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Lecture 13: Financial Instruments: Bonds and Futures

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On Monday, we applied the consumer theory framework to a model of intertemporal choice; on Wednesday, we applied that same framework to analyze choice under uncertainty and attitudes toward risk. Today we’ll wrap up our analysis of consumer theory by seeing how both of these combine to determine the demand for financial instruments such as bonds and futures.

There are various types of bonds, but the simplest is a promise to pay a certain amount on a certain date in the future. For example, a 10-year treasury note is essentially a piece of paper that will grant the holder of that piece of paper $€1000$ on a date 10 years from when it is issued.

The price of a bond is quoted as a percentage of its face value: for example, if a bond that pays $€1000$ when it comes due is currently selling for $€980$, it is said to be selling for 98. So: how much would an investor be willing to pay for such a piece of paper?

The answer depends on three factors:

Let’s approach each of these in turn.

Taking inflation into account: Nominal and real interest rates

An important consideration when thinking about consumption across time is whether there is likely to be inflation. For example, suppose all you spend money on is apples, and that today apples cost $€1.00$ each. Therefore, if you have $€100$, then at today’s price of apples, that could buy 100 apples.

Now let’s think about what would happen if you saved some of your money. Suppose you can earn 10% interest on any savings. If you saved all you $€100$, therefore it would grow to $€110$ in a year. If the price of apples is unchanged, you could therefore afford 110 apples. So far, so good.

However, what would happen if the price of apples increased – say, to $€1.08$? Then with your $€110$, you could only afford $110/1.08 \approx 102$ apples. In terms of purchasing power, your savings only grew by about 2% (from 100 apples to about 102 apples).

How can we incorporate this into our model? The intertemporal budget constraint we’ve been dealing with has been \(c_1 + {1 \over 1+r}c_2 = m_1 + {1 \over 1 + r}m_2\) where $r$ is the nominal interest rate (10% in the example above). Instead of thinking of the axes of an intertemporal choice model as “dollars in the present” and “dollars in the future,” let’s think of them as “consumption in the present” and “consumption in the future.” (So, in the simple example above, it would be “apples in the present” and “apples in the future”.) Let’s assume that consumption in the present has a price of $p_1$, and consumption in the future has a price of $p_2$. Let’s also assume that we can express the income stream $(m_1, m_2)$ in real terms: for example, that $m_1$ is the amount of consumption your period-1 income can buy in period 1, and $m_2$ is the amount of consumption your period-2 income can buy in period 2. (One interpretation of this is that your wages are indexed to inflation, which is a feature of some negotiated labor contracts.) Then the intertemporal budget constraint becomes \(p_1c_1 + {p_2 \over 1+r}c_2 = p_1m_1 + {p_2 \over 1 + r}m_2\) (Note that all we’ve done here is to add in the prices $p_1$ and $p_2$.) Inflation is what relates $p_1$ and $p_2$: that is, if $\pi$ is the inflation rate, then \(p_2 = (1 + \pi)p_1\) Substituting this in for $p_2$ gives us \(p_1c_1 + {p_1(1+\pi) \over 1+r}c_2 = p_1m_1 + {p_1(1+\pi) \over 1 + r}m_2\) Now every term has a $p_1$, so we can cancel all those terms, leaving us with \(c_1 + {1+\pi \over 1+r}c_2 = m_1 + {1+\pi \over 1 + r}m_2\) Lastly, let’s define the “real” interest rate $\rho$ as: \(1 + \rho = {1 + r \over 1 + \pi}\) Substituting this into the budget constraint gives us \(c_1 + {1 \over 1+\rho}c_2 = m_1 + {1 \over 1 + \rho}m_2\) This is just the exact same budget constraint as we started out with, only with the real interest rate $\rho$ in place of the nominal interest rate $r$. In short, inflation simply has the effect of reducing the slope of the budget line, making it less advantageous to save and also less onerous to borrow.

See interactive graph online here.

