an investment in real estate, or in the case of private equity, CF₀ might be the initial investment required to obtain the investment or meet the fund's first or only capital infusion; CF₁ through CF_T–1 are the actual or projected cash inflows if positive and cash outflows if negative, generated or required by the underlying investment. Positive cash flows are distributions from the investment to the investor, and negative cash flows are capital calls in which an additional capital contribution is required of each investor to the investment.

A CF_T may be the final cash flow when the investment terminates, the final cash flow received from selling or otherwise disposing of the investment, or a residual valuation, meaning some appraisal of the value of the remaining cash flows related to the investment. In the case of an appraised valuation of CF_T, the valuation should be designed to represent opinions with regard to the amount of cash that would be received from selling all remaining rights to the investment. The values are denoted here with the variable CF, which usually stands for cash flows, even though they may be hypothetical values or appraised values for the investments rather than actual cash flows.

Given all cash flows and/or valuations from period 0 to period T, the IRR is the interest rate that sets the left-hand side of Equation 3.9 to zero:

(3.9)

Another view of the IRR is that it is the interest rate that a bank would have to offer on an account to allow an investor to replicate the cash flows of the investment. In other words, if an investor deposited CF_t in a bank account at time t for each CF_t < 0 and withdrew CF_t from the bank account when CF_t > 0, and if the bank's interest rate on the account was IRR, then the bank account would have a zero balance after the last cash flow was deposited or withdrawn (CF_T).

3.3.2 Computing the IRR

In some simplified cases, such as investments that last only a few periods or investments in which most of the cash flows are identical (i.e., annuities), the IRR may be solved algebraically with a closed-form solution. In cases involving several different cash flows, the solution generally relies on a trial-and-error search performed by an advanced financial calculator or computer.

A simplified example to illustrate the trial-and-error method involves an investment that costs $250 million and lasts three years, generating cash inflows of $150 million, $100 million, and $80 million in years 1, 2, and 3, respectively. The IRR is found as that interest rate that solves the following equation:

(3.10)

The trial-and-error process selects an initial guess for IRR, such as 10%, and then searches for the correct answer: the IRR that sets the left-hand side of Equation 3.10 to zero. Inserting IRR = 0.10 (10%) into Equation 3.10 generates a present value of inflows equal to $279.11 million and a value to the entire left-hand side of $29.11 million. The objective is to have the value of the left-hand side of the equation equal to zero. In the case of this investment, a higher discount rate will generate a lower net value. If the next guess is an interest rate of 15%, the value of the left-hand side of the equation declines to $8.65 million. The process continues with as much precision as required. The IRR of this investment is 17.33% carried to the nearest basis point.

Advanced calculators and computer spreadsheets perform the trial-and-error process automatically. This solution of 17.33% for the IRR can be found on most financial calculators by inserting the cash flows (using cash flow mode) and requesting the computation of the IRR or in a spreadsheet with a function designed to compute IRR.

In this example, the trial-and-error process for finding the IRR works well because any increase in the discount rate lowers the present value of the cash inflows, and any decrease in the discount rate raises the present value of the inflows. The solution to the IRR problem is illustrated in Exhibit 3.1.

Because the IRR is the discount rate that sets the NPV of the investment to zero, the IRR is represented by the point at where the NPV curve crosses through the horizontal axis. This occurs between 17% and 18% on the figure, which corresponds to the previous solution of 17.33%. There is only one solution, and it is quite easily found. If a bank offered an interest rate of 17.33%, then an investor could deposit $250 million, and withdraw $150 million, $100 million, and $80 million after one, two, and three years, respectively; and the final account balance would be $0, ignoring rounding errors.

3.3.3 Interim Valuations and Four Types of IRRs

The primary reason for using the IRR approach is that regular valuations of the investment, such as daily market prices, are not available. An IRR can be performed on a realized cash flow basis or an expected cash flow basis. A realized cash flow approach uses actual cash flows through the termination of the investment to compute a realized IRR. An expected cash flow approach uses expected cash flows projected throughout the investment's life to compute an anticipated IRR. An IRR may be computed during an investment's life using both realized cash flows and either a current valuation or projections of future cash flows.

