Lecture 11

.title[
# Lecture 11
]
.subtitle[
## Climate Risk and Financial Instruments
]
.author[
### Ivan Rudik
]
.date[
### AEM 4510
]

---

``` r
if (!require("pacman")) install.packages("pacman")
pacman::p_load(
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)
options(htmltools.dir.version = FALSE)
knitr::opts_hooks$set(fig.callout = function(options) {
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knitr::opts_chunk$set(
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  echo = TRUE,
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)
  options
})
```

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---

# Roadmap

1. .hi[Weather markets (Schlenker and Taylor, 2021):] do traders forecast weather and climate?
2. .hi[Discounting] 
3. .hi[Municipal bond markets (Painter, 2020):] is sea level rise capitalized into municipal financing costs?
  - Marginal damages
4. .hi[Prediction markets (Meng, 2017):] what is the probability of environmental regulation?
5. .hi[Equity markets (Meng, 2017):] what is the financial impact of expected environmental regulation?
  - Marginal abatement costs

---

# Weather markets

<html><div style='float:left'></div><hr color='#EB811B' size=1px width=796px></html>
    
---

# Betting on the weather

The purpose of this lecture is to see whether financial markets price climate risk

Our first step: prove to ourselves that traders even recognize climate change

How?

Studying the market for weather derivatives
    
---

# Betting on the weather

Weather derivatives are a way for weather-exposed firms to manage climate risk
- Which kind of firms?

CME offers contracts based on weather indices in 13 cities (mostly US)

---

# Betting on the weather

Winter contracts are based on .hi[heating degree days (HDD)]
- HDD = max(0, `$65^\circ$`F - daily average temperature): how much colder than 65

Summer contracts are based on .hi[cooling degree days (CDD)]
- CDD = max(0, daily average temperature - `$65^\circ$`F): how much warmer than 65
  
--

Idea is `$65^\circ$`F is about where you would heat or cool a building to during the day

---

# Betting on the weather

At `$65^\circ$`F both contracts are at zero

Below 65 only HDDs matter, above 65 only CDDs matter

---

# Betting on the weather

How do the contracts work?

The contracts are bought and sold like regular assets, but the settlement price is explicitly based on the realized HDD/CDD and are sold in terms of months

For example for July:

`$$CDD_{july} = \sum_{d=1}^{31} CDD_{july,d} = \sum_{d=1}^{31} max(0, 65 - T_{july,d})$$`

where we sum over the days of July `$d$`

---

# Betting on the weather

CME contracts work where the settlement price is: 20 `$\times$` monthly CDD/HDD

They are a promise to pay, at the end of the month, 20 `$\times$` the *actual* CDD/HDD

If the July CDD contract is trading at 300 CDDs, the contract costs `$20 \times 300 = 6000$` to buy, or generates 6000 in revenue to sell

If *actual* July CDDs happened to be 330, a buy-side trader profits: `$20 \times (330 - 300) = 600$`

---

# Betting on the weather

Who might buy/sell weather contracts?

Natural gas suppliers in the Northeast

- Natural gas suppliers profit more when winter is cold (why?)

Do these firms want to buy or sell winter HDD contracts to manage risk?

---

# Mitigating risk: simple example

Selling a winter HDD contract is a bet that HDDs will be low `$\rightarrow$` winter will be warm

This bet pays off when natural gas revenues are low and helps manage risk

Example:

- The winter HDD contract is trading at 1250 HDDs

- At 1250 HDDs, NYSEG expects to generate $80 million in profit from its natural gas business

- An increase in HDDs of 1 increases natural gas profits by $80,000

---

# Mitigating risk: simple example

- The winter HDD contract is trading at 1250 HDDs
- At 1250 HDDs, NYSEG expects to generate $80 million in profit from its natural gas business
- An increase in HDDs of 1 increases natural gas profits by $80,000

If NYSEG sells `$Y$` winter HDD contracts, and the realized winter HDD is `$HDD_{actual}$`:

- Its futures market profits are: 
--
`$Y \times 20 \times (1250 - HDD_{actual})$`

- Its natural gas profits are: 
--
`$80,000,000 + 80,000 \times (HDD_{actual} - 1250)$`

Futures profits increase in `$HDD_{actual}$`, natural gas profits decrease

---

# Mitigating risk: simple example

Example:
- The winter HDD contract is trading at 1250
- At 1250 HDDs, NYSEG expects to generate $80 million in profit
- An increase in HDDs of 1 increases profits by $80,000

How many contracts should NYSEG sell if it wants to eliminate all risk?

Eliminate risk by setting sum of futures market profit and HDD-related natural gas profit to zero

`$$Y \times 20 \times (1250 - HDD_{actual}) + 80,000 \times (HDD_{actual} - 1250) = 0$$`

`$$Y = 4,000 \,\, \text{contracts}$$`

---

# Aggregating private information

Suppose now a trader has a private AI-based forecast for July weather

The July CDD contract is trading at 300

But the trader's AI forecast predicts 330 CDDs

Does the trader buy or sell?

The contract looks .hi[undervalued] so the trader buys July CDD contracts

---

# Aggregating private information

If the forecast is right and July CDDs are on average 330:

The trader receives: `$20 \times 330 = 6600$`

The trader paid: `$20 \times 300 = 6000$`

Profit per contract is:

`$$20 \times (330 - 300) = 600$$`

---

# Aggregating private information

What happens after the trader starts buying?

