A Mehod for Esimaing he Change in Terminal Value Required o Increase IRR Ausin M. Long, III, MPA, CPA, JD * Alignmen Capial Group 11940 Jollyville Road Suie 330-N Ausin, TX 78759 512-506-8299 (Phone) 512-996-0970 (Fax) www.alignmencapial.com November 2006 * Many hanks o Dr. Seven Kaplan, Neubauer Family Professor of Enrepreneurship & Finance a he Universiy of Chicago, for his helpful commens. 1
INTRODUCTION Alhough by definiion here is a close relaionship beween imes money earned (TME, defined as disribuions plus erminal valuaions divided by invesed capial) and inernal rae of reurn (IRR) for a privae marke invesmen, he relaionship is ordinarily quie hazy in he porfolios of mos insiuional invesors. This lack of precision is due o he naure of he IRR compuaion iself, since i is affeced o an unpredicable exen by he idiosyncraic blending of he unique weighs and iming of each of he individual cash flows ha make up he ypical privae equiy porfolio. However, wih some simplifying assumpions i is possible o represen he confused angle of capial invesed, capial disribued and erminal valuaion of a privae invesmen porfolio in erms of zero-coupon bond equivalency. The wo simplifying assumpions are: ha all invesed capial is invesed on he dae of he firs capial call and ha all disribuions and he erminal valuaion all have he same ending dae. Viewed as a zero-coupon bond in his way, a privae invesmen or a porfolio of privae invesmens has a precise relaionship beween TME and IRR. Alignmen Capial Group, (ACG) refers o his relaionship as he zero-coupon equivalen duraion (ZCED), which is an esimae of he dollar-weighed, ime-weighed average holding period of an invesmen or porfolio. ZCED, even hough i requires simplifying assumpions, can be adaped o esimae cerain aribues of he ypical privae equiy porfolio, including is change in IRR as a funcion of is change in TME and/or erminal valuaion, as described in deail below. METHODOLOGY The inernal rae of reurn (IRR) of a sream of cash flows (where capial invesed in period is denoed Inv and capial disribued in period or, in he case of he erminal valuaion, capial deemed o be disribued in period T - is denoed Dis ) is o choose r such ha: 1. T Inv = ( 1+ r) = 0 ( r) = 0 1+ T Dis I is obvious, in equaion 1 above, ha he magniudes of Inv and Dis and heir relaive iming all affec he oucome of he compuaion. I is no, however, obvious indeed, i is well beyond he scope of his research brief o wha degree each of hese elemens affecs he oucome. This is so because here is a heoreically infinie number of 2
permuaions of he weighs and iming of he various cash flows ha will resul in he same IRR, as illusraed in he example below: Dae of Cash Flow Scenario 1 Scenario 2 Scenario 3 1/31/1990 ($1,000,000) ($5,000,000) ($5,000,000) 1/31/1991 $1,200,000 $1,000,000 $0 1/31/1992 ($1,000,000) $1,000,000 $0 1/31/1993 $1,200,000 $1,000,000 $0 1/31/1994 ($1,000,000) $1,000,000 $0 1/31/1995 $1,200,000 $1,000,000 $0 1/31/1996 ($1,000,000) $1,000,000 $0 1/31/1997 $1,200,000 $1,000,000 $0 1/31/1998 ($1,000,000) $1,000,000 $0 1/31/1999 $1,200,000 $6,000,000 $25,824,763 Invesed Dollars(I) ($5,000,000) ($5,000,000) ($5,000,000) Reurned Dollars (R) $6,000,000 $14,000,000 $25,824,763 Dollars Earned (DE) $1,000,000 $9,000,000 $20,824,763 IRR 20.0% 20.0% 20.0% Times Earned (TE) = R/I 1.2 X 2.8 X 5.2 X However, i is possible o eliminae all of he uncerainies associaed wih he calculaion of IRR by reposiioning he daes of all he capial invesed o he dae of he firs capial call and all of he capial disribued o he erminal valuaion (ending) dae. These moves have he effec of ransforming he invesmen or he porfolio ino a zero-coupon bond wih only wo cash flows: one a he ouse of he invesmen and anoher a is erminaion. As a zero-coupon bond, he cash flows bear he relaionship o one anoher described in he rudimenary fuure value equaion below: 2. TME = ( 1+ IRR) n Solving equaion 2 for IRR, we obain: n 3. IRR = TME 1 1 The firs derivaive of equaion 3 is he insananeous rae of change of IRR given a sligh change in TME: 4. 1 dirr TME n 1 = dtme n Or, anoher way o express he same relaionship is: dirr 5. IRR TME, when TME is very small. dtme 3
