risk and return in environmental economics...robert pindyck (mit) risk and return june 2011 4 / 18....

RISK AND RETURN IN ENVIRONMENTAL

ECONOMICS

Robert S. Pindyck

Massachusetts Institute of Technology

June 2011

Robert Pindyck (MIT) RISK AND RETURN June 2011 1 / 18

Introduction

Environmental policy imposes social costs, yields expected socialreturn.

Like other private or public investments, return is uncertain.

Want to characterize risk/return tradeoff for environmentalinvestments.

Focus on climate change: long time horizon and considerableuncertainty.

Costly abatement would reduce GHG emissions now, and yielduncertain future benefits.

How important is reducing risk vs. expected benefits?

Two related questions:

Introduction

Introduction (Con’t)

Suppose under BAU global temperature is expected to increase,but with uncertainty that grows with time horizon.

Consider abatement policy to reduce expected rate of increase intemperature by a small amount.

This would yield future flow of uncertain benefits to society(uncertain because temperatures under BAU are uncertain).How does expected return from abatement policy compare tovariance of return?

Find the Sharpe Ratio for this policy, and see how it depends on:

Risk aversion, real GDP growth rate, discount rate, etc.Expected rate of temp. increase and variance of rate.

This would yield future flow of uncertain benefits to society(uncertain because temperatures under BAU are uncertain).

How does expected return from abatement policy compare tovariance of return?

Risk aversion, real GDP growth rate, discount rate, etc.

Expected rate of temp. increase and variance of rate.

Again, under BAU temperature will increase, with uncertaintythat grows with time horizon.

Compute “willingness to pay” (WTP) to reduce expected rate ofwarming and/or variance by some amounts.

WTP is maximum percentage reduction in current and futureconsumption society would give up to achieve that change.(Demand side of policy.)What is the trade-off between reducing expected rate of changeof temperature versus reducing the variance?What combinations of drift reduction and variance reductionyield the same WTP? Calculate “iso-WTP” curves.

Iso-WTP curve is social risk-return indifference curve. For agiven WTP, it describes “demand-side” policy tradeoff betweenrisk and return.

WTP is maximum percentage reduction in current and futureconsumption society would give up to achieve that change.(Demand side of policy.)

What is the trade-off between reducing expected rate of changeof temperature versus reducing the variance?What combinations of drift reduction and variance reductionyield the same WTP? Calculate “iso-WTP” curves.

WTP is maximum percentage reduction in current and futureconsumption society would give up to achieve that change.(Demand side of policy.)What is the trade-off between reducing expected rate of changeof temperature versus reducing the variance?

What combinations of drift reduction and variance reductionyield the same WTP? Calculate “iso-WTP” curves.

Basic Model

I use a simple model in which temperature follows an arithmeticBrownian motion (ABM), and reduces GDP growth rate.

Tt = anthropomorphic increase in temperature:

dT = αTdt + σTdz . (1)

Tt reduces real growth rate of consumption, gt :

gt = g0 − γTt , (2)

so process for gt is:

dg = −γαTdt − γσTdz ≡ −αdt − σdz . (3)

Consumption at a future time t is:

Ct = C0e∫ t0 g(s)ds = C0e

g0t− 12 αt2−σ

∫ t0 z(s)ds , (4)

Set C0 = 1.

Basic Model

gt = g0 − γTt , (2)

g0t− 12 αt2−σ

∫ t0 z(s)ds , (4)

Set C0 = 1.

Basic Model

gt = g0 − γTt , (2)

g0t− 12 αt2−σ

∫ t0 z(s)ds , (4)

Set C0 = 1.

Basic Model

gt = g0 − γTt , (2)

g0t− 12 αt2−σ

∫ t0 z(s)ds , (4)

Set C0 = 1.

Basic Model

gt = g0 − γTt , (2)

g0t− 12 αt2−σ

∫ t0 z(s)ds , (4)

Set C0 = 1.Robert Pindyck (MIT) RISK AND RETURN June 2011 5 / 18

Welfare Measure

CRRA utility. So at t = 0, welfare (under BAU) is:

1− ηE0

∫ ∞

1−ηt e−δtdt . (5)

So we will want an expression for E0(C1−ηt ).

