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Internal rate of return is a measure of the best private equity investment performance benchmark. In short, the internal rate of return is expressed as a percentage of the time-weighted returns. Internal rate of return using cash raised with (investment) the present value distribution (return on investment capital) the present value, present value of investment does not need to calculate the discount rate.
Improvements on Secant Method for Estimating Internal Rate of Return James Moten Jr. and Chris Thron Department of Accounting, Economics, and Finance and Department of Mathematics Texas A&M University – Central Texas Email: (jmoten,thron)@ct.tamus.edu ABSTRACT The Internal Rate of Return (IRR) of a stream of payments is often used for capital budgeting. IRR has no exact formula: typically, the secant formula is to compute a sequence of converging approximations. In this paper we provide improved formulas that are more accurate and require less computation than the secant formula. SUMMARY OF FORMULAS Symbol Explanation C0 = Initial cash outflow C1...CN = Cash inflows A = Sum of cash inflows = n=1…N Cn r1, r2 = IRR guesses (provided by the user) P1, P2 = Present value of cash inflows based on the IRR guesses: Pj = n=1…N (1+rj)-n∙ Cn for j-=1,2. kcorr1,kcorr2 = Secant formula correction factors (defined in the formulas below) In the formula for L(P1,P2) either base 10 or natural logarithms can be used, but all logarithms must be of the same type. Basic secant method formula The usual secant method formula is: IRRsecant = ( ) Exponential Interpolation formulas (1) The following formula requires only one NPV computation (as opposed to two for the secant method): ( ) ( ) IRRexpo1 = ( ) (2) The following formula requires two NPV computations (like the secant method): ( ) ( ) IRRexpo2 = ( )( ) Corrected secant formulas: (1) The following formula requires only one NPV computation (as opposed to two for the secant method): IRRcorr1 = ( ), where kcorr is a correction factor: = if P1 < C0 = if P1 > C0. The above formula has the following 99% confidence interval for true IRR: IRRcorr1 ± 0.2 |IRRcorr1 – IRRsecant1| (2) The following formula requires two NPV computations: IRRcorr2 = ( ) ( ), where and = if P1 < P2 < C0 = if P2 > P1 > C0 = if P2 > C0 > P1 The above formula has the following 99% confidence interval for true IRR: IRRcorr1 ± 0.4 |IRRcorr2 – IRRsecant2| PERFORMANCE COMPARISONS BETWEEN METHODS Statistical studies overview Statistical studies were performed in which the Secant Method and various alternative methods were applied to randomly-generated payment streams. The payment streams and random guesses were randomly generated according to the following parameters: One cash outflow followed by N inflows, where N was systematically varied from 5 to 30. Each inflow was uniformly distributed from 0 to 1. The true IRR was varied systematically between 1% and 10%. The initial IRR guesses used in the estimation formulas were uniformly distributed within a fixed percentage of the true IRR (25% for single-NPV, 50% for two-NPV). For each (N, IRR) combination, 100,000 different scenarios were created with different inflows and initial IRR guesses. To evaluate performance, contour plots were generated that showed the 99th-percentile levels for percentage errors of the different estimation methods for all (N, IRR) combinations in the range studied. Single-NPV methods simulation In this simulation, the IRR guess for each scenario was generated within 25% of the true value. Figure 1 compares the accuracy of three single-NPV methods: (a) Secant method; (b) Exponential interpolation; (c) Secant method with correction. The measures used to determine accuracy were the root mean square (RMS) error, 95% confidence level, and 99% confidence level for the absolute percentage error. Figure 1 Comparison of IRR estimation accuracy of secant method, exponential interpolation, and correct secant method Figure 1 shows that error measures for the secant method are roughly three and ten times larger than the corresponding for exponential interpolation and corrected secant method, respectively. The 2*RMS and 2.57*RMS levels are shown to assess the normality of the distributions. The actual confidence levels are slightly larger than those predicted by a mean-zero normal approximation, we conclude that the tails of the error distributions are slightly fatter than normal. Two-NPV method simulation In this simulation, the two IRR guesses for each scenario were generated independently within 50% of the true value. Figure 2 compares the accuracy of the three methods. Figure 2 99% confidence levels for IRR estimation percentage error, for different IRR estimation methods (based on two NPV’s) Figure 2 shows that error measures for the secant method are roughly 3.5 and 8 times larger than the corresponding for exponential interpolation and corrected secant method, respectively. The 99% confidence levels are considerably larger than those predicted by a mean-zero normal approximation, we conclude that the error distributions have numerous outliers. MATHEMATICAL DERIVATIONS Exponential interpolation Exponential interpolation is motivated by the fact that the NPV for inflows has the form NPV for inflows = n=1…N Cn (1+r)-n Since the NPV for inflows is essentially the positive combination of negative exponential terms, it is reasonable to approximate it by a single exponential: NPV for inflows K (1+r)- for some values of K and (to be determined) If the NPV for inflows is known at two different interest rates r1 and r2, then we have P1 K (1 + r1)- P2 K (1 + r2)- where P1 and P2 are the inflow NPV’s for r1 and r2 respectively. Taking the ratio of these two equations and solving for gives: log(P2 / P1) / log[ (1+rG1) / (1 + rG2) ]. Solving the first equation for K gives: K P1 (1 + rG1). Solve the following approximation for IRRexpo2 C0 K (1 + IRResxpo2)- to obtain: IRRexpo2 (C0 / K)1/ – 1 Substitution of K and followed by algebraic manipulation yields the formula for IRRexpo2. The formula for IRRexpo1 is a special case of IRRexpo2 in which r2 = 0 (so P2 = A). Secant method correction The correction term is based on the second-order Taylor Series approximation for natural logarithm: log(1 – x) x + x2/2 + O(x3) which is a good approximation when x is small. Setting x =1 – C0/P2 gives log(C0/P2) 1 – C0/P2 + (1 – C0/P2 )2/2 = (1/2) P2-2 (3P22 – 4P2C0 + C02) Similarly, setting x = 1 – P1/P2 gives log(P1/P2) 1 – P1/P2 + (1 – P1/P2 )2/2 = (1/2) P2-2 (3P22 – 4P1P2 + P12) Taking the ratio of these two equations gives ( )( ) ( ) ( )( ) Plugging into the exponential interpolation formula gives: ( )( ) ( )( ) ( )( ) Taking the first-order terms in the Taylor expansion in r1 and r2 gives ( )( ) ( )* ( )( )( ) + Which can be rearranged to give ( ) ( ) ( ( )) The factor kcorr is an empirical factor based on simulation. The correction terms used were chosen so as to minimize the 99% confidence error for the range of interest rates and guess errors simulated. The single-NPV formula may be obtained by setting r2=0, which corresponds to P2 = A. CONCLUSION The feasibility and accuracy of alternative methods for estimating IRR have been established. These methods are suitable for hand calculation, and are from 3-10 times more accurate than the secant method with comparable amounts of computation, over a broad range of scenarios. For the secant method with correction term, an estimate of the uncertainty has also been supplied. REFERENCES Cary, D. (2010) Calculating NPV and IRR with sub-annual and seasonal cash flow patterns. Journal of American Academy of Business 16,1 Fries, C.P. (2007)Mathematical Finance: Theory, Modeling, Implementation Wiley Publishing Megginson, W.L. & Smart, S.B. (2006) Introduction to corporate finance. Thompson Southwestern, United States Tang, S.L. & Tang, H.J. (2003) The variable financial indicator IRR and the constant economic indicator. The Engineering Economist 48,1