Discounting, Disagreement, and the Option to Delay

Abstract

Evaluating the benefits of investment projects related to climate change is complicated by disagreements surrounding the discount rate. It is widely known that greater discount-rate heterogeneity increases the weighted-average present value of investing if the weighted-average discount rate is held constant, which therefore reduces the minimum internal rate of return needed to justify now-or-never investment. A larger adjustment is appropriate for projects with longer lives. I extend this analysis in two directions, first by giving the decision-maker the option to delay investment. Greater discount-rate heterogeneity also increases the weighted-average present value of the option to wait and reevaluate investment in the future. This weakens the relationship between discount-rate heterogeneity and the optimal investment threshold—and in some cases actually reverses it. When discount rates are low and the project’s lifetime is short, increases in discount-rate heterogeneity can lead to tougher (not easier) optimal investment tests. The second extension examines the effect on these results of using different approaches to aggregate opinions about the discount rate, including the $\alpha$-maxmin, minimax-regret, and multi-utilitarian criteria.

Appendix

1.1 Proof of Lemma 1

Dixit and Pindyck (1994, pp. 130–132) treat the process for x as a random walk that can take steps of size $\Delta h = \sigma x \sqrt{\Delta t}$ up or down with respective probabilities

$$\begin{aligned} p = \frac{1}{2} + \frac{\mu \sqrt{\Delta t}}{2 \sigma } \;\;\; {\text {and}} \;\;\; 1-p = \frac{1}{2} - \frac{\mu \sqrt{\Delta t}}{2 \sigma }, \end{aligned}$$

for some short length of time $\Delta t$. Suppose the decision-maker waits $\Delta t$ units of time and then adopts a policy of investing the first time that $x \ge {\hat{x}}(t')$, where $t'$ is the time remaining until the investment option expires. If x is currently equal to ${\hat{x}}(t)$, then—provided $\Delta t$ is sufficiently small—the decision-maker invests after the $\Delta t$ units of time have elapsed if x increases and continues to wait if it decreases.³⁶ The present value of this policy therefore equals

$$\begin{aligned} \begin{aligned} PV [{\text {wait}}|r]&= \frac{1}{1+r \, \Delta t} \left( p \left( \frac{(1 - e^{-(r-\mu )T})({\hat{x}}(t) + \sigma {\hat{x}}(t) \sqrt{\Delta t})}{r-\mu } - I \right) \right. \\&\quad \left. \Bigg . + (1-p) F(t-\Delta t,{\hat{x}}(t)-\sigma {\hat{x}}(t) \sqrt{\Delta t};r) \right) , \end{aligned} \end{aligned}$$

(A-1)

where F is defined in Eq. (8). The Taylor series expansion of $F(t-\Delta t,{\hat{x}}(t)-\sigma {\hat{x}}(t) \sqrt{\Delta t};r)$ around $(t,{\hat{x}}(t))$ is

$$\begin{aligned} F(t-\Delta t,{\hat{x}}(t)-\sigma {\hat{x}}(t) \sqrt{\Delta t};r) = F(t,{\hat{x}}(t);r) - F_x(t,{\hat{x}}(t)-;r) \sigma {\hat{x}}(t) \sqrt{\Delta t} + O(\Delta t), \end{aligned}$$

where $F_x(t,{\hat{x}}(t)-;r)$ denotes the one-sided limit of $F_x(t,x;r)$ as $x \rightarrow {\hat{x}}(t)$ from the left. The value-matching condition implies that

$$\begin{aligned} F(t,{\hat{x}}(t);r) = \frac{(1 - e^{-(r-\mu )T}){\hat{x}}(t)}{r-\mu } - I, \end{aligned}$$

so that

$$\begin{aligned} F(t-\Delta t,{\hat{x}}(t)-\sigma {\hat{x}}(t) \sqrt{\Delta t};r) = \frac{(1 - e^{-(r-\mu )T}){\hat{x}}(t)}{r-\mu } - I - F_x(t,{\hat{x}}(t)-;r) \sigma {\hat{x}}(t) \sqrt{\Delta t} + O(\Delta t). \end{aligned}$$

