Investing and Financing-Type Projects: A Generalization to Investments

Transcription

1 Abstract Investing and Financing-Type Projects: A Generalization to Investments Joe Walker and Theodore Bos University of Alabama at Birmingham The capital budgeting literature occasionally makes a distinction between investing-type projects and financing-type projects. Since the one-period internal rate of return (IRR) formula is mathematically equivalent to the holding period rate of return (HPR), and since mirror-image investments and borrowings lead to the same IRR, the IRR must be reinterpreted in borrowing situations as a rate of borrowing rather than a rate of return. This paper introduces a formula for the one period rate of return on borrowing transactions based on an NPV/IRR derivation which can be used to generalize the return concept for any financial arrangement. This is important because short investments add unfamiliar layers of complexity to structured products such as credit default swaps and pure short 130/30 funds, and it is not generally recognized how to break out separate rates of return for these products which thereby hinders their optimization. Even the CFA Institute s Global Investment Performance Standards (GIPS, 2006a) does not identify these separate calculations or their need, and it relies on a net portfolio approach to calculating returns an approach that is incomplete in an age of rapid financial innovation.

2 1. Introduction The purpose of this paper is to apply a generalized rate of return calculation which accounts for non-normal cash flow investment cases. A background section will first explain the distinction between investment-type projects and financing-type projects 1. The financing-type projects/investments to be discussed here are not uncommon, and their modern counterparts (such as 130/30 funds and more particularly credit default swaps) have been alleged to have played significant roles in the recent world financial crisis. Furthermore, new products which would fit these financing-type patterns are practically guaranteed by the pace of financial innovation, and it behooves the finance profession to know how to properly calculate their returns since those are the single most important prerequisites to calculating their risk parameters. A dilemma will then be provided to showcase the intricacies of combining shorts and longs in the same portfolio. The theory section to follow will introduce a derivation of the holding period rate of return (HPR) from the NPV / IRR context. Some historical background on the early portfolio research will explain how short sale returns were interpreted in a portfolio context but not on an individual investment basis. Then an applications section will apply the formulas to the case of short sales, options, futures, borrowing with negative real rates of return, borrowing portfolios, and finally to the case of insurance. This section is designed to show the wide applicability of the new approach as well as its ease of application. A conclusion will follow. 2. Background Ross, Westerfield, and Jordan (RWJ, 2010, p ) provide a clear exposition of investing-type projects vs. financing-type projects in the corporate finance arena. They first show an investingtype project where the initial cash flow CF 0 is $100 and the cash flow at the end of period 1, CF 1, is +$130, thereby generating a positive internal rate of return (IRR) of 30%. The IRR is positive and the project makes money. Then they present a financing-type project B wherein the signs on the cash flows are reversed, where CF 0 = +$100 and CF 1 = $130. This scenario also generates a positive IRR of 30%, but the accept/reject rule has to be reversed because the project loses money. Now extend this financing-type example, naming it project B2, to the case wherein the cash flows are now CF 0 = + $100 and CF 1 = $90. The IRR is now 10%, yet this is a financing-type project which makes money vs. Project B above which, with a positive IRR of 30%, lost money. One way to make sense of this apparent paradox is to interpret the IRR strictly 1 See Ross, Westerfield, and Jaffe, 9 th ed., (2010, p. 146). See also Ivo Welch (2009, p. 78), Berk, DeMarzo 2 nd ed. (2011, p 161), and Brealy, Myers, Marcus (2007, p. 192) 5 th ed.. 1

