The most comprehensive educational resources for finance

The Mutual Fund Theorem and Covariance Pricing Theorems

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

Dynamic Hedging and Average Life

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,

Dynamic Hedging

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

Modeling Mortgage Prepayments and Valuing Mortgages

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

Callable Bonds and the Mortgage Prepayment Option

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

Backward Induction and Optimal Stopping Times

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

Uncertainty and the Rational Expectations Hypothesis

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

Quantifying Uncertainty and Risk

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

Will the Stock Market Decline when the Baby Boomers Retire?

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

Overlapping Generations Models of the Economy

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