This is just demand going up for this good, the contract

Higher demand pushes the July CDD price up: 300 `$\rightarrow$` 308 `$\rightarrow$` 323 `$\rightarrow$` 330

As the price rises, expected profit on the remaining trades gets smaller

The trader keeps buying until price is close to the forecast-implied value

---

# Aggregating private information

Now the market price itself contains this private information about weather

At 330, other traders can infer the market expects a much hotter July than before

This is how .hi[private forecasts become public information through prices]

Trading does two things at once:

1. Informed traders make profits on their private information
2. Prices become more informative about expected HDD/CDD through their trades

---

# Betting on the weather

Who else might participate in these markets (summer or winter)?

- Farmers
- Amusement parks
- Electricity utilities
- Snow plow services
- People who think they have better private information
- Gamblers

The market price should aggregate everyone's beliefs about weather

If traders actually internalize climate information, we should see weather derivative prices respond to weather and climate forecasts

---

# Betting on the weather

Schlenker and Taylor (2021) study whether the contract prices capitalize expected short-run weather, and long-run climate change

The first step is to see whether short-run weather is capitalized into the price

How do they do it?

1. Compute the .hi[weather anomaly:] how much warmer or cooler a day is relative to its average (accounting for overall warming over time)
2. Compute whether the change in the price from open to close on a given day is associated with weather anomalies

---

# Betting on the weather

Which weather forecast provides the biggest impact on weather futures prices?

Another way of asking this: how many days ahead is a weather forecast providing the most **new** information about weather

First lets see how good forecasts are at different lead times from 3 to 10 days ahead

---

# Betting on the weather

1-3 day forecasts are essentially perfect now

10 day forecasts have 40% "skill": 40% smaller error than if you just assumed temperature would be its long-run average

---

# Betting on the weather

Forecasts >10 days out have little skill `$\rightarrow$` little information value

Forecasts 1-3 days out have near-perfect skill, 
--
but 
--
their information is probably already capitalized by prior forecasts 5-7 days ago

---

# Betting on the weather

We should expect forecasts 3-10 days out to matter the most for contract prices

We should expect little to zero effect of 1-3 day forecasts, and 10+ day forecasts

---

# July CDD prices for Laguardia airport

July in NY averages about 400 CDDs

Each year from 2001-2020 (different color lines) differs in terms of .hi[actual CDDs] (price at 0), and .hi[expected CDDs] (prices to the left of 0)

---

# July CDD prices for Laguardia airport

What do you see?

In general, prices don't move much further than 10 days before the start of July `$\rightarrow$` consistent with short-run 10+ day forecasts not containing a lot of info

---

# July CDD prices for Laguardia airport

But if that's true, why are there diffrences aross years?

Differences across years can be from long-run trends, El Nino, etc

---

# July CDD prices for Laguardia airport

Once we near the actual month (about -40), forecasts are skillful about some days during July

They start trending toward their realized values (at 0)

From -30 to 0 we are .hi[in] the actual month and observe some of the realized CDDs

---

# Betting on the weather

So far we just eyeballed data, but now we want to actually compute whether the change in the price from open to close on a given day is associated with weather anomalies

Schlenker and Taylor estimate a regression model of how weather anomalies up to 1 week before some day `$t$`, and up to 3 weeks after day `$t$` affect the change in the contract price during day `$t$`

Let's think through the intuition before seeing the results

---

# Betting on the weather

Should weather anomalies 3 days ago (i.e. in the past) affect the change in the contract price today?

---

# Betting on the weather

Should weather anomalies in the future affect the change in the contract price today?

.hi[Yes!] Skillful forecasts should predict future weather anomalies, if traders use these forecasts then future weather anomalies should affect the current price change
- Suggests forecasts 10+ days ahead might not affect the price

---

# Futures prices predict future weather

X-axis: days before (left) and after (right) current trading day

Y-axis: change in contract price given a `$1^\circ$`C higher CDD anomaly

Black line: change in today's contract price given 1 CDD more in a day in the past (left of dashed line) or future (right)

Negative `$\tau$`s are past days / weather anomalies, positive `$\tau$`s are future days / weather anomalies

---

# Futures prices predict future weather

Does past weather affect changes in current prices?

Does future weather affect changes in current prices?

---

# Futures prices predict future weather

What does all this mean?

How can future weather affect today's prices or expectations?

Traders also are using the info correctly because prices are .hi[positively] associated with weather anomalies

---

# Futures prices predict future weather

Does the weather anomaly get fully capitalized?
- Is the market accurately pricing in future weather?

If so the .hi[total capitalization] of a 1 CDD anomaly should add up to 1

The sum (integral) of the values of the black line over all `$\tau$` equals 1 if the market is pricing correctly

---

# Futures prices predict future weather

Black line: the .hi[cumulative] sum of the previous black line

Adding up over all days gives a sum of .hi[1]

The market fully internalizes short-run weather!

It's fully internalized using forecasts up to 14 days ahead

---

# Futures prices and long-run climate

Weather futures capitalize short-run weather

What about long-run changes in climate?

If so, the long run trends in futures prices should match either:
1. Long run trends in weather
2. Predicted trends from climate models

---

# Futures prices predict long-run climate

Y-axis: CDD/HDD relative to the city average (0 is average)

Lines: contract price (dark green), actual weather CDDs/HDDs (red), climate model predicted CDD/HDDs (blue/neon)

What stands out?

---

# Futures prices predict long-run climate

Futures prices are strongly correlated with .hi[climate model predictions], weather to a lesser extent

---

# Discounting

---

# Motivating discounting: http://impactlab.org/map

At the end of the century we will have much more hot days in some places
<center>
<img src="files/11-climate-change.png" alt="" width="100%" />
</center>

---

# Motivating discounting: http://impactlab.org/map

At the end of the century we will have much fewer freezing days in others

---

# Motivating discounting: http://impactlab.org/map

This has massive implications for mortality
<center>
<img src="files/11-climate-damage.png" alt="" width="100%" />
</center>

---

# Motivating discounting

Some places are expecting to have huge gains in GDP from mortality risk

Others are expecting to have huge losses

This is all happening in 60-80 years

How do we compare these costs and benefits to those incurred today?