Denote F (C , g , 0) = E0(C1−ηt ), for t > 0. Write and solve

Kolmogorov eqn. for F . (See paper.) Get:

E0(C1−ηt )e−δt = e−δt+(1−η)g0t−1

2α(1−η)t2+16σ2(1−η)2t3

As t increases, E0(C1−ηt ) first decreases and then increases

without bound, so welfare integral must cover a finite horizon.

Welfare Measure

1− ηE0

∫ ∞

2α(1−η)t2+16σ2(1−η)2t3

Welfare Measure

1− ηE0

∫ ∞

2α(1−η)t2+16σ2(1−η)2t3

Welfare Measure

1− ηE0

∫ ∞

2α(1−η)t2+16σ2(1−η)2t3

Risk/Return for Incremental Abatement Policy

Consider abatement policy that reduces αT , and thereby reducesα = γαT , by a small amount.

Start with two periods, C0 = 1 and CT uncertain. Welfare(stochastic) is:

1− η

[1 + (C1−η

T e−δT )]

Abatement causes (at t = 0) small decrease in α, so return is:

rT = −∂WT

∂α= − 1

1− η

∂αC

1−ηT e−δT . (8)

Growth rate is g(s) = g0 − αs − σ∫ s0 dz = g0 − αs − σz(s) , so

C1−ηT = eg0T− 1

2 (1−η)αT 2−σ(1−η)∫ T0 z(s)ds , (9)

so return is:rT = 1

2T 2C1−ηT e−δT . (10)

Consider abatement policy that reduces αT , and thereby reducesα = γαT , by a small amount.Start with two periods, C0 = 1 and CT uncertain. Welfare(stochastic) is:

1− η

[1 + (C1−η

T e−δT )]

rT = −∂WT

∂α= − 1

1− η

∂αC

1−ηT e−δT . (8)

2 (1−η)αT 2−σ(1−η)∫ T0 z(s)ds , (9)

so return is:rT = 1

2T 2C1−ηT e−δT . (10)

1− η

[1 + (C1−η

T e−δT )]

rT = −∂WT

∂α= − 1

1− η

∂αC

1−ηT e−δT . (8)

2 (1−η)αT 2−σ(1−η)∫ T0 z(s)ds , (9)

so return is:rT = 1

2T 2C1−ηT e−δT . (10)

1− η

[1 + (C1−η

T e−δT )]

rT = −∂WT

∂α= − 1

1− η

∂αC

1−ηT e−δT . (8)

2 (1−η)αT 2−σ(1−η)∫ T0 z(s)ds , (9)

so return is:rT = 1

2T 2C1−ηT e−δT . (10)

Risk/Return (Two Periods)

Want expectation and variance of this return. Expected return is:

r eT = 1

2T 2E0(C1−ηT )e−δT = 1

2T 2e−ρ0T− 12 α(1−η)T 2+ 1

6 σ2(1−η)2T 3

where ρ0 = δ + (η − 1)g0.

Variance of return is:

V(rT ) = E0(r2T )− (r e

= 14T 4e−2ρ0T−α(1−η)T 2+ 1

3 σ2(1−η)2T 3[e

13 σ2(1−η)2T 3 − 1]

Then Sharpe Ratio is:

ST =r eT

SD(rT )=

13 (1−η)2σ2T 3 − 1

]−12

Note ST doesn’t depend on α. Both r eT and SD(rT ) grow by

factor 12α(1− η)T 2, so this cancels out of the ratio.

Also ST → 0 as T → ∞. Reason: SD(rT ) grows faster than r eT .

r eT = 1

2T 2E0(C1−ηT )e−δT = 1

2T 2e−ρ0T− 12 α(1−η)T 2+ 1

6 σ2(1−η)2T 3

where ρ0 = δ + (η − 1)g0.Variance of return is:

V(rT ) = E0(r2T )− (r e

= 14T 4e−2ρ0T−α(1−η)T 2+ 1

3 σ2(1−η)2T 3[e

13 σ2(1−η)2T 3 − 1]

ST =r eT

SD(rT )=

13 (1−η)2σ2T 3 − 1

]−12

r eT = 1

2T 2E0(C1−ηT )e−δT = 1

2T 2e−ρ0T− 12 α(1−η)T 2+ 1

6 σ2(1−η)2T 3

V(rT ) = E0(r2T )− (r e

= 14T 4e−2ρ0T−α(1−η)T 2+ 1

3 σ2(1−η)2T 3[e

13 σ2(1−η)2T 3 − 1]