Substituting this expression, together with the expressions for p and $1-p$, into Eq. (A-1) shows that

$$\begin{aligned}&PV [{\text {wait}}|r] \\&\quad = \frac{1}{1+r \, \Delta t} \left( \left( \frac{1}{2} + \frac{\mu }{2 \sigma } \sqrt{\Delta t}\right) \left( \frac{(1 - e^{-(r-\mu )T}){\hat{x}}(t)}{r-\mu } - I + \frac{(1 - e^{-(r-\mu )T})\sigma {\hat{x}}(t)}{r-\mu } \sqrt{\Delta t} \right) \right. \\&\qquad \left. + \left( \frac{1}{2} - \frac{\mu }{2 \sigma } \sqrt{\Delta t}\right) \left( \frac{(1 - e^{-(r-\mu )T}){\hat{x}}(t)}{r-\mu } - I - F_x(t,{\hat{x}}(t)-;r) \sigma {\hat{x}}(t) \sqrt{\Delta t} + O(\Delta t) \right) \right) \\&\quad = \frac{1}{1+r \, \Delta t} \left( \frac{(1 - e^{-(r-\mu )T}){\hat{x}}(t)}{r-\mu } - I + \frac{\sigma {\hat{x}}(t)}{2} \left( \frac{1 - e^{-(r-\mu )T}}{r-\mu } - F_x(t,{\hat{x}}(t)-;r) \right) \sqrt{\Delta t} + O(\Delta t) \right) \\&\quad = \frac{(1 - e^{-(r-\mu )T}){\hat{x}}(t)}{r-\mu } - I + \frac{\sigma {\hat{x}}(t)}{2} \left( \frac{1 - e^{-(r-\mu )T}}{r-\mu } - F_x(t,{\hat{x}}(t)-;r) \right) \sqrt{\Delta t} + O(\Delta t) \\&\quad = PV [{\text {invest}}|r] + \frac{\sigma {\hat{x}}(t)}{2} \left( \frac{1 - e^{-(r-\mu )T} }{r-\mu } - F_x(t,{\hat{x}}(t)-;r) \right) \sqrt{\Delta t} + O(\Delta t) \end{aligned}$$

as $\Delta t \rightarrow 0$.

1.2 Solution Algorithm When the Investment Option has a Finite Lifetime

I calculate an optimal investment policy using multiple identical grids in (t, x)-space, where t ranges from 0 to the number of years until the investment option expires and x ranges from 0 to an arbitrary upper bound which is large enough that I can assume investment occurs immediately along that boundary. Each grid gives the present value of the investment option for a different possible level of the discount rate. The present value corresponding to the discount rate $r_k$ is denoted $F_k(t,x)$.

There is no delay option along the boundary where $t=0$, so at these grid points the now-or-never solution applies. That is, at each grid point $(0,x_j)$ along this boundary, the present values equal

$$\begin{aligned} F_k(0,x_j) = \left\{ \begin{array}{ll} \frac{(1 - e^{-(r_k-\mu )T})x_j}{r_k-\mu } - I &\quad {\text{ if }} (0,x_j) \in {\text {investment}}\,{\text{ region}} , \\ 0 &\quad {\text{ if }}\, (0,x_j) \not \in {\text {investment}}\,{\text{region}}. \end{array} \right. \end{aligned}$$

That is, if $x_j$ is above the optimal now-or-never investment threshold then the decision-maker invests and the present value equals the investment payoff (calculated using the appropriate discount rate); if $x_j$ is below the threshold then the decision-maker does not invest and the present value equals zero. I then work forwards in t, filling in all of the grids simultaneously. In the waiting region the present values satisfy the partial differential equations

$$\begin{aligned} \frac{\partial F_k}{\partial t} = \frac{1}{2} \sigma ^2 x^2 \frac{\partial ^{2} F_k}{\partial x^{2}} + \mu x \frac{\partial F_k}{\partial x} - r_k F_k. \end{aligned}$$

(A-2)