We use a .hi[discount rate:] a value that tells us how much future dollars are worth in today's terms

---

# A simple example

Let `$r$` be the discount rate, so `$\beta = {1 \over 1+r}$` is the discount factor

Suppose we are considering two different projects that have costs and benefits that accrue differently over time

| Year | Project A Cost | Project A Benefit | Project B Cost | Project B Benefit |
|------|----------------|------------------|----------------|------------------|
| 0    | 10000          | 0                | 6000           | 0                |
| 1    | 1000           | 4000             | 0              | 1000             |
| 2    | 0              | 4000             | 0              | 3000             |
| 3    | 0              | 4000             | 0              | 3000             |

Project A has higher costs and benefits in nominal terms

---

# A simple example

**Project A:**

`$PV_A = \frac{4000}{1.05^1} + \frac{4000}{1.05^2} + \frac{4000}{1.05^3} - \frac{10000}{1.05^0} - \frac{1000}{1.05^1} = -59.39$`

**Project B:**

`$PV_B = \frac{1000}{1.05^1} + \frac{3000}{1.05^2} + \frac{3000}{1.05^3} - \frac{6000}{1.05^0} = 264.98$`

---

# What if the discount rate was 3%?

**Project A:**

`$PV_A = \frac{4000}{1.03^1} + \frac{4000}{1.03^2} + \frac{4000}{1.03^3} - \frac{10000}{1.03^0} - \frac{1000}{1.03^1} = 343.57$`

**Project B:**

`$PV_B = \frac{1000}{1.03^1} + \frac{3000}{1.03^2} + \frac{3000}{1.03^3} - \frac{6000}{1.03^0} = 544.09$`

---

# Discounting

Discounting results in us placing less value on costs and benefits that accrue in the future

A dollar 1 year from now is worth `$\beta = \frac{1}{1+r}$` dollars today

The timing of costs and benefits of projects can then sway which project has greater present value

---

# Return to Manne-Richels

We ignored the idea of discounting in our discussion of the Manne-Richels model

Our new problem with discounting is then:

`$$\min_{a_1} E[TC] = \underbrace{\frac{1}{2}a_1^2}_{\text{current cost}} + \beta\left[(1-p)\times \underbrace{0}_{\text{good state cost}} + p \times \underbrace{\frac{1}{2}(1-a_1)^2}_{\text{bad state cost}}\right]$$`

---

# Return to Manne-Richels

The first-order condition is:
`$$\frac{d E[TC]}{da_1} = a^*_1 - \beta p(1-a^*_1) = 0$$`

This gives us that:
`$$a^*_1 = \frac{\beta p}{1+\beta p}$$`

How does discounting affect our decisionmaking?

---

# Discounting and decisionmaking

`$$a^*_1 = \frac{\beta p}{1+\beta p}$$`

First, notice as `$r \rightarrow \infty$` we have `$\beta = {1 \over 1 + r} \rightarrow 0$`, we put less and less weight on the future

This means we do less abatement today in period 1!

That's intuitive, let's see what discount actually looks like graphically

What is the value of a future payment of $100?

---

# PV of $100

![higher discount rates mean lower value](11-slides-climate-finance_files/figure-html/discounting-1.png)
]

Things > 30 years in the future have basically no value with a 10% discount rate

At a 1% discount rate we value things 100 years in the future at almost half their value today
]

---

# Discounting

Why does this matter?

Lots of things (like climate change) have costs or benefits that occur .hi[far] in the future

e.g. the benefits of taking action against climate change will be mostly borne by future generations, decades from now

Depending on our choice of discount rate these costs and benefits can be substantial or trivial

---

# Discounting

1 million in damages in 200 years at a discount rate of r = 2% is worth 19,053 today

1 million in damages in 200 years at a discount rate of r = 8% is worth only 21 cents today

5 orders of magnitude difference!

This makes the choice of the discount rate one of the most important (and contentious) things about climate change policy

---

# Discounting: how do we choose?

How do we choose the discount rate?

This is just the real interest paid on certain investments

In a perfect market equilibrium, it is the productivity of capital

Why might this not be the rate we want to choose as a regulator?

---

# Discounting: how do we choose?

Issues with market rates:

Market rates don't reflect externalities

Super-responsibility of government: the government represents future generations as well as current generations (only current ones are represented in the market)

Dual-role of individuals: in political roles, people are more concerned about future generations than in their day-to-day behavior which determines the market rate

---

# Discounting: how do we choose?

With social discounting we determine the discount rate from economic and ethical considerations

Why should we discount the future?

First, .hi[time]: people are impatient

And .hi[growth/inequality]: all else equal, if someone is richer in 10 years, a dollar is worth more to them today than in 10 years (in utility terms)

---

# Ramsey Discounting

With a decent amount of math we can write the social discount rate `$r$` as a function of three terms:
`$$r = \delta + \eta \times g$$`

`$\delta$` is called the .hi[pure rate of time preference] or .hi[utility discount rate]: how much do we value future *utility*

`$\eta$` is the .hi[elasticity of marginal utility]: how quickly does marginal utility (benefit) decline in consumption (how severe are diminishing marginal returns)?

`$g$` is the .hi[growth rate]: how fast does consumption grow over time?

