Why IRR Cannot Replace Forward Rates for Cash Flow Accumulation
Summary
The document explains why the internal rate of return that equates a set of cash flows to their present value cannot generally be used as a single rate to accumulate those cash flows to a future date. Its example discounts payments with distinct forward rates across three annual periods and obtains an IRR of 3.964%. Accumulating each payment with that IRR produces a different year-three value than applying the relevant forward rates for the remaining periods.
The IRR is a single rate that summarizes the cash flow’s present-value relationship; it is not the rate applicable to each future interval. Treating it as an accumulation rate effectively assumes a flat rate over time, which need not match an upward or otherwise varying term structure. The answers connect forward rates to spot rates and show that multi-period accumulation depends on the rates for the specific intervals. The example illustrates the distinction, but does not develop broader bond valuation methods or address compounding conventions beyond its stated setup.
Key ideas
- An IRR equates the present value of a cash flow to its discounted payments.
- The IRR does not necessarily represent the rate for any particular future interval.
- Accumulating each payment with interval-specific forward rates can differ from using one IRR.
- Using IRR as a single accumulation rate amounts to assuming a flat yield curve.
- Forward rates connect spot rates across successive periods.
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Full text
# Using IRR to calculate future value of cashflow
# Using IRR to calculate future value of cashflow
Discounting a cashflow using given forward rates will result in the following present value:
PV = 102.875 = ${5\over (1+3\%)}$ + ${5\over (1+3\%)(1+4\%)}$ + ${105\over (1+3\%)(1+4\%)(1+5\%)}$
where 3%, 4% and 5% are the forward rates from years 0 to 1, 1 to 2 and 2 to 3, respectively.
The IRR corresponding to this cashflow is IRR = 3.964%
Can I use this IRR as an average rate to calculate the future value of the cashflow at the begining of year 3?
If positive, why is it different than the result achieved when using the forward rates above?
- Future value of cashflow using the IRR:
$FV_{year\ 3}$ = 115.603 = $5(1+3.964\%)^2$ + $5(1+3.964\%)$ + 105
- Future value of cashflow using the forward Rates:
$FV_{year\ 3}$ = 115.710 = $5(1+4\%)(1+5\%)$ + $5(1+5\%)$ + 105
This question came to be when I noticed that I end up with exactly the same present value when using either the IRR or the forward rates but end up with different future values when using the IRR or the forward rates.
## Answer by dm63 (score 2)
https://quant.stackexchange.com/a/32007
The IRR cannot be used to move the Present Value to a Future Value, because it doesn't represent the rate for that interval of time. It is a complicated average of rates for 1,2,3 years, not the rate for 3 years which you require.
## Answer by nathanesau (score 0)
https://quant.stackexchange.com/a/18561
The time $0$ forward rate from tme $n-1$ to time $n$ is
\begin{equation} 1 + i_0(n-1, n) = \dfrac{(1 + s_0(n))^n}{(1+s_0(n-1))^{n-1}} \end{equation}
where $s_0(n)$ is the $n$-year spot rate and $i_0(n-1, n)$ is the time $0$ forward rate from time $n-1$ to time $n$.
The term structure of interest rates must be increasing to avoid arbitrage opportunities. Suppose you have a dollar today. The accumulated value in one year is $1 + s_0(1) = 1 + i_0(0,1)$ and the accumulated value in two years in $(1 + s_0(2))^2$ = $(1 + i_0(1))(1 + i_0(i,2))$. You can't just use the IRR (that would assume a flat yield curve).Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.