Therefore, at each grid point $(t_i,x_j)$ I use a finite difference approximation of Eq. (A-2) to calculate the value of waiting, which I denote by ${\hat{F}}_k(t_i,x_j)$.³⁷ I then use the appropriate investment criterion to decide whether to invest or wait at this particular grid point, and then update all grids at this point simultaneously. The present value at this grid point is

$$\begin{aligned} F_k(t_i,x_j) = \left\{ \begin{array}{ll} \frac{(1 - e^{-(r_k-\mu )T})x_j}{r_k-\mu } - I &\quad {\text{ if }}\, (t_i,x_j) \in {\text {investment}}\,{\text{region}} , \\ {\hat{F}}_k(t_i,x_j) &\quad {\text{ if }}\, (t_i,x_j) \not \in {\text {investment}}\,{\text{ region}}.\end{array} \right. \end{aligned}$$

That is, if the decision-maker invests then the present value equals the investment payoff (calculated using the appropriate discount rate); if the decision-maker waits then the present value equals the value of waiting calculated using the finite difference approximation. I continue this process all the way to the boundary where $t_i=S$.

The various methods differ according to the investment criterion. For example, if the decision-maker attaches a weight $w_k$ to the discount rate $r_k$, the optimal now-or-never investment policy under the weighted-average approach is to invest at the grid point $(0,x_j)$ if and only if

$$\begin{aligned} \sum _k w_k \left( \frac{(1 - e^{-(r_k-\mu )T})x_j}{r_k-\mu } - I \right) > 0. \end{aligned}$$

For larger values of $t_i$, the grid point $(t_i,x_j)$ is in the investment region if and only if

$$\begin{aligned} \sum _k w_k \left( \frac{(1 - e^{-(r_k-\mu )T})x_j}{r_k-\mu } - I \right) > \sum _k w_k {\hat{F}}_k(t_i,x_j). \end{aligned}$$

In contrast, for the minimax regret criterion the optimal now-or-never investment policy is to invest at the grid point $(0,x_j)$ if and only if

$$\begin{aligned} x_j < \frac{2I}{\frac{1 - e^{-(r_1-\mu )T}}{r_1-\mu } + \frac{1 - e^{-(r_2-\mu )T}}{r_2-\mu }}. \end{aligned}$$

For larger values of $t_i$, the grid point $(t_i,x_j)$ is in the investment region if and only if $R^{\text {invest}}_{i,j} < R^{\text {wait}}_{i,j}$, where

$$\begin{aligned} R^{\text {invest}}_{i,j} \equiv \max _{k=1,2} \left( \max \left\{ 0, {\hat{F}}_k(t_i,x_j) - \left( \frac{(1 - e^{-(r_k-\mu )T})x_j}{r_k-\mu }-I\right) \right\} \right) \end{aligned}$$

is the maximum possible regret if the decision-maker invests and

$$\begin{aligned} R^{\text {wait}}_{i,j} \equiv \max _{k=1,2} \left( \max \left\{ 0, \left( \frac{(1 - e^{-(r_k-\mu )T})x_j}{r_k-\mu }-I\right) - {\hat{F}}_k(t_i,x_j) \right\} \right) \end{aligned}$$

is the maximum possible regret if the decision-maker waits.³⁸

1.3 The Relationship Between Regret and the Level of r

In the special case of a perpetual project, the regret from investing equals $(\sigma /2)\max \{0, f(r)\} \sqrt{\Delta t}$, where

$$\begin{aligned} f(r) = \beta (r) I - (\beta (r)-1) \frac{{\hat{x}}}{r-\mu }. \end{aligned}$$

Differentiating with respect to r shows that

$$\begin{aligned} f'(r) = \beta '(r) I + \left( \Big . \beta (r)-1 - (r-\mu )\beta '(r) \right) \frac{{\hat{x}}}{(r-\mu )^2}. \end{aligned}$$

Substituting in the expression for $\beta (r)$ and rearranging shows that

$$\begin{aligned} \beta (r)-1 - (r-\mu )\beta '(r) = \frac{ \left( \frac{-1}{2} - \frac{\mu }{\sigma ^2} \right) \left( \frac{2 r}{\sigma ^2} + \left( \frac{1}{2}-\frac{\mu }{\sigma ^2}\right) ^2 \right) ^{1/2} + \frac{r+\mu }{\sigma ^2} + \left( \frac{1}{2}-\frac{\mu }{\sigma ^2}\right) ^2}{\left( \frac{2 r}{\sigma ^2} + \left( \frac{1}{2}-\frac{\mu }{\sigma ^2}\right) ^2 \right) ^{1/2}}. \end{aligned}$$