ST =r eT

SD(rT )=

13 (1−η)2σ2T 3 − 1

]−12

r eT = 1

2T 2E0(C1−ηT )e−δT = 1

2T 2e−ρ0T− 12 α(1−η)T 2+ 1

6 σ2(1−η)2T 3

V(rT ) = E0(r2T )− (r e

= 14T 4e−2ρ0T−α(1−η)T 2+ 1

3 σ2(1−η)2T 3[e

13 σ2(1−η)2T 3 − 1]

ST =r eT

SD(rT )=

13 (1−η)2σ2T 3 − 1

]−12

r eT = 1

2T 2E0(C1−ηT )e−δT = 1

2T 2e−ρ0T− 12 α(1−η)T 2+ 1

6 σ2(1−η)2T 3

V(rT ) = E0(r2T )− (r e

= 14T 4e−2ρ0T−α(1−η)T 2+ 1

3 σ2(1−η)2T 3[e

13 σ2(1−η)2T 3 − 1]

ST =r eT

SD(rT )=

13 (1−η)2σ2T 3 − 1

]−12

Risk/Return – Continuous Time

Now welfare is

1− η

1−ηt e−δtdt

and return from small increase in α is:

r =∂W

∂α=

1−ηt e−δtdt

Expected return is:

re =∫ T

12 t2E0(C

1−ηt )e−δtdt =

12 t2e−ρ0t− 1

2 α(1−η)t2+ 16 σ2(1−η)2t3

Variance is V(r) = E0(r2)− (r e)2, so we need to find

E0(r2) = E0

(∫ T

1−ηt e−δtdt

Now welfare is

1− η

1−ηt e−δtdt

r =∂W

∂α=

1−ηt e−δtdt

Expected return is:

re =∫ T

12 t2E0(C

12 t2e−ρ0t− 1

2 α(1−η)t2+ 16 σ2(1−η)2t3

E0(r2) = E0

(∫ T

1−ηt e−δtdt

Now welfare is

1− η

1−ηt e−δtdt

r =∂W

∂α=

1−ηt e−δtdt

Expected return is:

re =∫ T

12 t2E0(C

12 t2e−ρ0t− 1

2 α(1−η)t2+ 16 σ2(1−η)2t3

E0(r2) = E0

(∫ T

1−ηt e−δtdt

Need expectation of products, G (C , g , 0) = E0(C1−ηi C

1−ηj ).

Write and solve Kolmogorov eqn., etc. (See paper.) Can show:

E0(r2) =∫ T

14 t2s2e−ρ0(t+s)− 1

2 α(1−η)(t2+s2)+ 112 σ2(1−η)2(t+s)3dtds .

Using eqns. (12) and (13), can find (numerically) expectation,SD, and Sharpe ratio for the cumulative return r .

To illustrate, calibrate against numbers in IPCC (2007).

Expected loss of GDP if T = 4◦C is 1% to 5%. I use 5%, whichimplies γ = .00025.E(T ) = 3◦C in 2100 implies αT = .03, so α = .0000075.5% prob of T ≥ 7◦C implies σT = .242, so σ = .000061.5% prob of T ≥ 10◦C implies σT = .424, so σ = .000106.I set g0 = .02, δ = 0, and η = 2 to 4. Also, T = 150 years.

1−ηj ).

E0(r2) =∫ T

14 t2s2e−ρ0(t+s)− 1

2 α(1−η)(t2+s2)+ 112 σ2(1−η)2(t+s)3dtds .

1−ηj ).

E0(r2) =∫ T

14 t2s2e−ρ0(t+s)− 1

2 α(1−η)(t2+s2)+ 112 σ2(1−η)2(t+s)3dtds .

1−ηj ).

E0(r2) =∫ T

14 t2s2e−ρ0(t+s)− 1

2 α(1−η)(t2+s2)+ 112 σ2(1−η)2(t+s)3dtds .

Expected loss of GDP if T = 4◦C is 1% to 5%. I use 5%, whichimplies γ = .00025.

E(T ) = 3◦C in 2100 implies αT = .03, so α = .0000075.5% prob of T ≥ 7◦C implies σT = .242, so σ = .000061.5% prob of T ≥ 10◦C implies σT = .424, so σ = .000106.I set g0 = .02, δ = 0, and η = 2 to 4. Also, T = 150 years.

1−ηj ).