3 as a rate of return for investing-type projects (negative CF 0 ) and to rename the IRR for financing-type projects (positive CF 0 ) as an internal rate of cost (IRC). Then note that although the IRC of project B2 is 10%, the corresponding IRR must be +10% since the rate of return to one party is simply the cost to the other. What particularly needs to be understood, though, is that Project A s IRR of +30% and Project B2 s IRR of +10% cannot be directly compared, and the dilemma below is constructed to explain this. 3. Dilemma Let an investor buy stocks for $1,100 at t=0 and sell the stocks at t=1 for $1,210. At the same time, let the investor short some other stocks for $100 and then cover them for $90 at the end of the period. It is assumed that there is no margin required and that the investor will have full use of the short sale proceeds, much like 130/30 funds do today. 2 The rate of return on the stock is +10% and the rate of return on the short is +10% since the investor made money on both. One could therefore argue that the rate of return on the portfolio is +10% no matter what the weights are, as long as they sum to one. On the other hand, one could argue that the net investment is $900 (= $1,000 $100), and that the net dollar return is $120 (= $110 + $10) to yield a 12% (= $120/$1,000) return. So why don t these two approaches mesh? 4. Theory What investment texts usually posit ad hoc is an HPR (r for short) formula where: r = [profit (+) or loss ( )] / + investment Since no justification is given for this formulation, it hinders understanding of the rate of return calculation in cases with positive initial cash flows. Therefore, the following derivation of the HPR is offered to set the stage for the rest of this discussion. Let the HPR produce a zero present value, PV, in the internal rate of return (IRR) sense so that, for the simple one-period case 3 PV = 0 = + CF 0 + CF 1 / (1 + r) (1) 2 The use of short sale proceeds are allowed in particular circumstances. It is interesting to note the literature on the new 130/30 and other 1X0/X0 mutual funds. See Lo and Patel (2008). In any case, we will show examples with margin and escrow shortly. 3 This paper will not deal with the multi-period case for reasons of space, time, and complexity that would detract from the immediate purposes at hand. Furthermore, while the one-period case is critical for portfolio generalization and therefore optimization, the multi-period case is not. 2

4 wherein r, the HPR, is shown to be just an IRR, the value of CF 0 is the cash flow at the beginning of the period, and CF 1 is the cash flow at the end of the period. Solving for r gives: r = (CF 1 + CF 0 ) / CF 0 (2) where the numerator, (CF 1 + CF 0 ), is the profit or loss and the denominator, CF 0, is the investment. Though the sign designation in the denominator of equation (2) appears unusual, the above two formulas lead to the same signs on the IRR (r) and profits with conventional investments. This formulation also avoids the accounting-like categories of profit, loss, and even investment which do not easily lend themselves to generalization in less standard applications. Applying equation (2) to buying a stock for $100 and selling it for $130 ( investing ) yields: r = ($130 $100) / ( $100) = + 30% Now consider the opposite case of borrowing $100 and repaying $130. Using equation (2), one obtains: r = ( $130 + $100) / ( $100) = + 30% Yet, clearly, this borrower did not gain 30% by borrowing even though the IRR calculation yields a positive rate of return on both an investment scenario as well as the borrowing scenario. Expediently, one could just reverse the signs of the quantities in the numerator for investments vs. borrowings. However, in addition to lacking context, it is not always clear which sign or signs to change those on the cash flows or the returns or (as we shall, see shortly) on the portfolio weights. The point is that investment rates of returns by themselves can be combined, and borrowing rates of returns by themselves can be combined; but to combine the two rates in a portfolio, one must be careful to translate the signs so that the return set contains either all investment rates of return or all borrowing rates of return in order to net them out. This translation or netting process is facilitated by the establishment of two complementary rate of return measures; namely, an investment rate of return (IROR) and a borrowing rate of return (BROR). This is not essentially different from than the creation of two (ad hoc) decision rules for investment-type projects versus financing-type projects. 4 The first formula will be the investment rate of return (IROR) formula: 4 Contact the authors for a fuller derivation of all results. 3

