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Improvements on Secant Method for Estimating Internal Rate of Return

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					              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

				
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Description: 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.