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Suppose I expect the return on my investment to follow some upward trend: $R_t = R_0 e^{\mu t}$, where $\mu > 0$. If I wish to compute the present value of these inflows, I would have $$ \int_o^\infty R_t e^{-\rho t}dt = \frac{R_0}{\rho - \mu}\ , $$ where $\rho > \mu$ is the project discount rate.

If I now pose the question, what is net-present value of these inflows if I delay my investment to some future time $t$, I see two possibilities:

  1. The "strict-present-value" approach: $$ \int_t^\infty R_s e^{-\rho s}ds = \frac{R_0 e^{-(\rho-\mu)t}}{\rho - \mu} =\frac{R_t e^{-\rho t}}{\rho - \mu}\ . $$
  2. The "already-there" approach: $$ \left(R_t \int_0^\infty e^{\mu s} e^{-\rho s}ds \right) e^{-\delta t} = \frac{R_t e^{-\delta t}}{\rho - \mu} \ , $$ where I introduce the risk-free rate $\delta < \rho$ to discount to the present.

The rationale behind the first approach should be clear. I arrived at the second by asking what an investor actually means when he asks if he should delay the investment: he is in fact placing himself in the future, where his return has some higher expected value $R_t$, and then he integrates out to infinity again from his new "$t=0$", but now needs to discount by a different rate to bring these values to the present.

These approaches are clearly commensurate if and only if $\delta = \rho$. Is there a reason to prefer one approach to the other?

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    $\begingroup$ The first approach looks more standard. How would you determine $\delta$? $\endgroup$
    – fesman
    Oct 13 '20 at 14:44
  • $\begingroup$ @fesman I had in mind something like this: investopedia.com/terms/r/risk-freerate.asp $\endgroup$
    – Anthony
    Oct 14 '20 at 10:50
  • $\begingroup$ The very first sentence in your linked article says the risk free rate is purely theoretical. You need to use real world discount factors (rates) to compare real world choices $\endgroup$
    – user50421
    Oct 14 '20 at 22:55
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    $\begingroup$ I would say the proper way to approach this is the second one, except $\delta$ does not need to be a risk-free rate. $\delta$ should somehow reflect your opportunity cost or your inter-temporal preference; it should answer the question "why am I delaying my investment until $t$?". So it might actually be a risk-free rate, or the cost of capital of your project $\rho$. The core of it is that it should reflect your opportunity cost over $[0,t]$. $\endgroup$ Oct 15 '20 at 9:36
  • $\begingroup$ @DaneelOlivaw Thank you, this confirms my intuition. Would you happen to know of a reference for this argument? $\endgroup$
    – Anthony
    Oct 15 '20 at 10:06
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Assume you apply a constant discount rate $\rho$ to your risky payoff and discount rate $\delta$ to riskless payoffs. The time $t$ value of your payoff stream is

$$\int_t^{\infty}R_se^{-\rho (s-t)}ds=\frac{R_t}{\rho-\mu},$$

where you need $\rho > \mu$. Within this framework the time $0$ value of your investment is

$$\frac{R_te^{-\delta t}}{\rho-\mu},$$ where I used the fact that this present value is known at time $0$. That is the correct valuation formula is the second one.

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  • $\begingroup$ The question asks “Is there a reason to prefer one approach to the other?” It does not ask for yet another formula derivation. $\endgroup$
    – user50421
    Oct 14 '20 at 22:57
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    $\begingroup$ @Stripedbass I am explaining what I think is the right way to do NPV math. I don’t think the question asks whether risk-free interest rates exist. $\endgroup$
    – fesman
    Oct 15 '20 at 5:48
  • $\begingroup$ @fesman You're right that introducing $\delta$ out of the blue would not make sense. But it seems to me that I need a risk-free rate to discount any outflows (economics.stackexchange.com/a/39936/23231) and it seems to me that there is a valid argument for additionally using this rate to discount the sum of future inflows, invested in the future. $\endgroup$
    – Anthony
    Oct 15 '20 at 7:31
  • $\begingroup$ @Anthony So are you assuming the payoff after $t$ is risky? $\endgroup$
    – fesman
    Oct 15 '20 at 7:53
  • $\begingroup$ @fesman Right. The payoff after $t$ is discounted by the project-rate $\rho$ (which may, with good reason, be high). But I could see no reason to additionally use $\rho$ to discount an investment which I have not yet made. $\endgroup$
    – Anthony
    Oct 15 '20 at 8:06
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The long and horrible answer is that the strict answer depends on the term structure of interest rates. But let's ignore that, for simplicity's sake ;-)

It also helps to demonstrate the theory if you can assume the security is liquid, because then you can make an arbitrage argument. Because then the deferred purchase based on the dividends/coupons received from year T (the waiting period) will be the T year forward. This forward should be priced, driven by the basis between your growth rate (Mu) and your cash rate (Delta). If cash plus forward does not equal spot, then there's an abritrage to be had. At least in theory, assuming a frictionless liquid market etc. etc.

Even if this does not hold in reality, it's hopefully evident that the discount rate morphs to being a function of both cash and risky. This is intuitive. By deferring the purchase, the investor is after all blending cash and risky returns in his payout schedule.

Below is a simple monkey model. If R1 is 1.00, then the current price will indeed be 1/(Rho-Mu) as per your first equation. And if payouts R grow by Mu, then the same Rho will produce a future price of Spot * (1+Mu)^T. The investor who defers receives cash returns (Delta, in red) through the deferment period. But from then, the amount of risky asset he can purchase at time T will be a function of the cash:risky basis, so his R values after purchase at time T must be pro-rated to this. Calculating the IRRs of these shows that the aggregate discount rate shifts from Rho towards Delta, the longer he defers purchase and thereby spends longer holding cash rather the risky asset.

So the quick answer to your question is (1) "somewhere in between" your two equations! And for short periods (ie a couple of years), the difference between this and Rho is insignificant. Likely a fraction of your forecast error of the value of Rho itself ;-)

enter image description here

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  • $\begingroup$ I’ve only been on quant stack a short time, but it’s quickly apparent that academics dominate discussion, and real world practitioners get bored and stop commenting or answering. Count the number of times you said “assuming xyz” or similar. Simplifications so your pet theories don’t have to deal with real world complexity. $\endgroup$
    – user50421
    Oct 15 '20 at 23:28
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    $\begingroup$ Maybe they do; but I like to think I've seen a thing or two in two decades down the coalface in the sellside broking industry. My concluding "pet theory" was "it doesn't really matter, does it?" If that's "academic", sorry but I am confused what you're saying here? $\endgroup$
    – demully
    Oct 16 '20 at 11:15
  • $\begingroup$ @demully Thank you for this excellent answer! It's helpful to know that it's not just a question of taste or NPV math, but that the passage of time has something to say here too. $\endgroup$
    – Anthony
    Oct 16 '20 at 11:35
  • $\begingroup$ I just laid off two PhD’s at my firm,, so I feel much better. Most clients are shifting to index funds, while wealthy clients want active management they can understand. No client demand for long winded equations. Tenure is irrelevant, clients rule! $\endgroup$
    – user50421
    Oct 16 '20 at 13:53

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