In this lecture, we use the overlapping generations model from the previous class to see, mathematically, how demographic changes can influence interest rates and asset prices. We evaluate Tobin's statement that a perpetually growing population could solve the Social Security problem, and resolve, in a surprising way, a classical argument about the link between birth rates and the level of the stock market. Lastly, we finish by laying some of the philosophical and statistical groundwork for...

Until now, the models we've used in this course have focused on the case where everyone can perfectly forecast future economic conditions. Clearly, to understand financial markets, we have to incorporate uncertainty into these models. The first half of this lecture continues reviewing the key statistical concepts that we'll need to be able to think seriously about uncertainty, including expectation, variance, and covariance. We apply these concepts to show how diversification can reduce risk...

In order for Social Security to work, people have to believe there's some possibility that the world will last forever, so that each old generation will have a young generation to support it. The overlapping generations model, invented by Allais and Samuelson but here augmented with land, represents such a situation. Financial equilibrium can again be reduced to general equilibrium. At first glance it would seem that the model requires a solution of an infinite number of supply equals demand...

This lecture continues the analysis of Social Security started at the end of the last class. We describe the creation of the system in 1938 by Franklin Roosevelt and Frances Perkins and its current financial troubles. For many democrats Social Security is the most successful government program ever devised and for many Republicans Social Security is a bankrupt program that needs to be privatized. Is there any way to reconcile the views of Democrats and Republicans? How did the system get into...

Standard financial theory left us woefully unprepared for the financial crisis of 2007-09. Something is missing in the theory. In the majority of loans the borrower must agree on an interest rate and also on how much collateral he will put up to guarantee repayment. The standard theory presented in all the textbooks ignores collateral. The next two lectures introduce a theory of the Leverage Cycle, in which default and collateral are endogenously determined. The main implication of the theory...

In order to understand the precise predictions of the Leverage Cycle theory, in this last class we explicitly solve two mathematical examples of leverage cycles. We show how supply and demand determine leverage as well as the interest rate, and how impatience and volatility play crucial roles in setting the interest rate and the leverage. Mathematically, the model helps us identify the three key elements of a crisis. First, scary bad news increases uncertainty. Second, leverage collapses....

This lecture continues the analysis of the Capital Asset Pricing Model, building up to two key results. One, the Mutual Fund Theorem proved by Tobin, describes the optimal portfolios for agents in the economy. It turns out that every investor should try to maximize the Sharpe ratio of his portfolio, and this is achieved by a combination of money in the bank and money invested in the "market" basket of all existing assets. The market basket can be thought of as one giant index fund or mutual...

This lecture addresses some final points about the CAPM. How would one test the theory? Given the theory, what's the right way to think about evaluating fund managers' performance? Should the manager of a hedge fund and the manager of a university endowment be judged by the same performance criteria? More generally, how should we think about the return differential between stocks and bonds? Lastly, looking back to the lectures on Social Security earlier in the semester, how should the CAPM...

Until now we have ignored risk aversion. The Bernoulli brothers were the first to suggest a tractable way of representing risk aversion. They pointed out that an explanation of the St. Petersburg paradox might be that people care about expected utility instead of expected income, where utility is some concave function, such as the logarithm. One of the most famous and important models in financial economics is the Capital Asset Pricing Model, which can be derived from the hypothesis that...

This lecture reviews the intuition from the previous class, where the idea of dynamic hedging was introduced. We learn why the crucial idea of dynamic hedging is marking to market: even when there are millions of possible scenarios that could come to pass over time, by hedging a little bit each step of the way, the number of possibilities becomes much more manageable. We conclude the discussion of hedging by introducing a measure for the average life of a bond, and show how traders use this...

I explain how, as a mathematical economist, I became interested in the practical world of mortgage securities, and how I became the Head of Fixed Income Securities at Kidder Peabody, and then one of six founding partners of Ellington Capital Management. During that time Kidder Peabody became the biggest issuer of collateralized mortgage obligations, and Ellington became the biggest mortgage hedge fund. I describe securitization and tranching of mortgage pools, the role of investment banks and...

Suppose you have a perfect model of contingent mortgage prepayments, like the one built in the previous lecture. You are willing to bet on your prepayment forecasts, but not on which way interest rates will move. Hedging lets you mitigate the extra risk, so that you only have to rely on being right about what you know. The trouble with hedging is that there are so many things that can happen over the 30-year life of a mortgage. Even if interest rates can do only two things each year, in 30...

