Justification for Reinvestment Assumption in NPV and IRR Formulas? Please help

Question

I am currently studying for CFA and getting confused about the reinvestment assumption that is claimed to be implicit in both the NPV and IRR formulas. From having done a lot of online research, it seems to be the case that many people think:

• NPV is true if and only if all interim cash flows are assumed to be reinvested at the discount rate

• IRR is true if and only if all interim cash flows are assumed to be reinvested at the IRR

The reason I am confused is because I cannot see an obvious reason as to why this is the case. From some of the stuff online, some people seem to justify the IRR reinvestment assumption in the following way:

Suppose initial investment is $\\\$200$, and from this I can expect to receive $\\\$150$ in year 1, $\\\$200$ in year 2, and $\\\$130$ in year 3.

In this case, IRR would be calculated by solving the following:

$-200 + \dfrac{150}{(1+IRR)}+\dfrac{200}{(1+IRR)^2}+\dfrac{130}{(1+IRR)^3} = 0$

Which gives an IRR of $61.7\%$. In order to prove that this can only be true if interim cash flows are reinvested at $61.7\%$, consider the following:

Interim cash flow 1 reinvested at IRR until end of period: $150(1+0.617)^2 = 392.2$

Interim cash flow 2 reinvested at IRR until end of period: $200(1+0.617) = 323.4$

Hence the year 3 value of the initial investment is $392.2 + 323.4 + 130=845.6$. This is equivalent to the initial investment compounded over the three years at the IRR $61.7\%$. In other words, $200(1+0.617)^3 = 845.6$. If one assumes a different rate of reinvestment for the interim cash flows, then the year 3 value of the initial investment will be different to what is implied by the IRR formula.

For example, suppose we chose to not reinvest the interim cash flows (i.e. our reinvestment rate is $0\%$), then we get the following:

Interim cash flow in year 1 not reinvested: $150$

Interim cash flow in year 2 not reinvested: $200$

Hence the year 3 value of the initial investment (when interim cash flows are not reinvested) is $150+200+130 = 480$. This gives a much lower CAGR of $33.9\%$ (i.e. $200(1+0.339)^3 = 480$.

Therefore whilst not featured as an explicit term in the formula, the formula implies that interim cash flows must be reinvested at the IRR in order for it to remain true.

Here is why I am confused:

Firstly, this kind of proof seems to assume that the IRR formula is returning a value that is the CAGR you can expect on your initial investment, which is not what I understood from reading about it. I thought IRR is just the discount rate that sets the NPV equal to 0, but this does not necessarily need to be a CAGR for the initial investment? For example, couldn't we just say that the IRR is the CAGR you can expect from your initial investment if and only if interim cash flows are reinvested. Otherwise, it is just the discount rate that sets the NPV equal to 0.

Secondly, I just don't understand why interim cash flows need to be reinvested - I can't get any intuitive grasp on this. Some resources are saying that by discounting a future cash flow to its present value using a discount rate, you are assuming that if you had that money today you could invest it at the discount rate. Therefore in order to maintain consistency, interim cash flows are assumed be reinvested at the discount rate in both the NPV and IRR formulas since they have a constant discount rate. I agree that by discounting future money back to the present using a discount rate we are saying that if we had that money today we could invest it at the discount rate, but just because I discount an interim cash flow to its present value using a discount rate, this does not imply that I need to reinvest this interim cash flow when the time comes. All that is happening when an interim cash flow is discounted using a discount rate is we are saying that it, as future money, is worth less to us now because of the opportunity cost of the discount rate. I do not see why reinvestment is necessitated by this.

Please can anyone help.

Answer 1

I think you understood the basic issue, that $IRR \ne CAGR$ even though many people think it is or should be. It is equal only when the reinvestment assumption is true. But this does not prevent you for using IRR (or CAGR, or TWR or other concepts of return) regardless of reinvestment as long as you realize that these concepts of return are not the same.

In Private Equity for ex. spectacular returns are obtained in the first few years of a fund and then they taper off as some acquired companies turn out much more difficult to turn around and sell, leading to $IRR \gg CAGR$, so you can't just look at IRR, or even worse compare private equity IRR to S&P 500 CAGR.

In Bonds it is similar. The IRR (under the name of YTM) is widely used but is not equal to the Horizon Return (the increase in wealth after investing for N years), except in the peculiar case where the reinvestment assumption holds. But in general the Horizon Return should be computed with actual or assumed reinvestment rates, and will be different from the YTM.

This is my view and there are also those who say we should teach that "there is no reinvestment assumption in the first place". My position is that the reinvestment assumption is not essential to IRR but IRR could be misleading in some situations, due to reinvestment issues. This post discusses the controversy How to calculate YTM in case coupon payments are reinvested at a different rate than the bond's coupon rate? It has some interesting references to the literature.