There are four types of IRRs based on the time periods for which cash flows are available. Although these terms are not uniformly defined in practice, they are useful for our purposes:

1. LIFETIME IRRS: A lifetime IRR contains all of the cash flows, realized or anticipated, occurring over the investment's entire life, from period 0 to period T. In other words, if in the context of Equation 3.9, time period 0 is the inception of the investment and time period T is the termination of the investment, then the IRR is a lifetime IRR.
2. SINCE-INCEPTION IRRS: A since-inception IRR is commonly used as a measure of fund performance rather than the performance of an individual investment. The cash flows that would then be used in Equation 3.9 are aggregate cash flows of a fund rather than a single portfolio company. The terminal (time period T) cash flow in this case is the appraised value of the fund's portfolio at time T rather than a liquidation cash flow. Interim cash flows represent actual fund-level cash flows from liquidated investments.
3. INTERIM IRRS: The interim IRR is a computation of IRR based on realized cash flows from an investment and its current estimated residual value. The key to an interim IRR is that generally T would not be the termination of the investment; thus, CF_T is an estimated value rather than a realized cash flow. The interim IRR can be calculated on an investment purchased subsequent to its inception.
4. POINT-TO-POINT IRRS: A point-to-point IRR is a calculation of performance over part of an investment's life. All cash flows are based on realized or appraised values rather than expected cash flow over the investment's projected life. Although any IRR is calculated from one point in time to another, a point-to-point IRR would typically not be used to refer to a lifetime IRR.

For IRRs computed over a time interval that begins after the investment's inception, the cash flow in time period 0, CF₀, would be either the first cash flow paid by an investor to acquire the investment or some valuation after the investment's inception, such as an appraisal. For IRRs computed over a time interval that ends prior to the investment's termination, the cash flow in time period T, CF_T, would be a valuation such as an appraisal or the sales proceeds at a date prior to the investment's termination. Three applications follow to illustrate lifetime, since-inception, and point-to-point IRRs.

3.4.1 Complex Cash Flow Patterns

For the purposes of this analysis of IRRs, a complex cash flow pattern is an investment involving either borrowing or multiple sign changes. A borrowing type cash flow pattern begins with one or more cash inflows and is followed only by cash outflows. An example of the borrowing pattern is when an investment such as a real estate project is sold and leased back. The divestment generates current cash at the cost of future cash outflows and may be viewed as a form of borrowing. A multiple sign change cash flow pattern is an investment where the cash flows switch over time from inflows to outflows, or from outflows to inflows, more than once. An example of a multiple sign change investment would be a natural resource investment involving (1) negative initial cash flows from purchasing equipment and land to set up an operation such as mining, (2) positive interim cash flows from operations, and (3) negative terminal cash flows from ceasing operation and restoration expenses. Exhibit 3.2 illustrates the complex cash flow patterns.

In the case of borrowing type cash flow patterns, there is a unique solution (i.e., there is only one IRR that solves the equation), but the IRR must be interpreted differently. In borrowing type cash flow patterns, a high IRR is undesirable because the IRR is revealing the cost of borrowing rather than the return on investment. Also, when a trial-and-error search is performed to find the IRR, any increase in the discount rate lowers the present value of the cash outflows rather than lowering the present value of the cash inflows, as would be the case in a simple cash flow pattern. Thus, the trial-and-error process must operate in a reverse direction from the simplified investment cash flow pattern. In other words, if the net value with a given discount rate is positive, the next IRR in the search should be lower rather than higher, as occurs in the case of a simplified cash flow pattern.

In the case of multiple sign change cash flow patterns, the problems are more troublesome. Whenever there is more than one sign change in the cash flow stream, more than one IRR may exist. In other words, two or more answers can probably be found using the IRR formula. In fact, the maximum number of possible IRRs is equal to the number of sign changes. When more than one IRR is calculated, none of the IRRs should be used. There is no easy way for the IRR model to overcome this particular shortcoming.

Consider a derivative deal that ends poorly for Investor A. The derivative required a $5,000 outlay from Investor A to the counterparty to open. In the first period, the derivative generates an $11,500 cash inflow to Investor A from the derivative's counterparty. The derivative then generates a cash outflow of $6,550 from Investor A at the end of the second period, at which point the derivative terminates. The derivative's cash flows from the perspective of Investor A are given in Exhibit 3.3, assigning period 0 to the first nonzero cash flow.