Note that we can write $\rho$ as \(\rho = {1 + r \over 1 + \pi} - 1 = {1 + r \over 1 + \pi} - {1 + \pi \over 1 + \pi} = {r - \pi \over 1 + \pi}\) which is approximately equal to $r - \pi$ as long as $\pi$ is fairly low.

Duration: Extending the Concept of “Present Value”

Often in economics we want to evaluate the present value of a stream of payoffs. Among many examples:

Let’s look at how to evaluate all of these by first looking at one future payoff; then several; then an infinite number.

Present value of a single payoff in the future

To think of the present value of a payoff in the future, think first of a concrete example: suppose you invested €1000 at 10% interest. After one year, it would grow to be worth $1.1 \times €1000 = €1100$; after two years, it would be worth $1.1 \times €1100 = 1210$; after three, $1.1 \times 1210 = €1331$; and so forth.

More generally, if you invested some present amount $v$ at an interest rate of $r$ (where $r = 0.1$ corresponds to a 10% interest rate), it would grow exponentially: \(\begin{aligned} \text{Initial value} &= v\\ \text{Value after one year } &= (1 + r)v\\ \text{Value after two years } &= (1+r)^2v\\ \text{Value after three years } &= (1+r)^3v\\ & \vdots\\ \text{Value after }t\text{ years } &= (1+r)^tv\\ \end{aligned}\) Now let’s flip the script, and think about how much you would need to invest to achieve some target amount in the future. Let’s define $PV$ as the amount of money we would need to invest in order to be worth some amount $x$ at some point $t$ years into the future. Then by the above equation we have \((1+r)^t \times PV = x\) solving for $PV$ by dividing both sides by $(1 + r)^t$ gives us \(PV(x,r,t) = {x \over (1 + r)^t}\) This relationship is visualized in the graph below. Try adjusting $x$ and $t$ (by dragging the dot) and $r$ (by adjusting the slider) to see how the present value of the future payoff changes:

See interactive graph online here.

Present value of a stream of payments

We’ve established that the value of a payment of $x$ received $t$ years in the future (written $x_t$) at interest rate $r$ is given by \(PV(x_t) = {x \over (1+r)^t}\) Logically, the present value of a series of payments received 1, 2, 3, and so on up to $t$ years in the future $(x_1, x_2, …, x_t)$ can be evaluated by just adding the present value of the payment received in each period: \(PV(x_1,x_2,...,x_t) = {x_1 \over 1 + r} + {x_2 \over (1 + r)^2} + \cdots + {x_t \over (1 + r)^t}\) For example, some bonds are called zero-coupon bonds (they only pay back their face value at the end of their duration), while others have a coupon of some amount that they pay to whomever is holding them at a particular time. So a bond that pays €1000 ten years from now, but has a 5% coupon of €50 per year, would have a stream of payments $(50,50,50,50,50,50,50,50,50,1050)$.

Evaluating an infinite stream of payoffs

Some bonds actually only have a coupon, and never pay their principal back. This is called a perpetuity: rather than pay a certain amount at a single point in the future, it pays a constant amount per period, forever. How much would something like this be worth?

Think about the present value of a stream $n$ payments of $x$ dollars each, starting one period in the future: that is, $x_0 = 0$ and $x_1 = x_2 = \cdots = x_n = x$. The present value of this stream is \(PV = \underbrace{ {x \over 1 + r} + {x \over (1 + r)^2} + \cdots + {x \over (1 + r)^n}}_{n \text{ terms in total}}\) You can see this for various values of $n$ in the graph below. Try dragging $n$ from 1 to 5 to see how the present value changes:

See interactive graph online here.

It’s clear that as $n$ increases, the present value changes by a lot at first, then by less and less. The mathematical formula for the limit of this expression as $n$ approaches infinity is \(PV = {x \over r}\) Indeed, if you check the “Show $PV$ for $n = \infty$” in the graph above, you can see this theoretical limit, which is pretty close to the $PV$ of $n = 5$ if $r$ is large!

