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, Rfcoll, can be expressed as
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, Rpcoll, 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:
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 Internal Rate of Return
The computation of traditional investment returns is not easy, but it is far easier than the computation of returns for some alternative investments. A main challenge with the analysis of some alternative investments is the lack of regularly observable market prices. Some alternative investments, such as private equity and private real estate, are analyzed using an internal rate of return approach. This approach has numerous potential complications and shortcomings. With the advantage of regular market prices, traditional investment analysis usually computes return as the change in price, net of fees, plus cash flows received (such as dividend or interest payments), divided by the initial price:
(3.8)
However, complications arise when prices cannot be regularly observed or when cash flows are received during the interim period, between the starting date and the ending date of the return observation. A major complexity related to these interim cash flows is that it is unclear how much return could be earned through their reinvestment. It is usually assumed that the intervening cash flows are reinvested in the same underlying investment, but this requires an interim price of that asset at the same time as the cash flows become available for reinvestment.
Since prices can be observed at least on a daily basis for most traditional investments, daily returns are easily computed from daily prices and daily cash flows. Returns over time periods in excess of one day with intervening cash flows can be computed as the accumulation of the daily returns within the time period. In other words, returns for longer time periods are formed from the daily returns of the days within the time period. Returns over time periods shorter than one day do not tend to have intervening cash flows, since dividends and interest payments are usually made on an end-of-day basis.
Despite challenges faced with various compounding assumptions and intervening cash flows, the returns of most traditional investments are made relatively straightforward when daily prices are available. However, return computations for investments that cannot be accurately valued each day generate challenges that are a primary topic of this chapter. For example, securities that are not publicly traded, such as private equity, do not have unambiguous daily valuations that can be used to compute daily returns. This section explains the application of the internal rate of return method to alternative investments and details the potential difficulties with interpreting and comparing internal rates of return.
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 CFT–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 CFT 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 CFT, 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:
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 CFt in a bank account at time t for each CFt < 0 and withdrew CFt from the bank account when CFt > 0, and if the bank's interest rate on the account was IRR, then the bank account would have a zero balance