This cash flow pattern changes signs twice, once from negative to positive and once from positive to negative. There are two IRRs: 3.82% and 26.20%. Both 3.82% and 26.20% satisfy the definition of the IRR because they set the present value of all cash inflows equal to the present value of all cash outflows. The net value of the present values of the cash inflows and outflows is illustrated in Exhibit 3.4. Note that the line crosses the horizontal axis twice, defining two different IRRs.

With the two IRR solutions 3.82% and 26.20%, there may be a temptation to think that the two IRRs can be somehow analyzed in unison to generate an intuitive feel for the derivative's attractiveness. But neither number is particularly useful, because the investment is really a combination of investing from period 0 to period 1 and borrowing from period 1 to period 2. In this particular case, the cash flow patterns have a positive net value between the two IRRs, using discount rates between 3.82% and 26.20%. But as a derivative, it is obvious that the cash flows to the other side of the derivative (the counterparty) would have the same numbers, but the signs of the cash flows would be reversed. In this case, the cash flows would be +$5,000, –$11,500, and +$6,550. From the counterparty's perspective, the IRR solutions would still be exactly the same at 3.82% and 26.20%. However, the deal's graph would appear as a mirror image, with negative net values between the two IRRs. As we would expect with a derivative deal, gains to one side of the contract would equal losses to the other side of the contract. Both sides would view the same IRRs because they used the same cash flows, but they would be looking at opposite cash flows and opposite net values. Therefore, using only the IRRs to decide if the derivative is beneficial is not possible.

3.4.2 Comparing Investments Based on IRRs

The previous section reviewed the difficulties of computing and interpreting IRRs when an investment offers a complex cash flow stream. But even if the investments being analyzed offer simplified cash flow streams (a cash outflow followed only by cash inflows), the IRR method of measuring investment performance has serious challenges. This section details the major challenges of comparing investments based on IRR.

The major challenge with comparing IRRs across investments occurs when investments have scale differences. Scale differences are when investments have unequal sizes and/or timing of their cash flows. When comparing investments with different scales, an investment with a higher IRR may be inferior to an investment with a lower IRR.

The following is a simple example that illustrates the problems that occur when comparing IRRs. Assume that a bank is offering high initial yields on a limited-time basis to induce investors to open a new account. Investors are allowed to open only one account. The example includes three types of accounts, each with the following interest rates and restrictions on time and amount:

• Account Type A: Receive 100% annualized interest for the first day on the first $10,000.
• Account Type B: Receive 100% annualized interest for the first year on up to $10.
• Account Type C: Receive 20% annualized interest for the first year on up to $10,000.

The IRR of alternatives A and B is 100%, whereas the IRR of alternative C is only 20%. However, alternative A has very small scale due to a time limitation of one day (timing), and alternative B has very small scale due to a cash flow size limitation of $10 (size). If annualized market interest rates are 5%, alternative A has a net present value of less than $30, and alternative B has an NPV of less than $10. Alternative C has an NPV of about $1,500, even though its IRR is only one-fifth that of the other two alternatives. The reason for this is that although all three alternatives have favorable IRRs, alternative C has much larger scale.

In this example, it is better to receive a lower rate on a large scale. In actual investing, scale differentials can be complex and subtle. In judging when a larger scale is worth a sacrifice in return, approaches to investments using the NPV method offer substantial potential in evaluating investment opportunities of different scales. But in alternative investments, especially private equity, IRR is the standard methodology, and scale differentials represent a challenge in ranking performance.

3.4.3 IRRs Should Not Be Averaged

Another challenge to using IRRs involves aggregation. Aggregation of IRRs refers to the relationship between the IRRs of individual investments and the IRR of the combined cash flows of the investments. Suppose that one investment earns an IRR of 15% and another earns an IRR of 20%. What would the IRR be of a portfolio that contained both investments? In other words, if the cash flows of two investments are combined into a single cash flow pattern, how would the IRR of the combination relate to the IRRs of the individual investments? The answer is not immediately apparent, because the IRR of a portfolio of two investments is not generally equal to a value-weighted average of the IRRs of the constituent investments. If the cash flows from two investments are combined to form a portfolio, the IRR of the portfolio can vary substantially from the average of the IRRs of the two investments.