A mortgage involves making a promise, backing it with collateral, and defining a way to dissolve the promise at prearranged terms in case you want to end it by prepaying. The option to prepay, the refinancing option, makes the mortgage much more complicated than a coupon bond, and therefore something that a hedge fund could make money trading. In this lecture we discuss how to build and calibrate a model to forecast prepayments in order to value mortgages. Old-fashioned economists still make...

This lecture is about optimal exercise strategies for callable bonds, which are bonds bundled with an option that allows the borrower to pay back the loan early, if she chooses. Using backward induction, we calculate the borrower's optimal strategy and the value of the option. As with the simple examples in the previous lecture, the option value turns out to be very large. The most important callable bond is the fixed rate amortizing mortgage; calling a mortgage means prepaying your remaining...

In the first part of the lecture we wrap up the previous discussion of implied default probabilities, showing how to calculate them quickly by using the same duality trick we used to compute forward interest rates, and showing how to interpret them as spreads in the forward rates. The main part of the lecture focuses on the powerful tool of backward induction, once used in the early 1900s by the mathematician Zermelo to prove the existence of an optimal strategy in chess. We explore its...

According to the rational expectations hypothesis, traders know the probabilities of future events, and value uncertain future payoffs by discounting their expected value at the riskless rate of interest. Under this hypothesis the best predictor of a firm's valuation in the future is its stock price today. In one famous test of this hypothesis, it was found that detailed weather forecasts could not be used to improve on contemporaneous orange prices as a predictor of future orange prices, but...

Where can you find the market rates of interest (or equivalently the zero coupon bond prices) for every maturity? This lecture shows how to infer them from the prices of Treasury bonds of every maturity, first using the method of replication, and again using the principle of duality. Treasury bond prices, or at least Treasury bond yields, are published every day in major newspapers. From the zero coupon bond prices one can immediately infer the forward interest rates. Under certain conditions...

In this lecture we move from present values to dynamic present values. If interest rates evolve along the forward curve, then the present value of the remaining cash flows of any instrument will evolve in a predictable trajectory. The fastest way to compute these is by backward induction. Dynamic present values help us understand the returns of various trading strategies, and how marking-to-market can prevent some subtle abuses of the system. They explain how mortgages work, why they're...

In the 1990s, Yale discovered that it was faced with a deferred maintenance problem: the university hadn't properly planned for important renovations in many buildings. A large, one-time expenditure would be needed. How should Yale have covered these expenses? This lecture begins by applying the lessons learned so far to show why Yale's initial forecast budget cuts were overly pessimistic. In the second half of the class, we turn to the problem of measuring investment performance, and examine...

Philosophers and theologians have railed against interest for thousands of years. But that is because they didn't understand what causes interest. Irving Fisher built a model of financial equilibrium on top of general equilibrium (GE) by introducing time and assets into the GE model. He saw that trade between apples today and apples next year is completely analogous to trade between apples and oranges today. Similarly he saw that in a world without uncertainty, assets like stocks and bonds...

Over time, economists' justifications for why free markets are a good thing have changed. In the first few classes, we saw how under some conditions, the competitive allocation maximizes the sum of agents' utilities. When it was found that this property didn't hold generally, the idea of Pareto efficiency was developed. This class reviews two proofs that equilibrium is Pareto efficient, looking at the arguments of economists Edgeworth, and Arrow-Debreu. The lecture suggests that if a...

Our understanding of the economy will be more tangible and vivid if we can in principle explain all the economic decisions of every agent in the economy. This lecture demonstrates, with two examples, how the theory lets us calculate equilibrium prices and allocations in a simple economy, either by hand or using a computer. In future lectures we shall extend this method so as to compute equilibrium in financial economies with stocks and bonds and other financial assets.

This lecture gives a brief history of the young field of financial theory, which began in business schools quite separate from economics, and of my growing interest in the field and in Wall Street. A cornerstone of standard financial theory is the efficient markets hypothesis, but that has been discredited by the financial crisis of 2007-09. This lecture describes the kinds of questions standard financial theory nevertheless answers well. It also introduces the leverage cycle as a critique of...

This lecture explains what an economic model is, and why it allows for counterfactual reasoning and often yields paradoxical conclusions. Typically, equilibrium is defined as the solution to a system of simultaneous equations. The most important economic model is that of supply and demand in one market, which was understood to some extent by the Ancient Greeks and even by Shakespeare. That model accurately fits the experiment from the last class, as well as many other markets, such as the...