(A-3)

The numerator is zero if and only if

$$\begin{aligned} \left( \frac{1}{2} + \frac{\mu }{\sigma ^2} \right) ^2 \left( \frac{2 r}{\sigma ^2} + \left( \frac{1}{2}-\frac{\mu }{\sigma ^2}\right) ^2 \right) = \left( \frac{r+\mu }{\sigma ^2} + \left( \frac{1}{2}-\frac{\mu }{\sigma ^2}\right) ^2 \right) ^2, \end{aligned}$$

which reduces to

$$\begin{aligned} 0 = \left( \frac{r-\mu }{\sigma ^2}\right) ^2. \end{aligned}$$

That is, the numerator in Eq. (A-3) equals zero when $r=\mu$. For all larger values of r it is strictly positive. It follows that $\beta (r)-1 - (r-\mu )\beta '(r) > 0$ for all possible values of the discount rate, so that f(r) is an increasing function of r.

1.4 Supplementary Material

1.4.1 Investment Payoff

If the benefit flow equals x at the time of investment then the assumption that it evolves according to geometric Brownian motion implies that the expected value of the benefit flow t years after investment equals $xe^{\mu t}$. The net present value of investment therefore equals

$$\begin{aligned} \int _0^T e^{-r t} \cdot xe^{\mu t} \, dt - I = x \int _0^T e^{-(r-\mu )t} \, dt - I = \frac{(1-e^{-(r-\mu )T})x}{r-\mu } - I. \end{aligned}$$

1.4.2 The Relationship Between Welfare and Net Present Value

Suppose a decision-maker maximizes the present value of a representative agent’s expected utility,

$$\begin{aligned} W = \int _0^\infty e^{-\delta t} {\mathbb {E}}_0 [u(c_t)] dt, \end{aligned}$$

where $u'(c) = c^{-\eta }$ for some constant $\eta \ge 0$ and consumption evolves according to the geometric Brownian motion

$$\begin{aligned} dc = \nu c \, dt + \phi c \, d\zeta . \end{aligned}$$

Consider a marginal project that generates perpetual benefits of x for T years, where x evolves according to the geometric Brownian motion

$$\begin{aligned} dx = \mu x \, dt + \sigma x \, d\xi \end{aligned}$$

and $(d\zeta )(d\xi ) = \rho \, dt$. As soon as the project is in place, the decision-maker’s objective function equals

$$\begin{aligned} W'&= \int _0^T e^{-\delta t} {\mathbb {E}}_0 [u(c_t + x_t)] dt + \int _T^\infty e^{-\delta t} {\mathbb {E}}_0 [u(c_t)] dt \\&= \int _0^T e^{-\delta t} {\mathbb {E}}_0 [u(c_t) + u'(c_t)x_t)] dt + \int _T^\infty e^{-\delta t} {\mathbb {E}}_0 [u(c_t)] dt \\&= W + u'(c_0) \int _0^T e^{-\delta t} {\mathbb {E}}_0 \left[ \frac{u'(c_t)}{u'(c_0)}x_t\right] dt \\&= W + c_0^{-\eta } \int _0^T e^{-\delta t} {\mathbb {E}}_0 \left[ (c_t/c_0)^{-\eta } x_t\right] dt. \end{aligned}$$

The properties of geometric Brownian motion imply that, when viewed from date 0, $y_t \equiv \log ((c_t/c_0)^{-\eta } x_t)$ is normally distributed with mean

$$\begin{aligned} {\mathbb {E}}_0[y_t] = \log x_0 + \left( \mu - \frac{\sigma ^2}{2}\right) t - \eta \left( \nu -\frac{\phi ^2}{2}\right) t \end{aligned}$$

and variance

$$\begin{aligned} {\text {Var}}_0[y_t] = (\sigma ^2 - 2\eta \rho \sigma \phi + \eta ^2\phi ^2)t. \end{aligned}$$