E0(r2) =∫ T

14 t2s2e−ρ0(t+s)− 1

2 α(1−η)(t2+s2)+ 112 σ2(1−η)2(t+s)3dtds .

Expected loss of GDP if T = 4◦C is 1% to 5%. I use 5%, whichimplies γ = .00025.E(T ) = 3◦C in 2100 implies αT = .03, so α = .0000075.

5% prob of T ≥ 7◦C implies σT = .242, so σ = .000061.5% prob of T ≥ 10◦C implies σT = .424, so σ = .000106.I set g0 = .02, δ = 0, and η = 2 to 4. Also, T = 150 years.

1−ηj ).

E0(r2) =∫ T

14 t2s2e−ρ0(t+s)− 1

2 α(1−η)(t2+s2)+ 112 σ2(1−η)2(t+s)3dtds .

Expected loss of GDP if T = 4◦C is 1% to 5%. I use 5%, whichimplies γ = .00025.E(T ) = 3◦C in 2100 implies αT = .03, so α = .0000075.5% prob of T ≥ 7◦C implies σT = .242, so σ = .000061.

5% prob of T ≥ 10◦C implies σT = .424, so σ = .000106.I set g0 = .02, δ = 0, and η = 2 to 4. Also, T = 150 years.

1−ηj ).

E0(r2) =∫ T

14 t2s2e−ρ0(t+s)− 1

2 α(1−η)(t2+s2)+ 112 σ2(1−η)2(t+s)3dtds .

Expected loss of GDP if T = 4◦C is 1% to 5%. I use 5%, whichimplies γ = .00025.E(T ) = 3◦C in 2100 implies αT = .03, so α = .0000075.5% prob of T ≥ 7◦C implies σT = .242, so σ = .000061.5% prob of T ≥ 10◦C implies σT = .424, so σ = .000106.

I set g0 = .02, δ = 0, and η = 2 to 4. Also, T = 150 years.

1−ηj ).

E0(r2) =∫ T

14 t2s2e−ρ0(t+s)− 1

2 α(1−η)(t2+s2)+ 112 σ2(1−η)2(t+s)3dtds .

Sharpe Ratio vs. σ. (g0 = .02, δ = 0)

0.5 1 1.5 2 2.5 3 3.5 4

x 10−4

Sharpe Ratio vs. σ, α=−4.6e−006, η1=1.5, η

2=2, η

3=4, T=150

η = 1.5

η = 2

η = 4

Willingness to Pay

What is risk/return tradeoff for WTP to change α and/or σ.

WTP applies to change in welfare, measured as expected valueof flow of discounted utility.

Consider policy to move from (α0, σ0) to (α1, σ1).Notation: w1 = WTP, ρ0 = δ + (η − 1)g0 anda(t) = −1

2α(1− η)t2 + 16σ2(1− η)2t3.

With no policy intervention, welfare is

1− η

∫ ∞

0e−ρ0t+a(α0,σ0,t)dt . (14)

With intervention, welfare is

1− η

∫ ∞

0(1− w1)1−ηe−ρ0t+a(α1,σ1,t)dt . (15)

Willingness to Pay

2α(1− η)t2 + 16σ2(1− η)2t3.

1− η

∫ ∞

0e−ρ0t+a(α0,σ0,t)dt . (14)

1− η

∫ ∞

0(1− w1)1−ηe−ρ0t+a(α1,σ1,t)dt . (15)

Willingness to Pay

Consider policy to move from (α0, σ0) to (α1, σ1).

Notation: w1 = WTP, ρ0 = δ + (η − 1)g0 anda(t) = −1

2α(1− η)t2 + 16σ2(1− η)2t3.

1− η

∫ ∞

0e−ρ0t+a(α0,σ0,t)dt . (14)

1− η

∫ ∞

0(1− w1)1−ηe−ρ0t+a(α1,σ1,t)dt . (15)

Willingness to Pay

2α(1− η)t2 + 16σ2(1− η)2t3.

1− η

∫ ∞

0e−ρ0t+a(α0,σ0,t)dt . (14)

1− η

∫ ∞

0(1− w1)1−ηe−ρ0t+a(α1,σ1,t)dt . (15)

Willingness to Pay

2α(1− η)t2 + 16σ2(1− η)2t3.

1− η

∫ ∞

0e−ρ0t+a(α0,σ0,t)dt . (14)

1− η

∫ ∞

0(1− w1)1−ηe−ρ0t+a(α1,σ1,t)dt . (15)

Willingness to Pay

2α(1− η)t2 + 16σ2(1− η)2t3.