5 Investment rate of return: IROR = (CF 1 + CF 0 ) / CF 0 (3) Note that it is identical to equation (2), but it is to be used only for investing situations. The second formula will be the borrowing rate of return (BROR) formula: 5 Borrowing rate of return: BROR = (CF 1 + CF 0 ) / CF 0 (4) and it is to be used only for borrowing-type situations. Therefore, the first (and incorrect) solution to the dilemma presented earlier resulted from naively combining the rate of return on an investment and the rate of return on a pure short sale. The solution is to realize that there was really only one investment (in the new sense), and that the other investment, the short sale, was not an investment (in the new sense) but was really a borrowing-type situation. 6 The usual portfolio investment rate of return is calculated as: portfolio investment rate of return = W A R A + W B R B (5) where W A is the investment portfolio weight of A, R A is the investment rate of return on A, W B is the investment portfolio weight of B, and R B is the investment rate of return on B. 7 Letting A denote the stock investment in the dilemma above, calculate the return on A using the investment rate of return formula because of the negative CF 0, R A = IROR = (CF 1 + CF 0 ) / CF 0 so one gets (+ $1,210 $1,100) / ( $1,100) = + $110 / + $1,100 = + 10%. 5 (In previous versions equations (3) and (4) were derived from two formulas: the NPV of investing and the NPV of borrowing. Again, contact the authors for more details. 6 As a matter of terminology simplification, the authors would like proffer the easier word, borrowment, in place of the awkward phrasing of borrowing-type situations. This would also serve as an effective counterpart to the word investment. 7 Net borrowment portfolios are possible too if more is borrowed than invested. In that case: portfolio borrowment rate of return = W A R A + W B R B where W A is the portfolio weight of borrowing A, R A is the return on borrowing A, W B is the portfolio weight of borrowing B, and R B is the return on borrowing. The sign of borrowment portfolio weights are positive for borrowments and negative for investments. 4

6 To calculate W A, note that the net initial portfolio cash flow, CF 0 (P) = CF 0 (A) + CF 0 (B) = $1,100 + $100 = $1,000, so W A = CF 0 (A) / (CF 0 (P) = ( $1,100) / ( $1,000) = +1.1 Now let B denote the short sale in the dilemma above. To calculate R B, note that if one looked at the short sale in isolation, the borrowment rate of return (due to the positive CF 0 ) would be calculated as follows: CF 0 = + $100, CF 1 = $90, and BROR = (CF 1 + CF 0 ) / CF 0 = ( $90 + $100) / + $100 = +10% But this positive BROR cannot be used in equation (5), because a net investment portfolio is being constructed. What is now needed is to calculate the corresponding IROR for this borrowment in order to include it into a net investment portfolio. Thus: R B = IROR = (CF 1 + CF 0 ) / CF 0 = ( $90 + $100) / $100 = $10 / +$100 = 10% Although the net result seems to be a mere change of signs, there is now a reasoned, logical context for doing so; furthermore, the new equations avoid any confusion over changing signs on the returns and the weights. To calculate W B, W B = CF 0 (B) / (CF 0 (P) = (+ $100) / ( $1,000) = 0.1 Thus the portfolio investment rate of return (R P ) is: portfolio investment rate of return R P = W A R A + W B R B portfolio IROR = +1.1 (+0.1) 0.1 ( 0.1) = = + 12% This solution can be seen as the investor shorting stock B for $100 and using the proceeds as well as the original $1,000 equity to make a total $1,100 investment in stock A. Thus, the investor makes a $110 dollar profit on stock A and a $10 dollar profit on shorting stock B to make a total profit of $120 on the original investment of $1,000 for a total return of +12%. 5