---

# Ramsey Discounting

Here's some alternative descriptions of how to think about these terms:

- The pure .hi["I want it now"] part of discounting
- Bigger `$\delta$` means we care less about future utility just because it comes later
]

---

# Ramsey Discounting

Here's the rest:

`$\eta$`: how much do we value poorer vs richer times/generations? Bigger `$\eta$` `$\rightarrow$` more averse to inequality over time
- `$\eta = - \frac{\partial U'(X)}{\partial X}{X \over U'(X)} =  - U''(X){X \over U'(X)}$`: tells us how .hi[concave utility] is, or how strong .hi[diminishing marginal utility] is
- If `$\eta$` is large, the utility value of an extra dollar falls quickly as consumption/wealth rises

`$g$`: how rich will we / future generations be compared to today?

---

# Ramsey Discounting

`$\eta$`: how much do we value poorer vs richer times/generations? Bigger `$\eta$` `$\rightarrow$` more averse to inequality over time

`$g$`: how rich will we / future generations be compared to today?

If future generations are richer (`$g > 0$`), a larger `$\eta$` puts .hi[less] value on future dollars relative to today

But if future generations are poorer (`$g < 0$`) a larger `$\eta$` puts .hi[more] value on future dollars relative to today

---

# Ramsey Discounting

`$$r = \delta + \eta \times g$$`

What this means is that if we have values for `$r$`, `$\eta$`, and `$g$`, we can compute the "correct" discount rate from a social perspective

How do we get values for these terms?

Two common approaches: .hi[descriptive] and .hi[prescriptive]

---

# Ramsey Discounting: the descriptive approach

The descriptive approach aims to calibrate the discount rate to the real world: market data can tell us how people value the future

We can observe `$g$` in the data / forecasts

We can sometimes estimate `$\eta$` from observed behavior over time

Once we pick a `$\delta$` we have our discount rate `$r$`

The descriptive approach generally chooses `$\delta$` so `$r$` matches market rates

Most philosophers and economists would probably not prescribe the descriptive approach

---

# Ramsey Discounting: the prescriptive approach

First we decide on the "correct" level of `$\delta$` and `$\eta$`

Then we observe `$g$` in the data / forecasts

That gives us `$r$`

---

# What's the utility discount rate?

Both approaches depend on us choosing `$\delta$`

What is the right value for `$\delta$`? 
--
This is a philosophical question

Ramsey (1928): placing different weights upon the utility of different generations is ‘ethically indefensible'

Harrod (1948): discounting utility represented a 'polite expression for rapacity and the conquest of reason by passion'

The above arguments are ethical arguments, so are typically used by those favoring the prescriptive approach

---

# What's the discount rate? Descriptive

The descriptive approach often results in the discount rate lying between 2 and 7% because that's where the market interest rate typically lies

`$g$` is observed and generally between 1 and 3%

`$\eta$` is then often estimated to be between 1 and 4

`$\delta$` is then 2-3% from reverse engineering the observed market rates

Quick example: `$\delta = 2\%, \eta = 2, g = 2\% \rightarrow r = 6\%$`

---

# What's the discount rate? Prescriptive

The prescriptive approach often results in `$\delta$` being zero or nearly zero for the ethical reasons described above:

What justification do we have for treating the well-being of future generations less than ourselves?

---

# What's the discount rate? Prescriptive

Choosing `$\eta$` also conveys ethical choices: how do we weigh the distribution of consumption across generations

Recall: `$r = \delta + \eta g$`

- `$\eta = 0$`: consumption in the future doesn't change our willingness to save/invest today (r is independent of g)

- `$\eta$` is large: if there is positive growth, we are .hi[less] likely to invest in the future (future generations will be rich anyway)

- `$\eta$` is large: if there is negative growth, we are .hi[more] likely to invest in the future (future generations will be poorer than today)

---

# Distributive justice

How should we think about well-being across generations?

These ethical views lead to different choices for `$\delta$` and `$\eta$`

---

# Distributive justice

Rawls' theory of justice applied here would set `$\delta = 0$` and `$\eta = \infty$`: fairness for all

More egalitarian perspectives with respect to:

.hi[intergenerational inequality] 
--
yields a larger `$\eta$` and larger `$r$` if growth is positive

---

# What do the experts think? Drupp et al. (2018)

---

# What do the experts think? Drupp et al. (2018)
 
<center>
<img src="files/11-experts-2.png" alt="" width="70%" />
</center>

---

# What do the experts think? Drupp et al. (2018)

---

# What do the experts think? Drupp et al. (2018)

---

# What do the experts think? Drupp et al. (2018)

---

# What do the experts think? Drupp et al. (2018)

---

# Discount rates were being significantly revised

---

# Discount rates were being significantly revised

---

# Discount rates were being significantly revised

---

# Discount rates were being significantly revised

---

# But that A-4 update was reversed in 2025

- revoked the .hi[November 9, 2023] Circular A-4 update
- reinstated the older .hi[September 17, 2003] Circular A-4
- so this shift toward lower federal discount rates was at least partly reversed
]

---

# Discount rates in the (very) long run

How should we think about discounting in the .hi[very] long run?

100, 200, 300 years into the future when we expect climate change impacts to be their worst?

Giglo, Maggiori, and Stroebel (2015) come up with a clever way to think about discount rates in the far future: looking at UK and Singaporean housing markets

---

# Discount rates in the (very) long run

In the UK and Singapore, properties are acquired via .hi[leasehold] or .hi[freehold]

- .hi[Leasehold:] temporary, pre-paid, tradable ownership contracts with maturities of 99-999 years, once it expires, you lose the property

- .hi[Freeholds:] same, but **perpetual** ownership, you never lose the property
    - Similar to how things work in the US

---

# Property prices, what do they tell us?

Imagine there are two properties A and B, .hi[identical in every way] 
--
except 
--
A is a leasehold with 500 years left until maturity and B is a freehold

Suppose we observe A selling for 900,000 dollars and B selling for 1,000,000 dollars

What do these prices mean? What value do they capture?

