# 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.

The time $0$ forward rate from tme $n-1$ to time $n$ is
$$1 + i_0(n-1, n) = \dfrac{(1 + s_0(n))^n}{(1+s_0(n-1))^{n-1}}$$
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).