7 5. History As background for the present day lack of theory-based short rates of return, one will not find any explicit references to short sale rates of return prior to the 1960 s for the simple reason that finance as a mathematical research discipline was just beginning. One of the earliest references to short sales was that of Markowitz (1952) who purposely excluded negative asset weights in his optimal portfolio constructions to specifically exclude short sales in his 1950 paper. Lintner (1965), however, did introduce short sales into the efficient frontier research simply by allowing the asset weights to be negative as well as positive. In other words, the short sales were incorporated merely as long sales with negative weights (or the rates of return on short sales were incorporated as rates of return on the corresponding long sales). As a result, all that was needed for the calculations were the returns on long sales, and the negative weights handled the fact that they were shorts; so a calculation for a short return in isolation was simply not needed. No margin or escrow was assumed for simplicity. If margin or escrow had been required, the set of (long) assets would have been augmented to include a set of margined short sale assets, making the analysis considerably more complex. But margin and escrow were not needed for the equilibrium theory, for as Markowitz noted, approximately a hundred percent of equilibrium theory is built on the assumption that you can short and use the proceeds. 8 It is also important to note that optimal asset weights are a hidden byproduct of the calculation of mean-variance points on the efficient frontier. These weights are rarely noted, but when they are, it is often found that the asset weights in the optimal portfolios have negative signs. In other words, allowing shorts usually results in the shorts being pervasive along the efficient frontier. 6. Applications A set of examples have been constructed below to more fully portray the application of the twoformula rate-of-return structure. They were chosen to show the range and diversity of borrowment problems that now easily succumb to an organized rubric. 6A. Short Sales The first application will be that of short sales where it will initially be assumed that there are no margins and no escrow and that the proceeds of the short sale can be used. This is not totally unrealistic as noted by Francis and Ibbotson (2002, p. 49) who maintain that margins are negotiable for substantial individuals and [that] institutional investors like Merrill Lynch or Citigroup can make short sales for their own account without posting any up-front guarantee money. Francis and Ibbotson go further and say, This allows the institutional investors to 8 Taken from the Markowitz (2006) interview. 6

8 enjoy initial cash inflows (instead of outflow) from their short sales. Thus, the one-period rate of return for a no-money-invested position is not defined. However, our proposed model does not agree with this conclusion. For example, assume a stock is shorted for $100 (CF 0 = + $100) and covered at the end of one period for $90 (CF 1 = $90). Since there is an initial positive cash flow, this is a borrowment according to the new definition, so the pure (no escrow, no margin) return is found by using: Borrowment ROR of short sale = (CF 1 + CF 0 ) / CF 0 BROR = ( $90 + $100) / + $100 = + 10% This is logical because the sign on the rate of return corresponds to the sign on the profit. Now let there be margin requirements (but no escrow requirements yet). Using, say, a 40% margin requirement, let the investor short the stock for $100 and put up $40 in margin money (CF 0 = + $100 $40 = + $60), and then let the investor cover one period later for $90 and recover the margin deposit (CF 1 = $90 + $40 = $50). Thus: BROR = (CF 1 + CF 0 ) / CF 0 BROR = ( $50 + $60) / + $60 = % Another example with not only margin but also escrow requirements comes from Jordan/Miller/Dolvin (page 77). They posit a short sale at $96 per share, a margin requirement of 60% (= $57.60), covering at $86.50, and paying dividends of $0.75 per share. The question asks for a rate of return on the investment. Our solution uses a cash flow framework. The initial cash flow CF 0 = + $96 (short sale proceeds) $96 (100% escrow) $57.60 (60% margin deposit) = $ The subsequent cash flow is CF 1 = + $96 (escrow return) $86.50 (cover) $0.75 (dividend) + $57.60 (margin recovery) = + $ Notice, though, how the institutional constraints transform this borrowment into an investment with a negative cash flow up front and a positive cash flow on the back end. 6B. Options Now using the IROR because of the initial negative cash flow (investment), we see that: investment ROR of short sale = (CF 1 + CF 0 ) / CF 0 IROR = ( $ $66.35) / ( $57.60) = % Buying a (pure) call or put option involves a negative initial cash flow, so the normal investment calculation applies (equation 3). For example, buying a call for $2 with strike price 7

