Valuing Deferred Investment Cash Flows Under Risky and Risk-Free Discounting

# NPV of Future Investment: Two Approaches?


# NPV of Future Investment: Two Approaches?












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:

- The "strict-present-value" approach: $$ \int_t^\infty R_s e^{-\rho s}ds = \frac{R_0 e^{-(\rho-\mu)t}}{\rho - \mu} =\frac{R_t e^{-\rho t}}{\rho - \mu}\ . $$

- 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?

## Answer by fes (score 2, accepted)

https://quant.stackexchange.com/a/58678

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.

## Answer by demully (score 1)

https://quant.stackexchange.com/a/58707

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 ;-)