There are two elegant ways of deriving this limit:

We’ll get back to this formula in Econ 51, when we talk about the value of an ongoing relationship…and the cost of betraying such a relationship…

Discounting future payoffs

One brief note bears mentioning. The above analysis refers only to the budget line: how much a certain amount of money would grow to in $t$ years. Preferences also depend on how far into the future we’re thinking about: you might care a great deal about your utility one year from now, but substantially less about your utility 30 years from now. Many models still posit that if $\beta$ is the factor describing how you feel about yourself one period from now, then $\beta^2$ would describe how you feel about yourself two periods from now, $\beta^3$ for three periods, and more generally $\beta^t$ would describe how you feel about your utility $t$ periods from now. This “exponential discounting” essentially says that your utility should be discounted in much the same way as compounded interest is; but this has been pretty well and truly debunked by behavioral economists.

Risk of Default

The third factor affecting a bond’s price is how risky it is: that is, how likely the issuer of the bond is to default on its obligations. This isn’t (yet!) much of an issue for debt issued by the U.S. government, but for corporate and municipal bonds the “safety” of the bond is a key determining factor in how much people are willing to pay for them.

Intuitively, your willingness to lend someone money depends on the likelihood with which you believe they will pay you back: the less you trust them, the more you’re going to want in interest.

The following chart shows the yield for the highest-rated corporate bonds (AAA) for the past 10 years, along with lower-rated (BAA) bonds. The lower-rated bonds usually have a yield that’s about one percentage point higher than the higher-rated ones:

Note that during periods of stress – the 2008 financial crisis and the 2020 COVID pandemic are shown on this graph – this gap increases dramatically. Crises don’t affect all companies equally: when a shock goes through the economy that increases the risk of companies going out of business, the ones that were already on shaky financial ground are much more likely to default on their bonds than those which are large and well-established enough to weather the storm.

How do we understand this risk in the context of our consumer model? One easy way would just be to assign a probability to the bond being repaid at all. Suppose that we let $\delta$ be the probability that the bond pays its face value when it comes due. This just multiplies the utility a consumer will get from the payoff if and when it happens. So instead of a utility of \(u(c_1,c_2) = v(c_1) + \beta v(c_2)\) we might have \(u(c_1,c_2) = v(c_1) + \delta \left[\beta v(c_2)\right]\) Basically what we’re saying here is that if $\beta$ is the weight that you place on your future utility, then $\delta \times \beta$ is that weight, adjusted for the fact that you may or may not get a payoff at all.

Of course, how the investor feels about that risk will depend on the curvature of the $v(c)$ function. The more risk-averse they are, the more sensitive their demand will be to the possibility of default.

Putting it all together

Let’s see how inflation, duration, and risk of default all contribute to a consumer’s demand for bonds.

From the inflation section and duration sections, can write the budget line for a bond that comes due $t$ years in the future as \(c_1 + \left({1 + \pi \over 1 + r}\right)^t c_2 = m_1 + \left({1 + \pi \over 1 + r}\right)^t m_2\) where $r$ is the yearly interest rate, and $\pi$ is the yearly inflation rate. (For simplicity, let’s assume that inflation is constant over the time period…much easier to do the math that way.)

From the default risk section, we can write the utility function as \(u(c_1,c_2) = v(c_1) + \delta \left[\beta v(c_2)\right]\) The solution to this problem yields some value of $c_1^*$ as a function of $r$, as well as the parameters $\pi$, $t$, $\delta$, and $\beta$. The homework question has you solve this problem for the Cobb-Douglas case $v(c) = \ln c$…but before you do that question, you should think about how you would imagine the demand curve for bonds to react to changes in each of those parameters…can you tell a story about each one, and why it would shift demand?

Futures and Prediction Markets

Most bonds, especially those issued by stable governments, are considered relatively safe investments: you’re definitely buying a set amount of money at some future date, and while things may happen between now and then that might make you regret your decision, the piece of paper is a piece of paper: as long as the issuer doesn’t default, you’ll be getting that value back in a few years.