Let's think about a simple example: you are a real estate investor deciding on purchasing a property to add to your rental portfolio in a competitive property market

---

# Property prices, what do they tell us?

A property makes sense to buy if its cost is less than its benefits

Houses are kind of like annuities:
  - Pay an upfront cost (mortgage)
  - Get a future stream of revenues (rental payments from renters)

Suppose buyers were competing for a property that has a net present value of $900,000, what market price would we expect someone to pay for this?

.hi[$900,000!] investors will compete, bidding higher and higher prices until it reaches the benefits of owning the property (same logic as why prices are the MB of regular goods in competitive markets)

---

# Discount rates in the (very) long run

The price of a house tells us the present value of the future stream of rental payments!

Now let's go back to the original example:

Suppose we observe A selling for 900,000 dollars and B selling for 1,000,000 dollars

What does the price difference between the two properties tell us?

---

# Discount rates in the (very) long run

Both properties are identical until year 500 when .hi-blue[poof], 
--
you no longer own property A but you still own property B

The difference in prices is telling us the present value of property B rental payments starting .hi[500 years from now]

The prices tell us about how the market discounts cash flows very, very far in the future, outside anyone's expected lifespan

---

# Why do discount rates change over time?

Discount rates for cash flows this year versus 500 years in the future may be different for a lot of reasons

- .hi[Changes in growth:] if growth slows down (e.g. from climate change), discount rates fall
    - The future is getting richer slower, so the future's marginal value of a dollar is higher than if growth did not slow
    
--

- .hi[Uncertainty:] if we are uncertain about future economic conditions determining the discount rate (e.g. climate change), the discount rate we should use is lower than the average (expected) discount rate

---

# Why do discount rates change over time?

Let's get a sense of how uncertainty over the proper discount rate matters

Suppose a hypothetical public transit project is going to impose 1 trillion dollars of costs in 100 years

Suppose the pure rate of time preference `$\delta = 1\%$`, and the elasticity of marginal utility `$\eta = 1$` so that the discount rate `$r = 1\% + 1 \times g$`

We think that in 100 years there is a 50% chance of crushing climate change or no climate change, so a 50% chance economic growth will either be 0% or 6%

What are the current expected costs of the project?

---

# Why do discount rates change over time?

First convert the two growth states into discount rates:

`$$r = 1\% + g$$`

If climate change crushes growth, then:

`$$g = 0\% \qquad \rightarrow \qquad r = 1\%$$`

If there is no climate change, then:

`$$g = 6\% \qquad \rightarrow \qquad r = 7\%$$`

---

# Why do discount rates change over time?

The current expected costs average over the two possible present values:

$$E[\text{Costs}] = 50\% \times \frac{\$1 \text{ trillion}}{1.01^{100}} + 50\% \times \frac{\$1 \text{ trillion}}{1.07^{100}}$$

$$E[\text{Costs}] = \frac{1}{2} \frac{\$1 \text{ trillion}}{1.01^{100}} + \frac{1}{2} \frac{\$1 \text{ trillion}}{1.07^{100}} = \$185 \text{ billion}$$

---

# Why do discount rates change over time?

Now compute the .hi[expected discount rate]:

`$$E[r] = 50\% \times 1\% + 50\% \times 7\%$$`

`$$E[r] = 4\%$$`

The costs at the .hi[expected discount rate] are:

$$\text{Costs at } E[r] = \frac{\$1 \text{ trillion}}{(1 + E[r])^{100}}$$

$$\text{Costs at } E[r] = \frac{\$1 \text{ trillion}}{1.04^{100}} = \$20 \text{ billion}$$

---

# Why do discount rates change over time?

The expected discount rate of 4% generated costs that were 10 times smaller than the actual costs!

This means that the expected discount rate is too high compared to the actual discount rate we should be using if we are uncertain about future discount rates

What discount rate should we use?

$$\frac{\$1 \text{ trillion}}{(1+r)^{100}} = \$185 \text{ billion} \qquad \rightarrow r = .017=1.7\%$$

---

# Why do discount rates change over time?

If we expected the future discount rate to be either 1% or 7%, the proper discount rate to use was actually 1.7%, .hi[not] the average 4%!

1.7% is called the .hi[certainty-equivalent] discount rate: the certain discount rate that delivers the same present value as the possible set of uncertain rates (1% and 7%)

.hi[Main takeaway:] Uncertainty about the future economic conditions governing the discount rate makes the discount rate we should be using lower than expected

---

# Discount rates in the (very) long run: United Kingdom

What are these long run discount rate?

In the UK:

- leases expiring within 100 years cost 17% less than a freehold
- leases expiring 150-300 years from now cost 5% less

Implies a discount rate of about .hi[2.6%]

]

---

# Discount rates in the (very) long run: Singapore

What are these long run discount rate?

In Singapore:

- leases expiring within 70 years cost 40% less than a freehold
- leases expiring 95-99 years from now cost 15% less

Implies a discount rate of about .hi[2.6%]

]

---

# Discount rates on rental payments

We can check the validity of these estimates by seeing whether .hi[rental payments] depend on the length remaining of the contract

There's no reason the rent you pay for your house should depend on how much longer the owner has property rights if your lease is short

---

# Discount rates on rental payments

Rental rates (mostly) do not depend on the remaining lease time!

]

---

# Municipal bond markets

---

# Bonds

> [Wiki] A bond is a type of security under which the issuer (debtor) owes the holder (creditor) a debt, and is obliged – depending on the terms – to repay the principal (i.e. amount borrowed) of the bond at the maturity date as well as interest (called the coupon) over a specified amount of time.

