Intertemporal Consumption, Investment, and Asset Pricing

Time is the hidden ingredient in every financial choice. As a student, you decide how much to spend today and how much to save for later. A company weighing an investment faces the same question: how can we wisely shift resources across time? This chapter gives you the tools to answer that. We start with a one‑person economy and build up to pricing bonds, stocks, and understanding why interest rates exist at all.

The Big Picture#

Every real financial choice—buying a house, issuing a bond, building a factory—ties today’s giving up something to tomorrow’s gains. To understand that tie, we need a clear way to talk about how people value time, how firms shift resources across time, and how markets price promises of future cash. By the end of this chapter, you’ll see that interest rates, bond yields, and stock prices are not random numbers. They come from a deep, unified logic of trading across time. We’ll build that logic step by step: starting with a single person on an island, and ending with the forces that keep the yields on different assets in harmony.

Autarky: Living on Your Own Endowment#

Imagine you are a farmer. You harvest all your food in the autumn. Your endowment—the crop—arrives in one big lump. But you must eat all year long. In a world with no neighbours, no banks, and no storage, you’d be stuck: you could only eat when the harvest comes. Having to consume exactly what you get at each point, with no way to shift resources through time, is called autarky.

Endowment: The bundle of resources (goods or income) that an individual receives at each point in time, before any trading or production.

Autarky: A state in which no trade, borrowing, or lending is possible, so each person must consume exactly their own endowment in every period.

Autarky is rarely the best you can do. You’d probably like a smoother pattern: eat less during the post‑harvest plenty, so you can eat more in the lean winter. But without a way to shift goods, you’re stuck. The first step out of autarky is to find a way to physically move goods through time.

Storage and Private Investment: Shifting Resources Across Time#

The simplest way to escape autarky is storage. If you can put some of today’s harvest in a dry barn, you can eat it later. Storage lets you turn one unit of today’s consumption into units tomorrow. The growth rate could be negative (spoilage) or positive (like planting seeds that multiply). More broadly, any action that uses today’s resources to create future resources is called investment.

Investment: The act of sacrificing current resources (consumption, labour, or cash) in exchange for a claim on future resources.

In our farmer’s world, suppose you can invest part of the harvest as seed. For every bushel you plant today, you get back bushels at the next harvest. This trade‑off is shown by an intertemporal production possibilities frontier (IPPF). It’s a curve that shows the most you can consume today () and tomorrow (), given your starting endowment and the investment technology available.

If you start with endowment now and later, and you can invest amount today to receive later, your consumption possibilities become

The IPPF slopes downward: investing more today cuts current consumption but boosts future consumption. Its slope, , is the marginal rate of transformation (MRT). The MRT shows how much extra future consumption you get by giving up one unit of current consumption. The shape depends on the technology. Usually, returns diminish: each extra unit invested produces a smaller increase in the future harvest. That makes the IPPF curved outward (concave).

With this technology, you can do better than autarky. You pick an investment level that shifts your consumption closer to what you prefer. But your choice is still tied to your own technology and preferences. To see how markets break that link, we first need a way to talk about preferences across time.

Atemporal Trade and the Marginal Rate of Substitution#

Even without thinking about time, you know how trade works: people swap apples for oranges until each person’s marginal rate of substitution (MRS)—the rate at which they’re willing to trade one good for another and stay equally happy—matches the market price ratio. Now we apply the same idea across time, treating consumption today and consumption tomorrow as two different “goods.”

Marginal Rate of Substitution (MRS): The amount of future consumption a person is willing to give up to obtain one extra unit of current consumption, holding overall satisfaction constant.

You can think of bundles of consumption today and tomorrow as . Your preferences can be shown by a utility function . Indifference curves connect all bundles that make you equally happy. Their slope at any point is the MRS. Most people are impatient: to willingly give up a unit of consumption today, they demand more than one unit tomorrow. So indifference curves are convex, and at a typical point the MRS is above in absolute value. For example, you might need units of future consumption to make up for losing unit now.