This section demonstrates the difficulty of aggregating IRRs, and the following extreme example illustrates the challenges vividly. Consider the following three investment alternatives:

The IRRs of the three alternatives are easy to compute because each investment simply offers two cash flows: one at time period 0 and one at time period 1. Using Equation 3.9, the IRR for a one-period investment is found by solving the equation 0 = CF₀ + CF₁/(1 + IRR), which generates the equation

Inserting the values for Investment A (CF₀ = –100, CF₁ = +110) generates the IRR of 10%, shown in the IRR column. Investments B and C both have CF₀ = –CF₁, so the IRRs of both Investment B and Investment C are 0%.

One might expect that combining Investment A with either Investment B or Investment C would generate a portfolio with an IRR between 0% and 10% because one investment in the portfolio would have a stand-alone IRR of 10%, as with Investment A, and the other would have a stand-alone IRR of 0%, as in the case of either Investment B or C. But IRRs can generate unexpected results, as indicated by the following analysis:

The computations simply sum the cash flows of two investments and compute the single-period IRR of the aggregated cash flows. The IRR of combining Investments A and B is –20%, and the IRR of combining Investments A and C is +20%. The IRRs of both combinations are well outside the range of the IRRs of the individual investments in each portfolio. What generates the unexpected result in this example is that Investments B and C begin with cash inflows and end with cash outflows (i.e., they are borrowing investments). But in practice, alternative investments, such as commodity or real estate derivatives and private equity, can have cash flow patterns sufficiently erratic to cause serious problems with aggregation of IRRs.

3.4.4 IRR and the Reinvestment Rate Assumption

Even if all the investments have simplified cash flow patterns without borrowing or multiple sign change problems, the IRR does not necessarily rank investments accurately. The use of the IRR to rank investment alternatives is often said to rely on the reinvestment rate assumption. The reinvestment rate assumption refers to the assumption of the rate at which any cash flows not invested in a particular investment or received during the investment's life can be reinvested during the investment's lifetime. If the assumed reinvestment rate is the same rate of return as the investment's IRR, then no ranking problem exists.

Suppose that Investment A offers an attractive IRR of 25% compared with the 20% IRR of Investment B. As previously discussed, it is possible that an investor would select Investment B over Investment A if investment B offers larger scale, meaning more money invested for longer periods of time. But if an investor who selects Investment A is able to invest additional funds at a 25% rate of return and is able to reinvest any cash flows from Investment A at the 25% rate, then the scale problem vanishes, and IRRs can be used to rank investments effectively. In practice, there would typically be no reason to assume that cash inflows could be reinvested at the same rate throughout the project's life, so ranking remains a problem. The reinvestment rate assumption is addressed by the modified IRR. The modified IRR approach discounts all cash outflows into a present value using a financing rate, compounds all cash inflows into a future value using an assumed reinvestment rate, and calculates the modified IRR as the discount rate that sets the absolute values of the future value and the present value equal to each other.

Extensions of the modified IRR methodology can be adapted to develop realized rates of returns on completed projects or for projects in progress. In the case of a private equity or private real estate investment with known cash flows since inception and with a current estimate of value, a realized or interim IRR can be calculated using the assumption that intervening cash inflows are reinvested at the benchmark rate.

3.4.5 Time-Weighted Returns versus Dollar-Weighted Returns

The purpose of this section is to provide details regarding time-weighted returns versus dollar-weighted returns. Briefly, time-weighted returns are averaged returns that assume that no cash was contributed or withdrawn during the averaging period, meaning after the initial investment. Dollar-weighted returns are averaged returns that are adjusted for and therefore reflect when cash has been contributed or withdrawn during the averaging period. The IRR is the primary method of computing a dollar-weighted return.

When evaluating the return of hedge funds, mutual funds, or any investment, it's important to recognize the distinction between the time-weighted return, which is similar to what is reported on performance charts in marketing literature and client letters, and the dollar-weighted return, which represents what the average investor actually earned; the two can be very different.