Bonds are assets, can be traded on secondary markets

---

# Bonds

Why do bonds exist?

Bonds are a way to raise money:
  - At time `$t$`, issue bonds with maturity `$T$`
  - Get money from creditors
  - Pay back interest/coupon over time between `$t$` and `$T$`
  - Pay back principal at some future date `$t+T$`
  
--

We will focus on .hi[municipal bonds (munis)] for pricing climate risk, why?

---

# Municipal bonds

Why munis for pricing climate risk?

Suppose Starbucks has coffee packaging plants in Miami

Miami is expecting disastrous sea level rise, what can Starbucks do to manage it?

Move its plants somewhere else away from the ocean

The city of Miami does not have the same option: it bears the full potential cost of sea level rise

---

# Municipal bonds

How do munis work?

Local governments issue munis for financing public projects (roads, infrastructure, etc)

Debt is typically paid back in a pre-specified way
  - .hi[General obligation bonds:] paid using tax revenue
  - .hi[Revenue bonds:] project-specific revenue (e.g. parking garage revenues)
  
--

Which are less risky for creditors?

---

# Municipal bonds

How do bonds work?

They have some .hi[face value] `$FV$`, the price paid when the bond matures

They pay out a .hi[coupon] (assume annually) `$C$`

They trade on the bond market at some .hi[price] `$P$` which will depend on:
  - The face value
  - The coupon
  - When the bond matures
  - *Other underlying economic conditions affecting risk of non-payment*

---

# Municipal bonds

The price of a bond should be equal to the expected NPV cash flow

If we hold a bond, we discount future payments because:
- Inflation
- Risk of non-payment
- Etc

---

# Municipal bonds

We can define the .hi[yield to maturity] `$y$` as:
`$$P = \left[\sum_{t=1}^T \frac{C}{(1+y)^t}\right] + \frac{FV}{(1+y)^T}$$`

`$y$` is the effective interest rate the investor is getting on a price `$P$` bond

If bonds issued by Ithaca and Syracuse have the same price `$P$`, but Ithaca's has a higher flow of coupons `$C$` and face value `$FV$`, what does that mean about their yields?

Ithaca's bonds have a higher yield `$y$`!

---

# Municipal bonds

We can define the .hi[yield to maturity] `$y$` as:
`$$P = \left[\sum_{t=1}^T \frac{C}{(1+y)^t}\right] + \frac{FV}{(1+y)^T}$$`

`$y$` is the effective interest rate the investor is getting on a price `$P$` bond

Ithaca's bonds have a higher yield `$y$`, what does this mean?

Ithaca's future cash flow is perceived as riskier than Syracuse's, greater future cash flows are trading at a lower value today
  
---

# Municipal bonds

Why would future climate risk be capitalized into munis? Examples for why?

1. If climate change (e.g. sea level rise) destroys infrastructure, raises municipal costs, raises risk of bankruptcy and non-payment of the bond
--

2. If climate change induces people to leave, this shrinks the tax base, makes it more difficult for the municipality to pay back the bond, raises risk of non-payment

Factors like these should be priced into the bond if traders understand climate risk

---

# Municipal bonds

Let's work with a simple one-period zero coupon example: `$C=0$`, `$T=1$`, `$FV = 105$`:
`$$P = \frac{105}{(1+y)^1}$$`

Suppose there is no climate change and the market yield is 5%, the price of the muni is:

`$$P = \frac{105}{1+.05} = 100$$`
---

# Municipal bonds

Now suppose we are considering the same muni, but there is a 7% chance that the city will be destroyed by sea level rise before next year
  - Additional 7% chance that the bond will not be paid
  
To bear this additional risk, traders will demand a higher yield (lower price)
  
---

# Municipal bonds 
  
We can solve for the new price:

`$$P = \frac{105}{1+.05} \times \underbrace{(1 - .07)}_{\approx 1/(1+.075)} = \frac{105}{1+.05} \times \frac{1}{1+.075}=93$$`

and the associated yield:

`\begin{align}
  \frac{1}{1+y} &= \frac{1}{1+.05} \times (1 - .07) = \frac{.93}{1.05} \\
  \Rightarrow y &= \frac{1.05}{.93} - 1 = .129
\end{align}`

---

# Municipal bonds

The additional 7% climate risk:
- Decreased the price by 7% from 100 to 93
- Increased the yield by 7.9 percentage points from 5% to 12.9%

As climate risk rises, traders demand greater yields to offset the chances of non-payment

We can measure the financial risks of climate change by looking at how places with different climate risk have munis with different yields

---

# Municipal bonds and sea level rise

Painter (2020) looks at how sea level rise (SLR) risk affects bond yields
  - Also looks at other things outside what we're doing in class
  
--

How should SLR affect bond yields of different maturities?

SLR is a slow phenomenon, will matter increasingly over the next century
--

- non-existent in the short run: short-term bonds shouldn't be affected
--

- only shows up in the long run: long-run bonds should be affected if investors care
  
---

# Municipal bonds and sea level rise

Climate risk: expected percent loss of city GDP

Where is the climate risk?

---

# Municipal bonds and sea level rise

What does the raw data say about climate/SLR exposed and non-exposed munis?