A much more volatile and risky form of financial instrument is a futures contract. This is essentially a bet about a future event – and depending on whether that event occurs or not, it could be worth a lot or a very little. This can be useful, especially if lots of people have bits of information which together form a market opinion on where prices are likely to go in the future; but can also open itself up to insider trading and corruption. Let’s first look at how futures markets can work well, and then analyze problems that may arise.

The wisdom of crowds

Suppose there is something, like a presidential election, which depends on how a lot of people are feeling about two candidates. Traditionally, leading up to an election, pollsters will gauge which candidate is ahead by contacting a sample of a few thousand voters, and then use what they know about those voters (where they live, how old they are, who they’ve voted for in the past, various demographic features) to try to predict the likely outcome of the election. These poll numbers rely on the ability of the pollster to extrapolate from a small sample size, and there are a number of ways in which they can and do go wrong.

However, in recent years, another kind of data has emerged: prediction markets like Kalshi and Polymarket allow anyone to trade on the outcome of an election, or anything else. On these sites, you can buy a contract that pays a dollar if an event happens, and one that pays a dollar if it doesn’t happen. Each of these contracts has a price. For example, I’m writing this on April 29, 2026; and checking Kalshi, I can see that for 25 cents you can buy a contract that pays a dollar if Gavin Newsom is the Democratic nominee for President in 2028. You can also buy a contract for 75 cents that pays a dollar if he isn’t the nominee. If we let state of the world 1 be the one in which Gavin Newsom is the nominee for president, and state of the world 2 be the one in which he is not, this means you can directly buy money in each of those two states of the world, at prices $p_1 = 0.25$ and $p_2 = 0.75$. The price ratio, $p_1/p_2 = 0.25/0.75 = 1/3$, reflects the fact that the market believes that the probability he will be the nominee is 1/3 the probability that he won’t.

Suppose you have $€300$ in your pocket that you could use to bet on this market. Since you haven’t yet bet, this is like saying you have the bundle $(300, 300)$: that is, you have €300. If you bought 800 shares of “yes” for 25 cents each, you would effectively “trade” to the bundle $(900,100)$. How? Well, you have to pay $€200$ for the shares, bringing the money in your pocket down to $€100$. If he’s the nominee (state of the world 1), you win $€800$, bringing your wealth up to $€900$. If he’s not (state of the world 2), you lose, and you’re stuck with $€100$.

Because you can buy any number of shares at these prices, your budget line is just the normal budget line we’re used to: \(0.25c_1 + 0.75c_2 = 300\) Note that the point $(300,300)$ lies along this budget line, because if you don’t bet anything, you consume €300 regardless of the outcome of the election.

How much you should bet (if anything) depends on the probability with which you believe Newsom will be the nominee, as well as your tolerance for risk. Let’s assume you believe he’ll win with probability $\pi$, and your value function is $v(c) = c^r$, as we had last class. In that case your expected utility function is \(\mathbb{E}[v(c)] = \pi c_1^r + (1-\pi) c_2^r\) We can then maximize this subject to the budget line we found above. For example, if $\pi = 0.5$ and $r = 0.5$, your optimal bundle would indeed be $(900,100)$, as shown in the following graph:

See interactive graph online here.

Try playing with $p$ (the price of a “yes” contract), $\pi$, and $r$. How does the strength of your belief affect how much you should bet? What about your risk aversion? Does the graph behave the way you would expect at this point? (I hope so!!!)

Note that your marginal rate of substitution with this utility function is \(MRS(c_1,c_2) = {\pi \over 1 - \pi}\times \left({c_2 \over c_1}\right)^{1 - r}\) Now, at your “endowment” of $(300,300)$, your MRS is therefore \(MRS(300,300) = {\pi \over 1 - \pi}\times \left({300 \over 300}\right)^{1 - r} = {\pi \over 1 - \pi}\) regardless of what $r$ is. Note that this is greater than the price ratio of 1/3 if \({\pi \over 1 - \pi} > {1 \over 3} \Rightarrow \pi > {1 \over 4}\) In other words, if you believe that Gavin Newsom is more than 25% likely to be the nominee, then you should buy at least some of the contracts saying he will.