Suppose there is a hedge fund that in year 1 starts with $100 million of AUM (assets under management). Let's further suppose that the hedge fund generates an average annual return of 20% for each of its first three years. With such a performance history, the hedge fund attracts quite a bit of new capital. Let's assume that the hedge fund attracts $200 million in new assets for year 4, another $200 million for year 5, and nothing in year 6. Unfortunately, the new capital does not help the hedge fund manager maintain the fund's stellar performance, and the manager earns 0% in years 4, 5, and 6. If we use time-weighted returns over this six-year period, the hedge fund manager has an average annual return of 9.5%:

In effect, the time-weighted return assumes that a single investment (e.g., $1) was made at the beginning of the period and was allowed to grow with positive returns and decline with negative returns until the end of the measurement period, with no cash withdrawals or additional contributions. The rate that equates the initial value with the accumulated value is the time-weighted average return, and it is somewhat near the arithmetic average annual return (in this case, 10% per year). The idea is that a single sum of money invested at the start of the first year and allowed to remain in the fund until the end of the last year would accumulate to the same value as if it had been invested at a fixed return of 9.5% per year, ignoring rounding.

But in practice, investors often contribute additional cash (i.e., make additional investments) or withdraw cash (e.g., liquidate part of the investment or receive cash distributions) during the time period under analysis. Their average returns depend on whether the amount of money invested was highest during the high-performing periods or during the low-performing periods. Dollar-weighted returns adjust the average annual performance for the amount of cash invested each year. In the case of the hedge fund, an investor who had much more cash in the fund in the early years than in the later years would earn more than an investor whose money was primarily invested in the last three years, when the fund generated 0% returns.

Dollar-weighted returns can be computed for each investor using investors' cash flows into and out of the hedge fund. The total cash flows into and out of the fund for all investors can be used as an indication of the performance of an average investor. The dollar-weighted return that individual investors experience depends on their cash contributions and withdrawals.

When the timing of the aggregated cash flows for the entire hedge fund is taken into account, the bulk of the hedge fund's assets earned a 0% return in years 4, 5, and 6. The example shows that only the first $100 million earned the great rates of return of the first three years. The $400 million that flowed into the hedge fund in years 4 and 5 earned a 0% return. When the timing of the aggregated cash flows is taken into account, the dollar-weighted return (solving for the IRR with cash flows reinvested) is only 4.3%. The IRR is found in this case with CF₀ = –100, CF₁ = 0, CF₂ = 0, CF₃ = –200, CF₄ = –200, CF₅ = 0, and CF₆ = +572.8; that is, CF₆ is found as: [(100 × 1.2 × 1.2 × 1.2) + 200 + 200].

Investment managers are best evaluated on time-weighted returns, as these managers should not be held accountable for the cash flow decisions of their investors. Investors should evaluate their own investment results using dollar-weighted returns based on the cash flows from their particular investment pattern.

3.5.1 Terminology of Waterfalls

The distribution of cash waterfalls has specialized terminology, and the terminology tends to differ between private equity and hedge funds. This section introduces most of the major terminology that is used in the remaining sections.

Cash inflows to a fund in excess of the costs of investment and the expenses of the fund represent the waterfall that is distributed to GPs and LPs. Excess revenue above expenses is referred to as cash flow or profit. In calculating profit, management fees are deducted, but any fees that are based on profitability are not deducted. (Management fees are usually deducted from the fund, regardless of profitability.)

Carried interest is synonymous with an incentive fee or a performance-based fee and is the portion of the profit paid to the GPs as compensation for their services, above and beyond management fees. Carried interest is typically up to 20% of the profits of the fund and becomes payable once the LPs have achieved repayment of their original investment in the fund, plus any hurdle rate.

A hurdle rate specifies a return level that LPs must receive before GPs begin to receive incentive fees. When a fund has a hurdle rate, the first priority of cash profits is to distribute profits to the LPs until they have received a rate of return equal to the hurdle rate. Thus, the hurdle rate is the return threshold that a fund must return to the fund's investors, in addition to the repayment of their initial commitment, before the fund manager becomes entitled to incentive fees. The term preferred return is often used synonymously with hurdle rate—a return level that LPs must receive before GPs begin to receive incentive fees.