In autarky, you are stuck with your endowment. Your MRS may not match the MRT from your own investment technology. But if you meet someone with a different endowment and preferences, you can trade. You swap some of your plentiful harvest‑time apples for their scarce winter apples at a price you both agree on. The price ratio between current and future goods is exactly the interest rate factor : one unit today trades for units tomorrow. The market gives you a universal exchange rate across time—a single slope that everyone can use.

Intertemporal Trade and the Fisher Separation Theorem#

Now put everything together. A person has an endowment, a private investment technology, and can borrow or lend at a risk‑free interest rate in a financial market. Lending one unit today gives you units tomorrow. Borrowing works in reverse: you promise to pay tomorrow and get one unit today. This creates a straight‑line budget with slope .

The breakthrough insight, known as the Fisher separation theorem, is that the optimal investment decision is completely independent of the individual’s consumption preferences.

Fisher Separation Theorem: In a perfect capital market, a person’s production (investment) decision can be separated from their consumption decision. Every investor, regardless of patience or utility function, should choose the investment level that maximizes the present value of their endowment.

Why? With interest rate , the value of any consumption stream is its present value. If you invest amount , your present value of wealth is

To maximize , you pick so that the extra return on the last unit of investment equals the market interest rate: . At that point, the MRT equals . That investment level gives the highest possible wealth, no matter what your utility function says. After you’ve maximised wealth, you can then borrow or lend to reach your favourite consumption point along the new budget line. That second step uses your personal MRS, but the first step is totally separate.

That separation is powerful. It means a firm with many shareholders—each with different patience—can still make one clear investment choice: take every project whose return beats the market interest rate. It also explains why stock markets can value companies without asking every shareholder how patient they are.

📝 Section Recap: When a perfect capital market exists, the investment decision (maximize present value) separates cleanly from the consumption decision (borrow or lend to match personal preferences). This is the Fisher separation theorem, a key idea in corporate finance.

The Term Structure of Interest Rates#

So far we’ve talked about “the” interest rate as one number for moving money across one period. In reality, markets offer different interest rates for different lengths of time. The term structure of interest rates (or yield curve) is the set of spot rates—the rates for risk‑free loans that pay a single sum at a specific future date—for all maturities.

Spot rate: The annualized interest rate on a risk‑free loan that pays a single lump sum at a particular future date, with no payments in between.

For example, a two‑year zero‑coupon bond might cost 90$100r_2$ satisfies

so . If the one‑year spot rate is , the term structure slopes upward. That often signals that markets expect short rates to rise in the future, or it may reflect a premium for locking up money for longer.

Why does the term structure matter? Because each future cash flow must be discounted at the spot rate that matches its timing. A payment promised in three years should be discounted with the three‑year spot rate, not the one‑year rate. That’s how you correctly turn a stream of payments into a present value.

📝 Section Recap: Different maturities have different interest rates; the set of spot rates for all maturities is the term structure. To price a future cash flow correctly, discount it with the spot rate that matches its timing.

Bond Pricing: Coupons, Face Value, and Yield to Maturity#

A bond is a promise to pay a stream of cash flows. Most bonds pay regular coupons—interest payments—until they mature, when the face value (the principal) is repaid. For example, a bond with face value 1{,}000C=$50T=3$50$1{,}050$ at the end of year 3.

Given the term structure of spot rates , the price of this bond is the present value of its promised payments:

In practice, people often quote a single number called the yield to maturity (YTM). The YTM is the constant discount rate that makes the present value of the bond’s promised cash flows equal to its market price:

Yield to Maturity (YTM): The single discount rate that, when applied to all of a bond’s promised cash flows, makes their present value equal to the bond’s current market price.

The YTM is a handy summary, but it blends all the spot rates together. When the term structure isn’t flat, bonds with the same maturity but different coupon patterns can have different YTMs. So think of YTM as a complex average of the spot rates, not as the exact return you’ll earn if you hold the bond to maturity. (You’d earn that return only if you could reinvest all coupons at the same YTM, which rarely happens.)