Climate(-exposed) bonds are 8 basis points more expensive to offer

11 basis points higher yield

---

# Yield

Comparing munis offered in the same state and year, controlling for other factors:

---

# Gross spread

They have higher gross spreads (higher underwriter search costs): 10-15bp higher for long-term bonds

---

# Maturities

Effect is larger for longer-maturity bonds: bonds maturing

---

# Credit ratings

SLR matters most for bonds with lower ratings:

Non-high grade munis costs are 50bp higher with higher climate risk

---

# What does this all mean?

We've seen that places more exposed to SLR:
- Must offer higher yields on long-term bonds, with yields increasing in time to maturing
- Incur higher gross spreads
- Must offer even higher yields if they do not have a high credit rating

Think about the long-run equilibrium of economic activity, where people live, etc

What does this suggest will happen?

---

# What does this all mean?

1. Capital is becoming more expensive in SLR-exposed cities
--

2. These cities will be less able to fund public projects (e.g. parks), making them less desirable
--

3. Marginal households who value these projects move elsewhere
--

4. Tax base shrinks `$\rightarrow$` feedback loop

Does this tell the whole story?

---

# What does this all mean?

Alternatively:

1. Capital is becoming more expensive in SLR-exposed cities
--

2. These cities will be less able to fund public projects (e.g. parks), making them less desirable
--

3. City funds adaptation projects (e.g. sea walls) to reduce exposure, decreasing yields and capital costs

In both cases, muni markets serve an important function for directing resources and people to the most productive areas

---

# Prediction markets

Prediction markets are where traders bet on binary outcomes

Will Republicans win the senate?

How does it work?

---

# Prediction markets

You can buy a share (asset) for whether the event will happen or not happen

The price of this share is:
 - 72c for Republicans winning
 - 32c for Democrats winning

---

# Prediction markets

After the election:
  - The shares for the winning side pay off $1 each
  - The shares for the losing side are worth $0
  
What does the prediction market tell us?

Let's think about the economics of the market

Let:
  - The cost of a share be `$c$` dollars
  - Your belief about the probability of an event happening be `$p$` percent

---

# Prediction markets

Based on your beliefs about the event, your expected profit from buying a share is: `$[p \times 1 + (1-p) \times 0] - c$`

You make a profit if `$p > c$`, you make a loss if `$p < c$`

Suppose `$p > c$`, what happens?

You expect a profit, you start buying shares...

This drives up the price `$c$`

This is true as long as `$p > c$`

---

# Prediction markets

Suppose `$p < c$`, what happens?

You expect a loss, you start selling your existing shares...

This drives down the price `$c$`

This is true as long as `$p < c$`

Individual profit motives always drive `$c$` toward `$p$`

---

# Waxman-Markey

The most important US climate policy of the 2000s was the 2009 American Clean Energy and Security Act: aka .hi[Waxman-Markey (WM)]

What did WM propose to do?

Set an annual cap on `$CO_2$` emissions that starts in 2012 and declines over time to:
- 83% of 2005 levels in 2020
- 58% of 2005 levels in 2030
- 17% of 2005 levels in 2050

---

# Waxman-Markey

WM allowed permits to be .hi[traded] 
--
and also .hi[banked and borrowed]
- .hi[Banked:] permits not used this year can be saved
- .hi[Borrowed:] can emit more than retired permits today on the promise of retiring the extra necessary permits in the future
- Borrowing had an 8% interest rate

---

# Waxman-Markey

How did WM allocate permits ([info here](https://grist.org/article/2009-05-15-waxman-markey-permit-proposal/))?

Most were freely allocated:

- .hi[35%] of permits go to electric utilities
--

- .hi[9%] of permits go to natural gas distributors
--

- .hi[1.5%] of allowances go to states to buffer users of home heating oil and propane
--

- .hi[15%] go to .hi-blue[energy-intensive, trade-exposed] industries
  - Over 5% energy intensity & 15% trade intensity `$\rightarrow$` free permits

Rest are auctioned or given to different government agencies

---

# Waxman-Markey: the history

June 26, 2009: Waxman-Markey passes the House of Representatives (219-212)
  - 211/255 Democrats vote yes, 8/176 Republicans vote yes
  - First cap and trade bill to be passed by congress!
  - Still needs to pass the senate
  
--

2009/2010 senate: 59 Democrats/independents, 41 Republicans

All Democrats and one Republican need to vote yes for it to pass the senate

It took until April 2010 to convince one Republican ... who was it?

---

# Waxman-Markey: the history

Lindsey Graham!

On Thursday April 22, 2010, after months of negotiation:

John Kerry, Joe Lieberman, and Lindsey Graham complete the senate-version of the bill

The unveiling of the bill was scheduled for Monday April 26, 2010
  
--

You won't believe what happens next

---

# Waxman-Markey: the history

Senate Majority Leader Harry Reid (Nevada) was up for re-election in 2010

In April, he was trailing his Republican challenger

On Thursday April 22, 2010 Senator Reid announces the Senate will start working on an immigration bill

This makes political sense: at the time Nevada was 30% Hispanic/Latino
  
--

The Senate calendar couldn't accommodate both climate and immigration legislation: Lindsey Graham thought it was cheap point scoring from Reid

---

# Waxman-Markey: the history

On April 13/19, 2010 the Arizona state house and senate voted to pass .hi[SB 1070:] the Support our Law Enforcement and Safe Neighborhoods Act 
--
aka the .hi[Show Me Your Papers Act]

SB 1070 required state law enforcement to ask suspicious people to present proof of legal immigration status
--
, it also made it a crime to not have immigration papers on hand

SB 1070 was .hi[incredibly] controversial

It was unclear whether governor Jan Brewer would sign it

---

# Waxman-Markey: the history

Friday April 23, 2010 Jan Brewer signs SB 1070 into law

This was seen by some as legalized racial profiling of the Latino population

Immigration becomes the focal point of congress

10PM Friday April 23, 2010: Graham's aide e-mails Lieberman's aide "Sorry buddy."