Why is this useful? Well, lots of people are forming opinions about Gavin Newsom right now. As events play out, and people talk to each other at church, or at bars, or at ball games, they get a sense of the mood of the electorate. And if lots of people think Gavin Newsom is getting traction, they’ll start bidding up the contract saying he’s likely to be the nominee. This will occur in real time, unlike polls which have a large lag and can’t respond quickly to current events.

Similarly, economic data can be predicted via these kinds of markets as well. In the movie Trading Places, the final scene centers around the futures market for frozen concentrated orange juice (FCOJ) responding to a crop report from the USDA on the orange harvest. But if orange farmers themselves could have been trading in the futures market for the price of oranges, the report wouldn’t have had much of an effect: because they know how their crops and their neighbors’ crops are doing, the farmers’ information could actually be better (in the aggregate) than a government agriculture report. When the official report came in, the market would have already priced in the current conditions, and it wouldn’t budge.

The role of information and the risk of insider trading

It’s one thing for a lot of small investors to place bets based on their perspective of a market. It’s quite another to have actors with insider information move markets. In Trading Places the villains of the film, the Dukes, have arranged to get an advanced copy of a crop report on the orange harvest. The report says the orange crop hasn’t been affected by frost, so the price of oranges will be low. The heroes of the film, Louis Winthorpe (played by Dan Aykroyd) and Billy Ray Valentine (Eddie Murphy), through some shenanigans involving a gorilla suit on an Amtrak train that have not necessarily aged well, intercept the report and give the Dukes a fake crop report saying that the price of oranges will be high. The Dukes, believing their insider information to be true, instruct their trader to go in and buy FCOJ futures at whatever price he can. Winthorpe and Valentine wait until the price has been bid up, and then offer to sell futures. (They don’t actually own any; this is essentially a promise to sell something that they will buy later.) The true information is revealed; the price plummets; Winthorpe and Valentine buy low, and the Dukes are ruined. It’s a classic scene.

More recently, though, insider trading has become a major problem on prediction platforms like Kalshi and Polymarket. A U.S. soldier who was involved in the extraction of Venezuelan president Nicolas Maduro has recently been charged with placing bets worth €400,000 based on his inside knowledge of the operation. In the runup to the U.S. bombing of Iran, a large number of well-timed bets on an imminent strike similarly earned some anonymous investors a great deal of money.

Even worse, bettors in prediction markets have an incentive to alter the probabilities of the events they’re betting on. A few days ago, a temperature gauge in Paris mysteriously spiked at just the same time as some curious bets were placed on their being an unusually high temperature – apparently, the bettors were able to change the temperature reading they were betting on. In a more sinister example, an Israeli reporter, Emanuel Fabian, received death threats because he reported accurately when an Iranian missile hit instead of being interceptors. The bettors reportedly had over €14 million riding on the outcome of that one bet.

Conclusion

This week we’ve seen how a model we developed from our preferences over simple things like apples and bananas can be generalized to analyze our preferences over much larger things like time and risk. We’ve seen how notions of “prices” and “opportunity cost” can be applied to abstract situations like “lotteries,” and how markets for these kinds of goods (like contracts in prediction markets) can be analyzed using the same basic building blocks that we use to analyze simpler goods.

This is obviously just a small sliver of what we can use this model for. In previous iterations of Econ 50, I’ve used this model to analyze how Chuck from Cast Away could optimally spend his time on a desert island, or how a gig worker chooses how much time spend working and how much to spend in leisure. All of these models, though, are variations on the same theme: in a world of tradeoffs, you have to balance the marginal benefits and costs of the options before you.

For the second half of the course we’ll look at a different type of problem: how much of a single good to produce. We’ll spend two weeks analyzing how firms choose how much of a good to produce, and then spend our last three weeks analyzing how societies, through markets, choose how much of a good to produce and consume.

For now, though, you’ve got a Checkpoint on Monday. Good luck!

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