Modified internal rate of return (`tvm.mirr`)¶

Returns the modified internal rate of return of a periodic cash flow series: the rate that grows the present value of the outflows, financed at a finance rate, into the future value of the inflows, reinvested at a reinvestment rate. Unlike the IRR, it states explicitly the rates at which money is borrowed and reinvested, and it always has a single value. The definition is that of the spreadsheet MIRR function.

Formula¶

For flows \(CF_0, \ldots, CF_n\) with the first at \(t = 0\):

\[ PV^{-} = \sum_{CF_t < 0} \frac{CF_t}{(1 + r_{\text{fin}})^{t}} \qquad FV^{+} = \sum_{CF_t > 0} CF_t \,(1 + r_{\text{re}})^{\,n - t} \]

\[ MIRR = \left(\frac{FV^{+}}{-PV^{-}}\right)^{1/n} - 1 \]

where \(r_{\text{fin}}\) is the finance rate and \(r_{\text{re}}\) the reinvestment rate, both per period.

Example¶

An investment of 120,000 followed by five years of income, financed at 10% and reinvested at 12%. The published values are 12.6094%; for the first three years only, −4.8045%; with reinvestment at 14%, 13.4759%.

from decimal import Decimal

import fairmark as fm

flows = [-120000, 39000, 30000, 21000, 37000, 46000]

r = fm.tvm.mirr(flows, finance_rate="0.10", reinvest_rate="0.12")
assert round(r.mirr, 6) == Decimal("0.126094")
assert abs(r.mirr - Decimal("0.12609413036590514583")) < Decimal("1e-15")
print(r.pv_negative_flows)  # -120000
print(r.fv_positive_flows)  # 217297.49504000

three_years = fm.tvm.mirr(flows[:4], finance_rate="0.10", reinvest_rate="0.12")
assert round(three_years.mirr, 6) == Decimal("-0.048045")

reinvest_14 = fm.tvm.mirr(flows, finance_rate="0.10", reinvest_rate="0.14")
assert round(reinvest_14.mirr, 6) == Decimal("0.134759")

Inputs¶

Field	Meaning	Default
`cash_flows`	Periodic cash flows, the first at \(t = 0\); at least two, with at least one positive and one negative	required
`finance_rate`	Rate paid on the money invested (applied to negative flows), decimal fraction per period	required
`reinvest_rate`	Rate earned on reinvested positive flows, decimal fraction per period	required

Outputs¶

Field	Meaning
`mirr`	Modified internal rate of return per period, decimal fraction
`pv_negative_flows`	Present value at \(t = 0\) of the negative flows, at the finance rate
`fv_positive_flows`	Value at \(t = n\) of the positive flows, at the reinvestment rate

Conventions (explicit, reported in `explain.conventions`)¶

Convention	Value	Default
Time origin	`cash_flows[0]` at \(t = 0\), as in `tvm.npv`	fixed
Negative flows	discounted to \(t = 0\) at the finance rate	fixed
Positive flows	compounded to \(t = n\) at the reinvestment rate	fixed
Finance and reinvestment rates	required inputs	none

Errors¶

Code	When
`FM-E001`	Fewer than two cash flows, or a field is missing or unknown
`FM-E003`	The finance rate or the reinvestment rate is −100% or lower; both are checked before any calculation
`FM-E006`	The flows have no positive or no negative value
`FM-E008`	Arithmetic overflow (inputs far outside a realistic range)

Accuracy¶

The present and future values use exact integer powers; measured against an independent 60-digit computation, these agree to about \(10^{-24}\) relative or better. The \(n\)-th root is computed as \(\exp(\ln(x)/n)\) with the decimal logarithm and exponential, which agree to about \(10^{-18}\) relative or better on typical inputs. The golden cases are checked to \(10^{-19}\).

Verified against¶

Published examples: ECMA-376 / Excel MIRR examples, 12.6094%, −4.8045% and 13.4759%.
Agrees with the MIRR function of Excel and LibreOffice, numpy_financial.mirr and pyxirr.mirr.

Modified internal rate of return (tvm.mirr)¶