9 of $120 and then covering when the stock price rises to $126 yields CF 0 = $2 and CF 1 = $126 $120 = + $6 for a return of: investment rate of return on buying a call option = (CF 1 + CF 0 ) / CF 0 IROR = (+ $6 $2) / ( $2) = + 200% However, (pure) call or put writing involves an initial positive cash flow which makes it a borrowment. For example, writing a call with a strike price of $120 for $2 gives CF 0 = + $2, and then covering when the underlying stock price rises to $126 gives CF 1 = ( $120 $126) = $6, a loss on the option offset by the + $2 sale, so one gets a BROR of 200%, a loss as shown below: borrowment rate of return on writing a call option = (CF 1 + CF 0 ) / CF 0 BROR = ( $6 + $2) / $2 = 200% Generalizing this example to account for, say, a 25% margin requirement when writing the call option (margin = 25% x $2 = $0.50) yields a net borrowment where CF 0 = $2 $0.50 = + $1.50 and CF 1 = + $120 $126 + $0.50 = $5.50 for a borrowment rate of: BROR = (CF 1 + CF 0 ) / CF 0 BROR = ( $ $1.50) / + $1.50 = %. A similar example can be constructed for buying a put. However, put writing involves a positive cash flow up front which by the definition makes it a borrowment. For example, writing a put with strike price of $120 for $2 gives CF 0 = + $2, and then covering when the underlying stock price hits $114 gives CF 1 = ( $120 + $114) = $6, a loss on the option offset by the + $2 sale, to yield a 200% BROR as shown below: borrowment rate of return on writing a put option = (CF 1 + CF 0 ) / CF 0 BROR = ( $6 + $2) / $2 = 200%. This example can easily accommodate a margin requirement. Although there is margin involved in writing put and call options, there is no escrow requirement. As an unusual consequence, a pure borrowment is therefore NOT regulated into an investment with option sales. 6C. Futures For futures, first assume a pure contract with no margin or escrow requirements. For example, at t=0 let the investor go long (promise to buy) gold for $500 / oz at t=1, only to see the spot 8

10 price rise to $600 / oz by that time. Therefore, CF 0 = $0, and CF 1 = + $600 $500 = + $100. In this (pure) situation, since there is no cash flow at t=0, either the investment or borrowment formulas (equations 3 or 4) can be used. Therefore, arbitrarily using equation (3) one gets: investment rate of return going long in futures = (CF 1 + CF 0 ) / CF 0 IROR = (+$100 + $0) / $0 = undefined. Generalizing this example to include a margin requirement of, say, $50, yields CF 0 = $50, and CF 1 = + $600 $500 + $50 = + $150. The institutional margin requirement transforms this transaction into an investment with the negative initial cash flow, so equation (3), the investment equation, is used to get: investment rate of return on going long on futures with margin IROR = (CF 1 + CF 0 ) / CF 0 IROR = (+ $150 $50) / ( $50) = + 200%. Similarly for a pure (no margin) short futures contract, let the investor go short, say, 1 ounce of gold for $500 / oz, and then let the spot price of gold be $600 / oz at t=1. Therefore, CF 0 = $0, and CF 1 = $600 + $500 = $100. The investor loses $100 for a dollar return of $100 on a zero investment so the return is again undefined. Similarly, we can use the investment rate of return on going short on futures since the initial cash flow is not positive as shown below: IROR = (CF 1 + CF 0 ) / CF 0 IROR = ( $100 + $0) / $0 = undefined. Including a margin requirement of, say, $50, results in CF 0 = $50, and CF 1 = $600 + $500 + $50 = $50. This now qualifies as an investment with the negative initial cash flow, so equation (3), the investment equation, is used to get: investment rate of return on going short on futures with margin IROR = (CF 1 + CF 0 ) / CF 0 IROR = ( $50 $50) / ( $50) = 100%. So a futures investment will, in practice, always be an investment in the usual sense of the word as well as in the sense used in this paper due to the negative initial cash flow, the margin, as there is no offsetting positive cash inflow as in the case of short sales. In fact, some authors have 9