A catch-up provision permits the fund manager to receive a large share of profits once the hurdle rate of return has been achieved and passed. A catch-up provision gives the fund manager a chance to earn incentive fees on all profits, not just the profits in excess of the hurdle rate. A catch-up provision contains a catch-up rate, which is the percentage of the profits used to catch up the incentive fee once the hurdle is met. A full catch-up rate is 100%. To be effective, the catch-up rate must exceed the rate of carried interest.

Vesting is the process of granting full ownership of conferred rights, such as incentive fees. Rights that have not yet been vested may not be sold or traded by the recipient and may be subject to forfeiture. Vesting is a driver of incentives. Vesting can be pro rata over the investment period, over the entire term of the fund, or somewhere in between, such as on an annual basis.

A clawback clause, clawback provision, or clawback option is designed to return incentive fees to LPs when early profits are followed by subsequent losses. A clawback provision requires the GP to return cash to the LPs to the extent that the GP has received more than the agreed profit split. A GP clawback option ensures that if a fund experiences strong performance early in its life and weaker performance at the end, the LPs get back any incentive fees until their capital contributions, expenses, and any preferred return promised in the partnership agreement have been paid.

3.5.2 The Compensation Scheme

A key element of the managerial compensation structure is the nature of the incentives that align interests between fund managers and their investors. Investors and fund managers have an agency relationship in which investors are the principals and fund managers are their agents. The compensation scheme is the set of provisions and procedures governing management fees, general partner investment in the fund, carried-interest allocations, vesting, and distribution. As with all agency relationships, compensation schemes should be designed to align the interests of the principals (the LPs) and the agents (the GPs) to the extent that the alignment is cost-effective. It is generally cost-ineffective to try to maximize the alignment of GP and LP interests. For example, requiring huge investments into the partnership by general partners might initially appear to be an effective method of aligning LP and GP interests. However, GPs with a large proportion of their wealth invested in a single fund may manage the fund in an overly risk-averse manner.

The partnership agreement provisions, as well as other terms and conditions, such as investment limitations, transfers, withdrawals, indemnification, and the handling of conflicts of interest, tend to look quite similar across fund agreements. Surprisingly, fund terms have been relatively stable across the market cycles. The explanation for this phenomenon is that both fund managers and their investors have sufficient negotiation power to reject terms sought by the other side that differ substantially from terms widely used in the market, but not so much leverage as to move the market in one direction or the other.

Management fees are regular fees that are paid from the fund to the fund managers based on the size of the fund rather than the profitability of the fund. The purpose of management fees is to cover the basic costs of running and administering the fund. These costs are mainly the salaries of investment managers and back-office personnel; expenses related to the development of investments; travel and entertainment expenses; and office expenses, such as rent, furnishings, utilities, and supplies. Management fees are nearly always calculated as a percentage of the net asset value of the fund, typically between 1% and 2.5% depending on fund size, but may taper off after the investment period or when a successor fund is formed. Although the management fee's general calculation is relatively simple and fairly objective, there are controversies surrounding the finer details.

The general partners' investment in the fund is the amount of capital they contribute to the fund's pool of capital. GPs typically invest a significant amount of capital in their funds, usually at least 1% of total fund capital, which is treated the same way as the capital contributed by limited partners. There are a number of reasons for this. For example, the GPs contribute a meaningful amount of capital to ensure their status as a partner of the fund for income tax reasons. More important, however, is that they contribute substantial personal wealth to the fund to help align the interests of fund managers and their investors. For all of the calculation examples that follow, the GPs' own investment in the fund is not being considered, because it has the same payoff as that of the limited partners. In other words, in this volume, the computations of the amount of cash being distributed to GPs ignore their ownership interest, since that ownership interest receives cash in the same manner as the LPs.

3.5.3 Incentive-Based Fees

Incentive-based or performance-based fees are a critical part of the compensation structure. Carried interest, as discussed earlier, is an incentive-based fee distributed from a fund to the fund's manager. The term carried interest tends to be used in private equity and real estate; the term incentive fee is more often used in hedge funds. Management fees are paid regardless of the fund's performance and therefore fail to provide a powerful incentive to produce exceptional investment results. Excessive and quasi-guaranteed management fees stimulate tentative and risk-averse behavior, such as following the herd. Consequently, the carried interest, meaning the percentage of the profit paid to fund managers, is the most powerful incentive to align interests and create value. The most common carried-interest split is 80%/20% (a.k.a. 80/20), which gives the fund manager a 20% share in the fund's net profits and is essential to attracting talented and motivated managers. These fees are asymmetric, as a fund manager shares in the gains of the investors, but does not compensate investors for any portion of their losses. (Note that the following examples ignore management fees for the sake of simplicity.)