In his research brief, we have assumed ha he desired change in IRR is 0.1% (10 basis poins): 6. IRR = 10bp or 0.10% The change in TME required o resul in a 10 basis poin change in IRR is herefore: 7..001 dirr = TME dtme Remember ha TME is he relaionship of oal disribuions plus he erminal valuaion (in he equaions below, Val = oal disribuions + erminal valuaion) and capial invesed (Inv in he equaions below): 8. Val TME = Inv Therefore we can describe he change in TME as a funcion of he change in erminal valuaion, assuming in his equaion he disribuions are a consan: 9. Val TME = Inv Thus, he change in valuaion required o resul in a 10 (is his 100?) basis poin change in he IRR of a zero-coupon bond is: 10. Val = TME * Inv 11..001n Val Inv TME n 1 1 Equaion 11 can be used o esimae he change in IRR resuling from a fracional change in he erminal valuaion of a ypical privae invesmen. We believe ha, excep for unusual circumsances, he resuls of his simple equaion should be wihin a few basis poins of he resuls of a complee calculaion using all of he cash flows in he porfolio. In general, he shorer he ime horizon of he invesmen, and he fewer he cash flows involved, he beer he esimae. 4
RESULTS We sampled weny invesmens from a porfolio using Equaion 11 above, calculaed he required change in TME and compared he resul o he acual change in TME as follows: Inv # Vinage n TME Acual Diff Prediced Diff Prediced - Acual 1 2004 1 0.89 0.10% 0.10% 0.00% 2 2004 1 5.79 0.00% 0.10% 0.10% 3 2003 2 0.85 0.00% 0.10% 0.10% 4 2003 2 0.98 0.17% 0.10% -0.07% 5 2002 3 1.02 0.21% 0.10% -0.11% 6 2001 4 0.95 0.14% 0.10% -0.04% 7 2001 4 1.65 0.15% 0.10% -0.05% 8 2000 5 1.52 0.20% 0.10% -0.10% 9 1998 7 1.34 0.26% 0.10% -0.16% 10 1997 8 1.30 0.20% 0.10% -0.10% 11 1997 8 1.40 0.15% 0.10% -0.05% 12 1996 9 1.86 0.18% 0.10% -0.08% 13 1995 10 3.09 0.04% 0.10% 0.06% 14 1994 11 2.37 0.10% 0.10% 0.00% 15 1993 12 3.41 0.06% 0.10% 0.04% 16 1989 16 1.58 0.14% 0.10% -0.04% 17 1989 16 2.18 0.10% 0.10% 0.00% 18 1988 17 1.74 0.10% 0.10% 0.00% 19 1988 17 2.12 0.10% 0.10% 0.00% 20 1988 17 3.11 0.05% 0.10% 0.05% 21 1988 17 3.52 0.02% 0.10% 0.08% 22 1987 18 1.82 0.12% 0.10% -0.02% 23 1987 18 4.26 0.03% 0.10% 0.07% 24 1987 18 4.46 0.04% 0.10% 0.06% 25 1986 19 1.21 0.19% 0.10% -0.09% 26 1986 19 1.58 0.13% 0.10% -0.03% 27 1986 19 1.86 0.11% 0.10% -0.01% 0.11% Mean -0.01% 0.07% Sd Dev 0.07% The mean error of he esimae (i.e., he prediced change in IRR minus he acual change in IRR) is only abou 1 basis poin. The large sandard deviaion of he error of he esimae relaive o he mean conribues o he lack of correlaion beween n and he error of he esimae, as indicaed in he following regression: 5
0.15% 0.10% 0.05% Prediced - Acual 0.00% 0 2 4 6 8 10 12 14 16 18 20-0.05% y = 2E-05x - 0.0004-0.10% R 2 = 0.0411-0.15% -0.20% n CONCLUSION The use of simplifying assumpions (all capial is invesed on he firs capial call dae; all capial reurned and he valuaion are a he ending dae, ransforming he invesmen ino a zero-coupon bond) is sufficien o enable he analys o esimae fairly accuraely he change in IRR o be expeced from small changes in TME, even for older invesmens wih a hisory of complex cash flows. 6
Alignmen Capial Group is a full-service privae equiy consuling firm based in Ausin, Texas. The firm s mission is o undersand privae equiy as an asse class in a porfolio conex, and hus o assis our cliens in making opimal invesmen decisions. Ausin Long is a co-founder of Alignmen Capial Group. His responsibiliies include performing due diligence on invesmen managers, providing sraegic porfolio managemen advice and conducing original research. www.alignmencapial.com 7