11 actually specified margin as the investment, although without being explicitly aware of the context shown in this paper. 9 6D. Borrowing at a Negative Real Rate Another application that may help the reader see the value of the two-formula structure to generalize to unusual situations is that of taking out a loan in a period of inflation so high that the real rate of return is negative. Say an individual borrows $100 at 5% while the inflation rate is 7%. Then CF 0 = + $100 and CF 1 = $100*1.05 / 1.07 = $98.13 in real terms. Using BROR (equation 4 above) one gets BROR = ( $ $100) / $100 = = %. If one (erroneously) used the IROR, one would get IROR = ( $ $100) / ( $100) = 1.87%. So the borrowing ends up taking place at a positive real rate of return of %. Notice the absence of any margin or fees and the consequent similarity to a (pure) and profitable short sale one initially borrows but eventually profits by covering the borrowed amount (paying back the loan with less real purchasing power). On this note, one could observe the recent negative U.S. T-bill rates as borrowing yielding a positive real rate of return to the government. 6E. A Borrowment Portfolio Similar to the dilemma explained above, consider the case wherein a borrower takes out a oneperiod loan A for $1000 at 5% and then makes a one-period loan B for $100 at 10% while the ex post inflation rate turns out to be 8%. As there is an initial net cash inflow, this constitutes a net borrowment portfolio. Therefore, two (real) borrowment rates of return need to be calculated. For loan A: BROR = (CF 1 + CF 0 ) / CF 0 = ( 1000*1.05/ ) / 1000 = % real borrowment rate of return. For loan B: BROR = (+100*1.10/ ) / ( 100) = 1.85% real borrowment rate of return for lending the $100. Then calculate: portfolio borrowment rate of return = W A R A + W B R B (6) where W A is the portfolio weight of borrowing A, R A is the return on borrowing A, W B is the portfolio weight of borrowing B, and R B is the return on borrowing B. Therefore, W A = 1000/900 = +1.11; W B = 100/900 = 0.11; R A = %; R B = 1.85%. 9 Although Breeden (1997) criticizes those who use margin as the basis in their calculation of rates of future returns. 10

12 Portfolio BROR = 1.11*2.78% 0.11*( 1.85%) = = % Thus this net borrowment portfolio leads to a profit at a rate of %. 6F. GIPS Example. A practitioner-generated example comes from GIPS 10 which shows an extant portfolio consisting of $110 stocks and $10 short call options at the beginning of the period. Valuations of the stocks and options are $117 and $15 respectively at the end of the period. The GIPS example then asks for a portfolio rate of return. Thus, for the stock, CF 0 = $110, and CF 1 = + $117; while for the short call options, CF 0 = + $10, and CF 1 = $15. The net initial cash flow is $100, so this portfolio is a net investment. Thus equation (3) and the IROR s are applicable. Let the stock be investment A and the short call options be investment B. For the stock the IROR is calculated due to the initial negative cash flow as: IROR = (CF 1 + CF 0 ) / CF 0 = (+ $117 $110) / ( $110) = % = R A. For the short call, while the initial cash flow is positive and it would in isolation be a borrowment, in this portfolio it must be calculated as an lror as: IROR = (CF 1 + CF 0 ) / CF 0 = ( $15 + $10) / ( + $10) = + 50% = R B. The weights are W A = 110/100 = 1.1 and W B = 10/100 = 0.1. Thus using equation (5) yields: Portfolio IROR = 1.1*(6.364%) + ( 0.1)*50% = 7% 5% = 2% The IROR formula yields the same answer as GIPS provides, though the IROR procedure also calculates the individual rates in a general framework. The GIPS answer was obtained by just looking at the net return versus the net investment which is always true ex post. Yet in a priori situations where one has yet to optimize the investment weights, the GIPS approach cannot be used. Also, the GIPS example handles only the intra-holding period rate of return for an ongoing portfolio and does not provide guidance on how to handle the entire holding period rate of return complete with escrow and fees. 6G. Insurance A final scenario to be understood is that (pure) insurance is in essence a borrowment. What normally makes insurance an investment rather than a borrowment in the real world is the 10 As shown in the GIPS Handbook (2006a) and also on the GIPS (2006b) website FAQ s. 11