3.5.4 Aggregating Profits and Losses

In the case of multiple projects within private equity funds, two approaches are used for determining profits and distributing incentive fees. Carried interest can be fund-as-a-whole carried interest, which is carried interest based on aggregated profits and losses across all the investments, or can be structured as deal-by-deal carried interest. Deal-by-deal carried interest is when incentive fees are awarded separately based on the performance of each individual investment.

Participating in every investment's profit, or deal-by-deal carried interest, can be problematic because the general partner can make profits on successful investments while having little exposure to unsuccessful transactions. As the limited partners take the bulk of the capital risk, this approach significantly weakens the alignment of interests. A fund-as-a-whole carried-interest approach protects the interests of the LPs but may be less effective in attracting talented managers. The fund-as-a-whole scheme may entail the risk of frustrating the fund managers, as their rewards may be deferred for years until all deals can be aggregated. Carried-interest distribution is typically one of the most intensively negotiated topics. The amount of the payment is often not as much of an issue as the timing of the payment. In practice, carried-interest schemes include elements of both approaches in order to circumvent their respective limitations.

3.5.5 Clawbacks and Alternating Profits and Losses

Clawbacks are relevant to funds that calculate carried interest on a fund-as-a-whole basis. The idea of typical clawback provisions is that incentive fees distributed to managers are returned when a firm experiences losses after profits so that the total incentive fees paid, ignoring the time value of money, are equal to the incentive fees that would be due if all profits and losses had occurred simultaneously. Funds experience early profits and late losses in two primary instances. In private equity funds, it is possible that a few of the projects in which the fund has invested may successfully terminate and generate large cash inflows and profits to the fund. Other projects may fail at a later date, thereby generating large losses or write-offs. An important issue when a fund experiences large gains early in its life, followed by subsequent losses, is whether incentive fees paid on the early profits will be returned to the LPs.

Another instance in which losses follow profits is more common in the hedge fund industry, where market conditions or managerial decision-making can cause strategies to be highly successful in one time period and then highly unsuccessful in a later time period. In this case, the fund earns high profits followed by large losses.

In both cases, it is possible that incentive fees, or carried interest, could be paid during the earlier profitable stage, even though subsequent losses could cause the investment to have no profit over its entire lifetime. Thus, a limited partner could end up paying incentive fees for an investment that lost money over its lifetime. Clawback provisions are designed to address this problem for limited partners.

The goal of clawback provisions is to protect the economic split agreed between the GP and LPs. The clawback provision is sometimes called a giveback or a look-back, because it requires a partnership to undergo a final accounting of all of its capital and profit distributions at the end of a fund's lifetime. Clawback provisions are the opposite of vesting. Vesting of fees is the process of making payments available such that they are not subject to being returned.

A clawback provision is a promise to repay overdistributions, but such a promise is only as good as the creditworthiness of the GP. The GP is normally organized as a limited liability vehicle with no assets other than the interest in the fund. In the partnership agreements of many funds, the clawback provision simply binds the GP and requires his or her cooperation and financial support.

The sentiment that clawbacks are worthless is not uncommon. Situations arise in which LPs are unable to receive the clawbacks they are owed. Attempting to enforce the clawback provisions may lead to years of litigation without resulting in any return of cash. The simplest and, from the viewpoint of LPs, most desirable solution is to ensure that the GP does not receive carried interest until all invested capital has been repaid to investors. With this approach, however, it can take several years before the fund's team sees any gains, and it could be unacceptable or demotivating to the fund managers. An accepted compromise for securing the clawback obligation is to place a fixed percentage of the fund manager's carried interest proceeds into an escrow account as a buffer against potential clawback liability.

Clawbacks typically refer to GP clawbacks, or corrective payments to prevent a windfall to the fund manager. However, it is also possible for LPs to receive more than their agreed percentage of carried interest. Consequently, some partnership agreements also address so-called LP clawbacks.

