style="font-size:15px;"> For very small returns, we can roughly think of Rm = ∞and ln(1 + R) as being equal to R: as R → 0 then R → Rm = ∞ 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 R0,T, which can be expressed as being equal to the following product in terms of the returns of the asset over the sub-periods (Rt):
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:
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 Returns Based on Notional Principal
Much investment analysis centers on the concept of the rate of return, defined as the rate at which an asset changes value (with any interim cash flows, such as dividends, considered). As a rate, a return is usually expressed as a portion or percentage of the asset's starting value. However, alternative investing often includes assets for which there is no clear starting value other than perhaps zero. Examples can include derivative contracts, such as forward contracts and swaps. This section describes some of the mathematics and modeling designed to address issues that arise when there is a zero starting value, or no clear starting value, to a contract.
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
