The Value of Waiting to Invest

1. The Value of Waiting to Invest Robert McDonald Daniel Siegel

2. Introduction: Paper brings out the importance of waiting to invest assuming there are risk-averse investors with diverse portfolios. The paper derives formulae for the value of the option to invest in an irreversible project when project value and cost of investing are stochastic. The paper provides a realistic means of incorporating risk aversion considerations into the timing problem and show that timing considerations are important.

3. General Investment Problem

4. Solving the valuation problem: If invest or else defer. When then PV is << expected present value of payoff When T= ∞ So Choose a boundary B to maximize Subject to 1a and 1b

5. Choose a boundary B’ to maximize << Expected PV of payoff (3) << Value of the opportunity

6. where where Important>>> Or else growth rate of project value is expected to be greater than discount rate.

7. Optimal scrapping: Payoff from scrapping is V-F where Where c* is less than 1 Firm scraps project when project value < scrap value Thus the firm can benefit from increases in V-F but is protected from decreases

8. Computing the correct discount rate: μ is the rate at which future payoffs are discounted. Actual rate of return on investment opportunity : Obtained using Ito’s derivative of option value equation 4>>

9. In CAPM: The above quadratic equation has the solution where

10. δv is that part of the required return on V that is forgone by receiving only the price increases in V. δvthen  cost to holding the option and the option is worth less δf is that part of the return on F that is forgone by receiving only the price increases in F. δfthen  increase in gain from deferral and value of option increases.

11. Jumps in Vt Positive probability that Vt can take discrete jump to 0. Investment is worthless in that case. If Poisson event is uncorrelated with the market portfolio and V, F is constant then the expected present value of the payoff with uncertain expiration date is < same as (4) Discount rate is μ+λ in (5)

12. Doing the Ito’s derivative of option value equation 4 >> now gives As the only risk is

13. Loss per dollar of V if the project were undertaken at V/F =1 and not waiting for optimal time

15. Investment decision of a monopolist: example

16. <<< inverse demand curve η is the price elasticity of demand and θt is the demand shift parameter following the stochastic process <<< instantaneous profits. Maximized profits are t are then: Present value of expected maximized profits is where From the CAPM eqn (9) <<< eqn 24 becomes δv is the Payout ratio of the installed project

17. Conclusion • Investment timing considerations are important. Suboptimally adopting a project with zero NPV can lead to a loss of 10 to 20 % of the project’s value. • It is optimal to defer investing until the PV of benefits is twice that of the investment costs. • Limitations are brought forth: • Brownian motion for V and F is credible when they are present values. • For prices that are not present values, a mean reverting process would represent that prices tend to equilibrium prices. • Reversibility is included in the analysis. If a project is more reversible it will depreciate faster. Hence high δv . Timing is worthless in such cases.