📝 Section Recap: A bond’s price is the present value of its promised coupons and face value, discounted with spot rates. The yield to maturity is the single constant rate that makes that present value equal to the market price.

Equity Pricing: The Present Value of Dividends#

A share of stock is a piece of ownership in a company. It gives you a right to a stream of dividends—cash payments the company distributes to owners. Using the same time‑value idea, a share’s fair price today should be the present value of all expected future dividends:

where is the dividend at time , and is the appropriate risk‑adjusted discount rate. (For now, we’ll assume risk can be captured by a single rate .)

That sum looks infinite, but it becomes very simple if dividends are expected to grow at a constant rate forever. This is the Gordon growth model. If is the next dividend and , the price simplifies to

This formula captures the three key drivers of stock value: the expected dividend (), the required return (), and the growth rate (). Small changes in or can cause big price moves. That’s why stock markets watch earnings growth and interest rates so closely.

📝 Section Recap: A stock’s price is the present value of its expected future dividends. Under constant growth, the simple formula shows how required returns and growth prospects drive value.

Price‑Earnings Ratios: A Quick Valuation Lens#

Instead of dividends, analysts often talk about the price‑earnings (P/E) ratio—the share price divided by earnings per share. The P/E ratio isn’t just a mood gauge; it comes straight from the dividend discount model. If a company pays out a fraction of its earnings as dividends (), and earnings grow at rate , then

The P/E is high when the payout ratio is high, the required return is low, or growth is high. Since the payout ratio is usually between and , the P/E can be roughly for a company that pays out all earnings. But in reality, retained earnings help fuel growth. The point is that P/E is not a separate method; it’s just a rearrangement of the present‑value logic you already know.

📝 Section Recap: The P/E ratio is the present‑value model in a compact form. It reflects growth expectations, payout policy, and the required rate of return—all in one number.

The Equation of Yield Across Assets#

We’ve seen that bonds and stocks are both priced by discounting future cash flows. In a smooth market with no arbitrage, two assets that promise the exact same future payoffs must sell for the same price today. If they didn’t, you could buy the cheap one and sell the expensive one, locking in a risk‑free profit with no net investment. This no‑arbitrage principle forces the risk‑adjusted yields on similar assets to be equal.

No‑arbitrage principle: In well‑functioning markets, assets that offer identical future cash flows must trade at the same price, preventing risk‑free profits.

Take a one‑year risk‑free bond and a one‑year risk‑free bank deposit. If the bond offered a higher yield, investors would avoid deposits and rush into bonds. That would push the bond price up until its yield matched the deposit rate. So in equilibrium, all risk‑free instruments with the same maturity have the same yield. For risky assets, differences in expected yields compensate for differences in risk. But the same logic holds: any two assets with identical risk must have the same expected return. Otherwise, arbitrageurs would quickly wipe out the gap.

This unifying idea—that all financial claims are priced by discounting expected future cash flows at a rate that, after adjusting for risk, lines up across markets—ties the chapter together. Whether you’re pricing a government bond, a corporate bond, or a stock, you’re solving the same basic equation: price equals the present value of future cash.

📝 Section Recap: In arbitrage‑free markets, assets with identical payoffs must have identical prices and thus the same yield. This principle ensures consistency across all asset classes.

Summary#

We began with a lone farmer stuck in autarky and ended with a world where every financial instrument—from a storage shed to a company’s stock—fits into one framework. The big idea: financial markets exist to move resources through time, and they do it by pricing future cash flows. The interest rate is the price of time. The Fisher separation theorem shows that in perfect markets, anyone can value an investment by maximizing present wealth, no matter their patience. Then we saw how the term structure gives us a full set of prices for money at every horizon, which we used to price bonds and stocks. Finally, the no‑arbitrage principle makes sure all these prices fit together without conflict.