3.5.6 Hard Hurdle Rates

A hurdle rate, or preferred return, specifies that a fund manager cannot receive a share in the distributions until the limited partners have received aggregate distributions equal to the sum of their capital contributions as well as a specified return, known as the hurdle rate. In other words, a hurdle rate specifies a return level that LPs must receive before GPs begin to receive incentive fees. This section details hurdle rates and discusses a hard hurdle rate. A hard hurdle rate limits incentive fees to profits in excess of the hurdle rate.

The sequence of cash distributions with a hard hurdle rate is as follows:

• Capital is returned to the limited partners until their investment has been repaid.
• Profits are distributed only to the limited partners until the hurdle rate is reached.
• Additional profits are split such that the fund manager receives an incentive fee only on the profits in excess of the hurdle rate.

3.5.7 Soft Hurdles and a Catch-Up Provision

A soft hurdle rate allows fund managers to earn an incentive fee on all profits, given that the hurdle rate has been achieved. Returning to the example of a one-year $10 million fund with a hurdle rate of 10% and profits of $2 million, a soft hurdle rate of 10% allows the fund manager to receive 20% of the entire $2 million profit, or $400,000. As long as the resulting share to the limited partners allows a return in excess of the hurdle rate, then the hurdle rate can be ignored in terms of computing the incentive fee. The limited partners receive $1.6 million, which is a 16% return.

The soft hurdle in this case allows the fund manager to receive an incentive fee on the entire profit. A soft hurdle has a catch-up provision that can be viewed as providing the fund manager with a disproportionate share of excess profits until the manager has received the incentive fee on all profits. The sequence of cash distributions with a soft hurdle rate is as follows:

• Capital is returned to the limited partners until their investment has been repaid.
• Profits are distributed only to the limited partners until the hurdle rate is reached.
• Additional profits are split, with a high proportion going to the fund manager until the fund manager receives an incentive fee on all of the profits.

Once the fund manager has been paid an incentive fee on all previous profits, additional profits are split using the incentive fee. This is called a catch-up provision.

3.5.8 Incentive Fee as an Option

Incentive fees are long call options to GPs, who receive the classic payout of a call option: If the assets of the fund rise, they receive an increasing payout, and if the assets of the fund remain constant or fall, they receive no incentive fee. The underlying asset is the fund's net asset value, and the time to expiration of the option is the time until the next incentive fee is calculated, at which time a new option is written for the next incentive fee. In the absence of a hurdle rate, the strike price of the call option is the net asset value of the fund at the start of the period or the end of the last period in which an incentive fee was paid, whichever is greater. The GPs pay for this call option by providing their management expertise.

A hurdle rate may be viewed as increasing the strike price of the incentive fee call option. A hurdle rate increases the amount by which the net asset value of the fund must rise before the fund manager receives an incentive fee. The higher the hurdle rate, the lower the value of the call option.

As a call option, incentive fees provide fund managers with a strong incentive to generate profits. The call option moves in-the-money when the net asset value of a fund rises to the point of providing a return in excess of any hurdle rate. The call option moves out-of-the-money when the net asset value of the fund falls below the point of providing a return in excess of any hurdle rate.

When the option is below or near its strike price, the incentive fees provide the fund manager with an incentive to increase the risk of the fund's assets. The effect of increased risk is to increase the value of the call option. If the risks generate profits, the fund manager can benefit through high incentive fees. If the risks generate losses, the effect on the fund manager is limited to receiving no incentive fee, ignoring clawbacks.

When the incentive fee call option is deep-in-the-money, the fund manager benefits less from an increase in the risk of the underlying assets. The consequences of net asset value changes to the fund manager are more symmetrical when the option is deep-in-the-money, meaning when large incentive fees are likely. Risk aversion may motivate the fund manager to lessen the risk of the underlying assets when the incentive fee option is deep-in-the-money.

It can be argued that the multifaceted incentives generated by the optionlike character of incentive fees are perverse. The LPs prefer fund managers to take risks based on market opportunities and the risk-return preferences of the LPs. However, incentive fees can motivate fund managers to base investment decisions on the resulting risks to their personal finances. In summary, incentive fees can cause decisions involving risk to be based on the degree to which an option is in-the-money, near-the-money, or out-of-the-money.