What YTM and IRR Do—and Do Not—Assume About Reinvestment

# Answer by Count (score 3)


# Does IRR (and therefore YTM) assume that all cashflows are reinvested at the IRR (or YTM)? If so, how does IRR the formula show this?












There are many articles I have read recently that say the reinvestment of interim cashflow idea in the IRR is a fallacy though I am not sure who to believe since so many resources, for example Investopedia, says: about YTM "it is the internal rate of return (IRR) of an investment in a bond if the investor holds the bond until maturity, with all payments made as scheduled and reinvested at the same rate."

Articles I am referencing in regards to the fallacy of the reinvestment assumption: "The controversial reinvestment assumption in IRR and NPV estimates: New evidence against reinvestment assumption" (Arjunan and Kannapiran), "The IRR, NPV and the Fallacy of the Reinvestment Rate Assumptions" (Lohmann), and "The Internal Rate of Return and the Reinvestment Fallacy" (Keane).

Thanks

## Answer by Count (score 3)

https://quant.stackexchange.com/a/69352

### Calculating the YTM

The yield to maturity (YTM) is often used as a yield measure. The YTM of a bond is defined as the solution of the equation: $$ P_d=\sum_{t=1}^T\frac{C_t}{(1+r)^t} $$ Where $P_d$ is the bond's dirty price. When calculating the YTM, you don't have to worry about the reinvestment assumption. For instance, assume that you have a 6 year bond with a notional of \$100 that pays an annual interest rate of 5%. If the bond's dirty price is \$94.17, you can calculate the YTM as the solution of the equation: $$ 94.17=\frac{5}{1+r}+\frac{5}{(1+r)^2}+\frac{5}{(1+r)^3}+\frac{5}{(1+r)^4}+\frac{5}{(1+r)^5}+\frac{105}{(1+r)^6} $$ You cannot solve this equation analytically, but you can derive the solution numerically (for instance use Excel solver). The result is $r=6.1928\%$ (rounded).

### Interpretation of the YTM

Now you know that the YTM of that bond is $6.1928\%$, fine. However, the main question for an investor if he uses the YTM as a yield measure is: If I hold the bond until maturity, will the "true" return be equal to the YTM?

The answer is yes, if and only if

- The coupon payments prior to maturity can be reinvested at the YTM.

- The investor holds the bond to maturity.

Let's see why this is the case.

- You invested \$94.17 by buying the bond.

- At maturity you get \$100 notional, therefore your capital gain is \$5.83.

- The interest that you receive is \$30 ($\\\$5 \cdot 6$).

- Reinvestment of interim cashflows: You invest the \$5 from the first year for 4 years at the YTM. You get

$5 \cdot 1.061928^4 -5\approx 1.75$.

Then you invest the \$5 from the second year for 3 years at the YTM and get

$5 \cdot 1.061928^3 -5\approx 1.36$

and so on. The total capital gain from the reinvestment of interim cashflows at the YTM is \$5.05.

So, your total dollar return is \$5.83+\$30+\$5.05=\$40.88 and the total future dollars that you get are \$94.17+\$5.83+\$30+\$5.05=\$135.05. Calculating the return yields $$ \left(\frac{135.05}{94.17}\right)^{\frac{1}{6}}-1=6.1928\% $$ which is equal to the YTM.

### Summary

To calculate the YTM you don't need to worry about reinvestment income. However, ex-post, if you cannot reinvest at the YTM your realized return will not be equal to the YTM! Do the math, reinvest the interim cashflows at 4% or 12 %. Your realized return will not be equal to the YTM. In this particular example, the effect from reinvesting the cashflows is relatively small, only 12.35%, but if you have bond with long maturity this effect is no longer negligible. For example, assume that the bond has a maturity of 30 years and the bond's dirty price is \$83.91, then your YTM is again 6.1928%. However, your reinvestment income will be equal to \$287.97 which accounts for 60.93% of the future dollars that you will receive. If you can only reinvest the interim cashflows at let's say 2%, your realized return will be significantly less than 6.1928%.

I hope you find this answer helpful.

