<p>To be clear, we are assuming that the nominal investment value is a constant <span class="math-container">$K$</span>. As such, if we calculate the NPV in the future, that future NPV will rise.</p>
<p>Of course, we do not work with future NPV’s, we are making a decision now, and so we need to discount that future NPV by the discount rate, to get an apples-to-apples comparison.</p>
<p>I will now look at the sub-questions.</p>
<ol>
<li>An investment <span class="math-container">$K$</span> in the future is not the same thing as an investment <span class="math-container">$K$</span> now. We can buy a zero coupon bond with maturity <span class="math-container">$t$</span>, and meet the future outflow of size <span class="math-container">$K$</span> with an bond that costs <span class="math-container">$Ke^{-r_bt}$</span>, where <span class="math-container">$r_b$</span> is the mathematical discount rate on the bond. (Quoted yields follow different conventions, so you need to adjust quoted yields for the calculation.)</li>
<li>The reason I would discount future outflows at the bond rate is because that is the mechanism to translate current cash to future cash flows in a “guaranteed” fashion. You could use the project discount rate, but you are assuming reinvestment at the project discount rate, which is awkward if the project is only implemented in the future. The exact answer will depend on conventions for the NPV calculation, but there are pros and cons of each definition.</li>
</ol>
<p>Which bond to use? If one is cautious, one sticks to “risk-free” sovereign bonds. (Disclaimer: all bonds have risks.) If one feels lucky, one can take credit risk with the investment, raising the bond yield. However, default risk then needs to be taken into account in the NPV analysis.</p>
<p>Since one might not find a zero coupon bond, one needs to calculate a zero coupon curve from observed bond yields. This is non-trivial, but descriptions would be found in most textbooks of fixed income analysis.</p>

CC BY-SA 4.0
2020-09-28T08:44:34.207Z
Thank you. The comment about discounting a future NPV was the crucial point that I had been missing.
Anthony
CC BY-SA 4.0
2020-09-27T16:24:10.027Z
2020-09-27T16:24:10.027Z
Brian Romanchuk
<p>It is clear to me why the benefits accruing from an investment should be discounted. What I wish to understand is what the correct way discount the investment itself is.</p>
<p>Assuming zero inflation, consider that at a personal discount rate of <span class="math-container">$\rho$</span>, the net present value of an investment <span class="math-container">$K$</span> can be computed as
<span class="math-container">$$
 \frac{R}{\rho - \mu} - K\ ,
$$</span>
where <span class="math-container">$R$</span> is the expected revenue flow with trend <span class="math-container">$\mu$</span>. If this quantity is positive, the investment is a good one. But suppose I pose the question of how much I stand to gain by delaying the investment to some later time <span class="math-container">$t$</span>, say. The return term above is modified to <span class="math-container">$R_0 e^{\mu t}/(\rho - \mu)$</span>, and if I leave <span class="math-container">$K$</span> as is, this new NPV will be strictly greater than the old value: a speculative bubble. I hence have two questions.</p>
<ol>
<li>Should <span class="math-container">$K$</span> be discounted when computing the NPV of the investment at some future time? If so, what would the rationale be? I can understand the time value of money applied to the return, but why should a future investment (i.e. a future cost) be discounted?</li>
<li>If the answer to (i) is in the affirmative, would it make sense to use a different discount rate for <span class="math-container">$K$</span> than <span class="math-container">$\rho$</span>? It seems to me that costs should be discounted differently (more gently) than returns.</li>
</ol>

CC BY-SA 4.0
2020-09-27T15:47:19.930Z
I’m not sure that I understand your comment about R having a trend. Your equation implicitly assumes that R is constant. In any event, you discount future investments because you can buy a bond in the meantime, and so your future investment can be larger. You can use the bond discount curve for this discounted, which differs from the project discount rate.
Brian Romanchuk
CC BY-SA 4.0
2020-09-27T15:54:00.740Z
@BrianRomanchuk Thank you for the comment, I have corrected the question accordingly. I think I understand what you say about investing in a bond in the meantime, but do you have a reference for the "bond discount curve"? I would like to understand how it is used.
Anthony
CC BY-SA 4.0
2020-09-27T16:00:17.503Z
@BrianRomanchuk or do you simply mean that I discount $K$ as $K e^{-\delta t}$, where $\delta < \rho$ is the bond discount rate?
Anthony
<reference-request><discount-rate>
Rationale and Methodology for Discounting the Investment in an NPV Calculation
Anthony
# Rationale and Methodology for Discounting the Investment in an NPV Calculation