Graham formally states he refuses to delay a climate bill for immigration and abandons his legislation

---

# Waxman-Markey: the probabilities

---

# Waxman-Markey: the probabilities

Markets almost never thought WM was a favorite to pass!

Even around Graham-Kerry-Lieberman announcement, probabilities were only 20%

---

# Waxman-Markey: the probabilities

The prediction market prices tell us the market's expectation of climate legislation being implemented

The legislation itself tells us how much emissions must be reduced

With one more piece of information - the .hi[costs] of the legislation - we can back out the `$CO_2$` MAC

---

# Equity markets

We can back out the costs of the legislation from .hi[stock return] data

Let's write down a simple model of stock returns and firm value to see how

- `$P_t:$` probability of WM passing
- `$V^{WM}_{i,t}:$` the value of firm `$i$` on day `$t$` if WM .hi[passes]
- `$V^{-}_{i,t}:$` the value of firm `$i$` on day `$t$` if WM .hi[does not pass]
- `$V_{i,t}:$` the expected value of firm `$i$` on day `$t$` based on the probability of WM passing:
  - `$V_{i,t} = P_t \times V^{WM}_{i,t} + (1 - P_t) \times V^{-}_{i,t}$`
  - `$V_{i,t}$` is the prediction market weighted average
  
To keep it simple, assume nothing else is changing besides WM probabilities

---

# Equity markets

`$$V_{i,t} = P_t \times V^{WM}_{i,t} + (1 - P_t) \times V^{-}_{i,t}$$`

Define `$E^{WM}_i$` as the percentage effect of WM on firm value: `$E^{WM}_i = \frac{V^{WM}_{i,t} - V^{-}_{i,t}}{V^{-}_{i,t}}$`

`$E^{WM}_i$` tells us how much firm value changed as a result of WM going into effect

Rewrite `$V_{i}$` in terms of `$E^{WM}_i$`:

`$$V_{i,t} = V^{-}_{i,t} \times (1 + P_t \, E^{WM}_i)$$`

The firm value on day `$t$` is the value of the firm if WM doesn't pass `$(V^{-}_{i,t})$`, but scaled up by the effect of WM `$(E^{WM}_i)$` times the chances WM passes `$(P_t)$`

---

# Equity markets

`$$V_{i,t} = V^{-}_{i,t} \times (1 + P_t \, E^{WM}_i)$$`

.hi[Stock returns] `$r_{i,t}$` are the change in log firm value: 
`$$r_{i,t} = \ln V_{i,t} - \ln V_{i,t-1} = \ln \frac{V_{i,t}}{V_{i,t-1}}$$`

We can write this as:

`$$r_{i,t} = \ln \frac{V^{-}_{i,t} \times (1 + P_t \, E^{WM}_i)}{V^{-}_{i,t-1} \times (1 + P_{t-1} \, E^{WM}_i)}$$`

---

# Equity markets

`\begin{align}
  r_{i,t} &= \ln \frac{V^{-}_{i,t} \times (1 + P_t \, E^{WM}_i)}{V^{-}_{i,t-1} \times (1 + P_{t-1} \, E^{WM}_i)} \\
  r_{i,t} &= \ln \frac{V^{-}_{i,t}}{V^{-}_{i,t-1}} \ln \frac{(1 + P_t \, E^{WM}_i)}{(1 + P_{t-1} \, E^{WM}_i)} \\
  r_{i,t} &= \ln \frac{V^{-}_{i,t}}{V^{-}_{i,t-1}} \times \left[ \overbrace{\underbrace{\ln (1 + P_t \, E^{WM}_i)}_{\approx P_t \, E^{WM}_i}}^{\ln (1+x) \approx x} - \overbrace{\underbrace{\ln (1 + P_{t-1} \, E^{WM}_i)}_{\approx P_{t-1} \, E^{WM}_i}}^{\ln (1+x) \approx x} \right] \\
  r_{i,t} &= P_{t} E^{WM}_i - P_{t-1} E^{WM}_i + \underbrace{\ln \frac{V^{-}_{i,t}}{V^{-}_{i,t-1}}}_{\substack{\ln 1 = 0 \\ \text{if not changing}}}
\end{align}`

---

# Equity markets

`$$r_{i,t} = P_{t} E^{WM}_i - P_{t-1} E^{WM}_i = (P_t - P_{t-1}) \times E^{WM}_i$$`

If no other determinants of firm value are changing from day-to-day, the stock return tells us the effect of WM on firm value scaled by the change in the probability that WM happens

We have data on stock returns `$r_{i,t}$`, we have data on market expectations `$P_t, P_{t-1}$`, we can then get `$E^{WM}_i$`

---

# Equity markets: getting MAC

WM generally has a bigger negative effect on firm valuations the more energy intensive the firm

---

# Equity markets: getting MAC

At an energy intensity of 5%, WM reduces firm value by about 6%!

This is the present value of the expected costs of having to buy permits in an auction versus getting them for free

How do we get the MAC (i.e. the implied permit price)?

---

# Equity markets: getting MAC

- `$\tau$`: permit price / MAC
- `$E_t$`: emissions
- `$r$`: interest rate

The present value of the expected costs is: `$.06 = \sum_{t=0}^\infty \frac{\tau \times E_t}{(1+r)^t}$`

If we have forecasts about emissions and the interest rate, we can solve for `$\tau$`!

---

# Equity markets: getting MAC

Using the listed decline in emissions and a 5% interest rate gives us these MACs

In 2010 MACs were...quite low!

At most `$18/tCO_2$`, about 1/10 the current best estimate of MD, 1/3 the best estimate in 2010

What does this imply about WM?

If MAC < MD, then the cap is .hi[below] the socially efficient level