The properties of the normal distribution imply that

$$\begin{aligned} {\mathbb {E}}_0 \left[ (c_t/c_0)^{-\eta } x_t\right] = e^{{\mathbb {E}}_0[y_t] + \frac{1}{2} {\text {Var}}_0[y_t]} = {\mathbb {E}}_0[x_t] \exp \left( -\left( \eta \nu + \frac{1}{2} (2\eta \rho \sigma \phi - \eta (\eta +1)\phi ^2) \right) t \right) . \end{aligned}$$

Therefore “with-project” welfare equals

$$\begin{aligned} W' = W + c_0^{-\eta } \int _0^T e^{-r t} {\mathbb {E}}_0[x_t] dt, \end{aligned}$$

provided that the present value is calculated using the discount rate

$$\begin{aligned} r = \delta + \eta \nu + \eta \rho \sigma \phi - \frac{1}{2}\eta (\eta +1)\phi ^2. \end{aligned}$$

As the initial expenditure reduces overall welfare by $u'(c_0)I$, undertaking the project changes overall welfare by

$$\begin{aligned} W' - W - u'(c_0)I = c_0^{-\eta } \left( \int _0^T e^{-r t} {\mathbb {E}}_0[x_t] dt - I\right) . \end{aligned}$$

That is, the change in welfare equals the product of the current marginal utility of consumption and the project’s NPV.

1.4.3 Partial Differential Equation for the Option Value

Let F(t, x) denote the present value of the right to invest in the project at some future date, where t is the number of years until the investment option expires and x is the flow of benefits the project would generate if it were currently in place. Suppose that there is no possibility that investment will occur over the next dt years, so that after this increment of time has elapsed the project rights will be worth $F(t-dt,x+dx)$. Itô’s Lemma implies that

$$\begin{aligned} F(t-dt,x+dx) = F(t,x) + \left( \frac{1}{2} \sigma ^2 x^2 F_{xx}(t,x) + \mu x F_x(t,x) - F_t(t,x) \right) dt + \sigma x F_x(t,x) d\xi . \end{aligned}$$

As ${\mathbb {E}}[d\xi ]=0$, the expected value of the project rights equals

$$\begin{aligned} {\mathbb {E}}[F(t-dt,x+dx)] = F(t,x) + \left( \frac{1}{2} \sigma ^2 x^2 F_{xx}(t,x) + \mu x F_x(t,x) - F_t(t,x) \right) dt. \end{aligned}$$

As no benefit flows are generated prior to investment, the present value of the project rights is

$$\begin{aligned} F(t,x)&= e^{-r \, dt} {\mathbb {E}}[F(t-dt,x+dx)] \\&= (1-r \, dt) \left( F(t,x) + \left( \frac{1}{2} \sigma ^2 x^2 F_{xx}(t,x) + \mu x F_x(t,x) - F_t(t,x) \right) dt\right) + o(dt) \\&= F(t,x) + \left( \frac{1}{2} \sigma ^2 x^2 F_{xx}(t,x) + \mu x F_x(t,x) - F_t(t,x) - rF(t,x) \right) dt + o(dt). \end{aligned}$$

Taking $dt\rightarrow 0$ shows that F must satisfy the partial differential equation

$$\begin{aligned} F_t(t,x) = \frac{1}{2} \sigma ^2 x^2 F_{xx}(t,x) + \mu x F_x(t,x) - rF(t,x). \end{aligned}$$

(A-4)

1.4.4 Valuing Perpetual Project Rights When the Discount Rate is Known

Suppose the investment option never expires and that the precise value of r is known with certainty. The former assumption means that there is no explicit time dependence in the decision-maker’s problem, so that the value of the project rights and the optimal investment threshold will both be independent of t. Substituting the option value F(x) into Eq. (A-4) shows that F(x) satisfies the ordinary differential equation

$$\begin{aligned} 0 = \frac{1}{2} \sigma ^2 x^2 F_{xx}(x) + \mu x F_x(x) - rF(x) \end{aligned}$$

whenever x is less than the arbitrary investment threshold ${\hat{x}}$. The project rights are worthless if $x=0$, because the properties of geometric Brownian motion mean that the project’s benefit flow will always be zero. In contrast, at the investment threshold the value of the project rights equals the net present value of investing. Thus, F satisfies the boundary conditions $F(0)=0$ and

$$\begin{aligned} F({\hat{x}}) = \frac{(1-e^{-(r-\mu )T}){\hat{x}}}{r-\mu } - I. \end{aligned}$$