(Asked by: username_0 on 2020-09-27)

It is clear to me why the benefits accruing from an investment should be discounted. What I wish to understand is what the correct way discount the investment itself is.

Assuming zero inflation, consider that at a personal discount rate of $\rho$, the net present value of an investment $K$ can be computed as
$$
\frac{R}{\rho - \mu} - K\ ,
$$
where $R$ is the expected revenue flow with trend $\mu$. If this quantity is positive, the investment is a good one. But suppose I pose the question of how much I stand to gain by delaying the investment to some later time $t$, say. The return term above is modified to $R\_0 e^{\mu t}/(\rho - \mu)$, and if I leave $K$ as is, this new NPV will be strictly greater than the old value: a speculative bubble. I hence have two questions.

1. Should $K$ be discounted when computing the NPV of the investment at some future time? If so, what would the rationale be? I can understand the time value of money applied to the return, but why should a future investment (i.e. a future cost) be discounted?
2. If the answer to (i) is in the affirmative, would it make sense to use a different discount rate for $K$ than $\rho$? It seems to me that costs should be discounted differently (more gently) than returns.

**Tags:** reference-request, discount-rate
> **Comments:**
> **username_1**: I’m not sure that I understand your comment about R having a trend. Your equation implicitly assumes that R is constant. In any event, you discount future investments because you can buy a bond in the meantime, and so your future investment can be larger. You can use the bond discount curve for this discounted, which differs from the project discount rate.  
> **username_0**: @BrianRomanchuk Thank you for the comment, I have corrected the question accordingly. I think I understand what you say about investing in a bond in the meantime, but do you have a reference for the "bond discount curve"? I would like to understand how it is used.  
> **username_0**: @BrianRomanchuk or do you simply mean that I discount $K$ as $K e^{-\delta t}$, where $\delta < \rho$ is the bond discount rate?  

---

## 1 answers

### Answer by username_1 on 2020-09-27. Score: 2 | Accepted answer

To be clear, we are assuming that the nominal investment value is a constant $K$. As such, if we calculate the NPV in the future, that future NPV will rise.

Of course, we do not work with future NPV’s, we are making a decision now, and so we need to discount that future NPV by the discount rate, to get an apples-to-apples comparison.

I will now look at the sub-questions.

1. An investment $K$ in the future is not the same thing as an investment $K$ now. We can buy a zero coupon bond with maturity $t$, and meet the future outflow of size $K$ with an bond that costs $Ke^{-r\_bt}$, where $r\_b$ is the mathematical discount rate on the bond. (Quoted yields follow different conventions, so you need to adjust quoted yields for the calculation.)
2. The reason I would discount future outflows at the bond rate is because that is the mechanism to translate current cash to future cash flows in a “guaranteed” fashion. You could use the project discount rate, but you are assuming reinvestment at the project discount rate, which is awkward if the project is only implemented in the future. The exact answer will depend on conventions for the NPV calculation, but there are pros and cons of each definition.

Which bond to use? If one is cautious, one sticks to “risk-free” sovereign bonds. (Disclaimer: all bonds have risks.) If one feels lucky, one can take credit risk with the investment, raising the bond yield. However, default risk then needs to be taken into account in the NPV analysis.

Since one might not find a zero coupon bond, one needs to calculate a zero coupon curve from observed bond yields. This is non-trivial, but descriptions would be found in most textbooks of fixed income analysis.

> **Comments:**
> **username_0**: Thank you. The comment about discounting a future NPV was the crucial point that I had been missing.  

---

