When deciding whether to invest in a project an investor first will compare investment or sunk costs to the expected profit and based on this decision will decide what to do. Depending on the specifics of the project calculating of sunk cost and expected profit might be rather different and will play the main role in the decision to invest, wait and invest later or not to invest at all. More detailed consideration of the standard NPV rule: to invest if present value of cash flow is greater than sunk cost will show that some projects cannot be simply estimated using this idea. For the irreversible projects such as building a factory or buying an option NPV method may not be proper because it does not take into consideration the opportunity cost of waiting for new information, and, then investing. In other words, if investor knows that the price of the product producing on the factory will go down or the product will not be sold at all, because of some new competitive product, he will most likely choose not to build it at all. Now, different investment opportunities may be taken into consideration, for example, building the factory in steps or start using the factory for a different use. 
Let us calculate a value of the project using regular NPV rule and NPV rule that takes into consideration time effect or this opportunity to wait and invest later. These calculation have been done by many researchers, but Pindick and Dixit in their book Investment Under Uncertainty propose very easy way to compare different results. We will just use their idea but with a simpler numbers and show how different ways of computing provide different results. On this simple example we can learn the main idea of the cost of waiting and how the investment project can have completely different value if we wait. 
Now, consider the factory that can produce some product. There is F=100 the sunk cost of building the factory and the price of the product is P=$10 today. In a year there are two possible options: the price will go up to $15 with probability 1/2 or will go down to $5 with probability 1/2 and then stay at these levels. Let us calculate net present value of the project if invested today (expected price will be $10, because 1/2·15+1/2·5=10 and the interest r=0.1 for easy calculations): NPV=100+Σ0[10/(1+r)t]=$10 notice that in this case we have summation from 0 to infinity because the process starts on the step one, it means we invest in project right now. Using our parameters we can see that NPV if invested today is equal $10. 
Now let us calculate net present value, but in this case we do not invest right now, but wait and invest only if the new price is 15, which happens with probability 1/2: NPV=0.5[100/(1+r)+ Σ1[10/(1+r)t]]=$30 we have to discount the value of the sunk cost and also sum not from zero as in the first case but from 1, since we wait for a year. Thus, we can see that NPV if we wait is bigger by $20, this value is the value of flexibility or option to wait. The same thing can be computed for threestep process and more. 
Now we can see the difference in the calculations and gain if we wait. This gives an investor new option  to wait and gather information, so his expected profit will be higher. Of course, the real world problems are much more difficult and different economists and mathematicians are trying to work on bringing to reality financial models. Assumption that the price of the product will go up to 15 or down to 5 is very unrealistic and, even though, it gives an idea on how to plan our investment it can hardly be used in real world applications. 
Among different financial models there is one that is implemented right now in most of financial analyses. The prices of products in the competitive market or the price of the stock can be represented as a geometric Brownian motion. Brownian motion was named after biologist Brown and is studied in molecular physics. The idea is very simple. The big particles of some liquid were put in different solution and Brown had been looking at the motion of particles. Though, it was random because of the large number of independent chaotic collision with molecules of the solution, he could notice some certain trajectories of the motion, so in the end he can see that the particle had traveled some distance from the initial point. The same analogy was noticed in the stock prices: since all the jumps of price are independent of time (the same idea of memoryless particles: the stock does not know what happened to it before) and there is large number of players in the market the price of the stock can be described as Brownian motion. The assumption of geometric Brownian motion is usual in financial literature. 
Ones of the first who introduced the idea of the value of waiting to invest were Robert McDonald and Daniel Siegel. They suggested that the expected discounted cash flow would be described by the following formulas: dV=αVdt+σVdz, where z  standard Wiener process, which is characterized only by its current value. In other words, to make forecast for this process one needs only the present value of the process and Wiener process has independent increments, which means that changes in the process happen independently. All these assumptions are reasonable for various financial processes 
. Another step in obtaining the optimum rule for investment is taking into consideration Bellman's Principle, which states: An optimal policy has property that, whatever the initial actions, the remaining choices constitute an optimal policy with respect to the sub problem starting at the state that results from the initial actions. This idea was expressed by Bellman in the following expression: Ft(xt)=max{π(xt,u)+1/(1+r)ε[Ft+1(xt+1)], where u is characteristics of investment decision (for example, u can be a step when the project terminated), x is the current state of the project and F(x) is the expected cash flows, which is equal to maximized sum of the present profit and discounted future value. The problem of investor is to maximize F, so he must choose u in such way that F will be the most optimal from the position of cash flow. In the twoperiod example described above present profit is the difference between the expected value and sunk cost if invested today, the continuation value is the discounted net present value if we waited till the second step. 
Now, combining equations (1)(3) and with the use of stochastic transformations Bellman equation becomes: 1/2σ2V2F’’(V)+αVF’rF=0 which is second order stochastic differential equation, which can be solved by imposing proper boundary conditions. The general solution for the equation is: F(V)=AVβ where β is a coefficient, which can be found from the boundary conditions. 

Skipping all mathematical calculations we can draw a graph  dependence F of V (Graph 1, F(V) for different values of σ) and see the optimal investment rule or in other words we can see when F(V) is greater than the sunk cost I. To avoid calculations we will use already computed data by Pindick and Dixit and demonstrated in their book Investment Under Uncertainty. We set I=1,r=0.04,δ=0.04 and σ=0.2. δ is pretty reasonable payout rate and σ is standard deviation of rate of return (on stock market is about 20 percent). Now, we can see that the standard NPV rule to invest when V is equal to I is simply inappropriate and the value of V must at least twice as large as the sunk cost. F(V)=1/4V2 shows that V has to be at least 2 in order to F(V) equal to sunk cost. 
The importance of the following result has been noticed by many economic researchers, because it modernizes the old NPV principle of investing as long as present value of cash flow is greater than sunk cost. This new approach shows how the value to wait can change the initial value of the project and, thus, change the option to invest. More deep analyses of the model can show different investment rules for various initial parameters. 
Though, the described model has several assumptions, the main assumption of presenting net present value as a geometric Brownian motion is the most important one and has been implemented in the financial field for a while. Empirical works by financial institutions have shown that such assumption lets investors obtain reasonable results and plan the investment in advance. This technology has been also used in reducing risk on the portfolios when hedging. The obtained results can be easily implemented in the options pricing theory and were applied by Pindick and Dixit in the works. With all the assumptions the model shows realistic results and have been used by many financial institutions since 1980s. 
References 1. Investment Under Uncertainity, Avinash Dixit and Robert Pindick, Princeton University Press, 1994 2. Investment timing, Robert McDonald and Daniel Siegel, The Quarterly Journal of economics, v.111, 1986