This combination of ordinary differential equation and boundary conditions has solution $F(x)=Ax^\beta$, where

$$\begin{aligned} \beta = \frac{1}{2} - \frac{\mu }{\sigma ^2}+\sqrt{\frac{2 r}{\sigma ^2} + \left( \frac{1}{2}-\frac{\mu }{\sigma ^2}\right) ^2} \end{aligned}$$

and A is chosen such that

$$\begin{aligned} A {\hat{x}}^\beta = F({\hat{x}}) = \frac{(1-e^{-(r-\mu )T}){\hat{x}}}{r-\mu } - I. \end{aligned}$$

That is,

$$\begin{aligned} A = \left( \frac{(1-e^{-(r-\mu )T}){\hat{x}}}{r-\mu } - I\right) {\hat{x}}^{-\beta }, \end{aligned}$$

so that the investment option is worth

$$\begin{aligned} F(x) = \left( \frac{(1-e^{-(r-\mu )T}){\hat{x}}}{r-\mu } - I\right) \left( \frac{x}{{\hat{x}}}\right) ^{\beta } \end{aligned}$$

in the waiting region (that is, for all $x < {\hat{x}}$).

1.4.5 Optimal Investment Threshold When the Discount Rate is Known

The decision-maker chooses ${\hat{x}}$ in order to maximize the value of the investment option, $Ax^\beta$. Equivalently, ${\hat{x}}$ maximizes

$$\begin{aligned} A = \left( \frac{(1-e^{-(r-\mu )T}){\hat{x}}}{r-\mu } - I\right) {\hat{x}}^{-\beta }. \end{aligned}$$

The first-order condition for this problem is

$$\begin{aligned} 0 = \left( \frac{1-e^{-(r-\mu )T}}{r-\mu }\right) {\hat{x}}^{-\beta } - \beta \left( \frac{(1-e^{-(r-\mu )T}){\hat{x}}}{r-\mu } - I\right) {\hat{x}}^{-\beta -1}. \end{aligned}$$

Solving this equation for ${\hat{x}}$ shows that the optimal investment threshold is

$$\begin{aligned} {\hat{x}} = \frac{\beta I}{\beta -1} \cdot \frac{r-\mu }{1-e^{-(r-\mu )T}}. \end{aligned}$$

Equivalently, immediate investment is optimal when the present value of the project’s future benefits equals

$$\begin{aligned} \frac{(1-e^{-(r-\mu )T}){\hat{x}}}{r-\mu } = \frac{\beta I}{\beta -1}. \end{aligned}$$

1.4.6 Achieving an Arbitrary Present Value of Future Benefits

An initial benefit flow of x implies the present value of future benefits equals an arbitrary positive constant B if and only if

$$\begin{aligned} x = \frac{B(r-\mu )}{1-e^{-(r-\mu )T}}. \end{aligned}$$

The sensitivity of this level of x to the discount rate r is measured by

$$\begin{aligned} \frac{\partial x}{\partial r} = \frac{B(1-(1+(r-\mu )T)e^{-(r-\mu )T})}{(1-e^{-(r-\mu )T})^2} = \frac{B(1-(1+y)e^{-y})}{(1-e^{-y})^2}, \end{aligned}$$

where $y=(r-\mu )T$. As $e^y > 1+y$ for all $y>0$, it follows that x is an increasing function of r. Note also that

$$\begin{aligned} \frac{\partial }{\partial y} \left( \frac{\partial x}{\partial r} \right) = \frac{B e^{-y} \left( y-2+(y+2)e^{-y}\right) }{(1-e^{-y})^3}, \end{aligned}$$

which is positive for all $y>0$. This shows that x is more sensitive to r when y is larger (that is, when r is larger and when T is larger).