1− η

∫ ∞

0e−ρ0t+a(α0,σ0,t)dt . (14)

1− η

∫ ∞

0(1− w1)1−ηe−ρ0t+a(α1,σ1,t)dt . (15)

Willingness to Pay

Equate W1 and W2 to get WTP:

w1 = 1−[G (α1, σ1)G (α0, σ0)

] 1η−1

, (16)

where G (α0, σ0) =∫ ∞0 e−ρ0t+a(α0,σ0,t)dt, and likewise for

G (α1, σ1).

So given starting values of α and σ we can calculate WTP todecrease α and/or decrease σ.

Willingness to Pay

Equate W1 and W2 to get WTP:

w1 = 1−[G (α1, σ1)G (α0, σ0)

] 1η−1

, (16)

where G (α0, σ0) =∫ ∞0 e−ρ0t+a(α0,σ0,t)dt, and likewise for

G (α1, σ1).

So given starting values of α and σ we can calculate WTP todecrease α and/or decrease σ.

Iso-WTP Curves

Iso-WTP curves, i.e., combinations of α′ and σ′ for which theWTP is again w1, describe risk-return tradeoff.

Find (numerically) combinations of α′ and σ′ that satisfy

G (α′, σ′) = G (α1, σ1) . (17)

Can also obtain combinations of α′ and σ′ for which WTPequals some arbitrary number, w . From eqn. (16), findcombinations that satisfy

G (α′, σ′) = (1− w)η−1G (α0, σ0) . (18)

Iso-WTP Curves

G (α′, σ′) = G (α1, σ1) . (17)

G (α′, σ′) = (1− w)η−1G (α0, σ0) . (18)

Iso-WTP Curves

G (α′, σ′) = G (α1, σ1) . (17)

G (α′, σ′) = (1− w)η−1G (α0, σ0) . (18)

Iso-WTP Curves (Con’t)

Iso-WTP curve is social risk-return indifference curve.

For a given WTP, it describes “demand-side” policy tradeoffbetween risk and return.

Figure shows example of iso-WTP curve.

Parameters are g0 = .02, δ = 0, η = 2, and T = 300 years.Starting drift and volatility are α0 = .0000075 andσ0 = .000061 (Point A). WTP to reduce α to zero but leave σunchanged (Point B) is .01815.Curve shows other combinations of α and σ that (relative tostarting values α0 and σ0) also have WTP of .01815.

Parameters are g0 = .02, δ = 0, η = 2, and T = 300 years.

Starting drift and volatility are α0 = .0000075 andσ0 = .000061 (Point A). WTP to reduce α to zero but leave σunchanged (Point B) is .01815.Curve shows other combinations of α and σ that (relative tostarting values α0 and σ0) also have WTP of .01815.

Parameters are g0 = .02, δ = 0, η = 2, and T = 300 years.Starting drift and volatility are α0 = .0000075 andσ0 = .000061 (Point A). WTP to reduce α to zero but leave σunchanged (Point B) is .01815.

Curve shows other combinations of α and σ that (relative tostarting values α0 and σ0) also have WTP of .01815.

Iso-WTP Curve (η = 2, g0 = .02, δ = 0)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 10−4

−1.5

−0.5

1x 10−5

WTP = 0.01815

Iso-WTP in “Uncertain Outcomes...”

Different because in that model starting point varies along curve.Ending point is E (T ) = 0.

1.5 2 2.5 3 3.5 4

SD(T) (° C)

2.12° C

WTP = .0141

WTP = .0113

MRS = −6.47

MRS = −7.30

MRS = −1.97

MRS = −0.88

MRS = −1.45

MRS = −2.01

Conclusions

Objective is to characterize risk/return tradeoff forenvironmental investments.

Incremental policy: Like other investments, can calculate SharpeRatio. How does it depend on characteristics of “investment?”

Non-incremental policy: Can (at some cost) change α and/or σ.Find WTP for this policy.

Iso-WTP curve: social risk-return indifference curve thatdescribes “demand-side” policy tradeoff between risk and return.

Cost of reducing α or σ? If linear or convex, can determineoptimal risk-return policy mix.

Conclusions

risk and return in environmental economics...robert pindyck (mit) risk and return june 2011 4 / 18....

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