13 reserve requirement that, similar to the legal constraints on short sales and option sales, transforms a theoretical borrowment into a real-world investment. However, the legal constraints are exactly what the credit default swaps were designed to avoid, and thus credit default swaps fit the (pure) borrowment definition. Even in the insurance industry, perpetual (fire) insurance is a rare but highly illustrative example of a net borrowment (see Howard and Solberg (1958, p. 70)). Perpetual fire insurance is written on a continuous basis, one comparatively large premium or deposit paid at the outset buying insurance for an indefinite period. 7. Summary and Conclusion What this paper provides is a general rate of return formulation that works in all situations a formulation that is theoretically justified and that does not simply appear ad hoc. Included in this is the first explicit recognition that the HPR is nothing more than the IRR that comes from the one-period NPV formula. As a consequence of this formulation, the same sign on the (percentage) rate of return occurs regardless of whether one has an investment with a positive NPV versus a borrowment with a negative NPV. The corporate finance solution is to introduce a second accept/reject rule for financing-type situations, but this paper generalizes this concept to the investment arena by providing an explicit rate of return formula for borrowments as well as investments. Borrowments are by no means rare, and they are typified by examples such as (pure) short sales, all option writing, (pure) futures writing, borrowment portfolios, (pure) insurance, or any of the other modern financial engineering products that are at the center of the current controversies over banking and credit crises. In addition to providing explicit formulas for rates of return for short positions in isolation, the formulation herein anticipates possible missteps in incorporating short positions into (long or short) portfolios. Simply put, then, this paper defines borrowing situations as borrowments as distinct from investment situations; and it therefore introduces a borrowment rate of return formula to complement the traditional investment rate of return formula, in strict analogy to the investmenttype projects and the financing-type projects in corporate finance. Finally, this framework easily accommodates the CFA GIPS standards and even their 2006 agenda on improving leverage calculations, explanations, and disclosure. 11 These formulas ought to be especially useful for the never-ending evolution of exotic securities which is characterized by the wrapping of shorts into portfolios and the end-runs around the restrictions on short positions. 11 See Howland (2005), The continuing evolution of GIPS has attracted global support, and more recent developments are designed to address the more exotic funds offered by the alternative investment industry. 12

14 REFERENCES Berk, Jonathan and Peter DeMarzo, Corporate Finance, 2 nd edition, 2011, Boston: Prentice-Hall. Bodie, Z., A. Kane, and A. Marcus, Investments, 7 th edition, 2008, New York: McGraw Hill. Brealy, Richard A., Stewart C. Myers and Allen J. Marcus, Fundamentals of Corporate Finance, 5 th edition, 2007, New York: McGraw-Hill/Irwin. Douglas Breeden, "Some Common Misconceptions about Futures Trading," quoted on CFA Institute, Global Investment Performance Standards (GIPS ) Handbook, Second edition CFA Institute, GIPS Guidance Statement on Calculation Methodology, 2006, revised. Francis, J. C. and R. Ibbotson, Investments: A Global Perspective, 2002, Upper Saddle River, New Jersey: Pearson. Howard, William M., and Harry J. Solberg, Perpetual Fire Insurance, Journal of Finance, Vol. 13, No. 1 (Mar. 1958), pp Howland, Anthony, DIY solutions, Funds Europe, May 2005, p. 62. Jordan, B.D., T.W. Miller, and S.D. Dolvin, Fundamentals of Investments, 6 th edition, McGraw- Hill/Irwin, New York, Lintner, J., The Valuation of Risky Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets, The Review of Economics and Statistics, Vol. 47, No. 1. (Feb.,1965), pp Lo, A. W., and P. N. Patel, 130/30: The New Long-Only, Journal of Portfolio Management, Vol. 34, No. 2 (Winter 2008), pp Markowitz, H., Portfolio Selection, The Journal of Finance, Vol. 7, No. 1. (Mar., 1952), pp Markowitz, H., interview 13

15 American Finance Association, History of Finance (2006), transcript of full text of Harry Markowitz Interview at Rady School of Management at the University of California San Diego by Steve Buser (edited for clarity and readability). Ross, S.A., R.W. Westerfield, and J. Jaffe, Corporate Finance, 9 th edition, 2010, New York: McGraw-Hill Irwin. Welch, Ivo, Corporate Finance: An Introduction, 2009, Pearson: Boston. 14