## Answer by Alper (score 2)

https://quant.stackexchange.com/a/69345

The reinvesting-of-interim-cash-flows-at-IRR (RICFI) assumption is neither required for nor has any impact on the calculation of IRR of a project, or a debt instrument's yield-to-maturity (YTM), IRR's application to debt instruments. I think the Investopedia article you have mentioned is a bit misleading in that respect. If mentioned at all, the RICFI assumption is in essence only a reminder that a "project" with some early cash flows such as a coupon bond is subject to more reinvestment risk than another one with predominantly late cash flows such as a discount bond.

Those trying to match certain liabilities rather towards the end of a project or thinking that interest rates may decline somewhat early in the life of a project may consider a project with front-loaded cash flows as having reinvestment risk. Yet others who might think the interest rates might rise or even better opportunities might come up somewhat early in the life of a project might view such projects as rather presenting reinvestment opportunities and prefer them. If such issues are important for an investor, s/he might consider assessing a project in more depth under different scenarios with more variables but bringing in additional variables and forecasts to an assessment is likely to make it more complicated and may actually reduce its usefulness if not really needed.

If one is trying to decide on one or more limited-life projects with high enough IRRs, computing their durations might help in identifying the level of reinvestment risk/opportunity associated with them. Debt instruments such as bonds are nearly always subject to duration analysis and I am a bit surprised this technique is not used more frequently when analyzing financial aspects of projects with limited lives other than debt instruments.

Addition / further comments

Some of the later answers to the OP seem to suggest that either the RICFI assumption should be included in stating the IRR of a project (or YTM of a debt instrument) or, implicitly, the project’s IRR is recalculated after the use of interim cash flows are explicitly accounted for until the end of a project. This is in general unnecessary and misleading at least for the reasons below.

First, suppose we have two projects providing the same total net cash inflow over the same number of years but one of them provides some of the positive cash flow earlier than the other. According to the time value of money concept, the project with more positive cash flows early in the project should be ranked higher. The IRR equation based on original cash flows ensures this is the case. That is, contrary to one of the claims in one of the answers, the IRR gets realized as “promised” as long as the payments are made on time.

Second, what one does with the interim cash flows is external to a given project. As soon as scenarios on how the interim cash flows might be used are included in the calculation of an IRR, we would be calculating the IRR of a different project. The egregiousness of the focus on the handling of the interim cash flows can be exemplified by thinking about how the interim cash flows should be accounted for if they were to be immediately spent by a person or paid out as dividends by a company. Only an IRR calculation based on original cash flows would account for such events correctly. In addition, if we can assume the spent or distributed interim cash flows have provided the expected returns, then we can safely assume that this is the case for all interim cash flows given a certain project.

As I said before, the RIFCI assumption is in general superfluous. Tools such as duration analysis might be more helpful than a mere “fair warning” statement such as RIFCI if and when actually needed. I think the treatment of interim cash flows should be included in a project’s IRR calculation only if this is critical to the investor in deciding between projects and the use of interim cash flows and their returns can be estimated with some certainty. However, if this is done, one would be calculating the IRR of a different project. More importantly, unless really needed, the project's analysis would likely get more complicated and uncertain without being actually more useful.

## Answer by SuavestArt (score 2)

https://quant.stackexchange.com/a/69348

There's no reinvestment assumption when IRR is computed for a series of cash flows. IRR can be found using numeric methods which have no built-in assumptions.

## Answer by emot (score 0)

https://quant.stackexchange.com/a/69351

The reinvestement assumption is not required to calculate the IRR, BUT to say some investment gave you $X\%$ equal to IRR rate, it is required that the interim cashflows are reinvested at the IRR rate. For illustration purpose consider this investment:

5 year investment, initial cost = 100, annual positive payments of +5 at the end of each year and the last payment at 5Y of +105.This investment has IRRR of 5%.

If you don't reinvest the cashflows (or reinvest at 0%) you get 125\$ from 100\$ invested. Therefore the rate of return of this project is $(125/100)^{(1/5)}-1=4.56\%$ which is lower than the IRR of 5%. But if you reinvest each interim cashflow at the IRR, then you get $5*(1+5\%)^4+5*(1+5\%)^3+5*(1+5\%)^2+5(1+5\%)^1+105=127.63\\\$$ which gives you $(127.63/100)^{(1/5)}-1=5\%$ rate of return, which is exactly the IRR. Therefore to achieve IRR of 5% you have to reinvest the interim cashflows at the IRR.

Hope that helps.