Economics 1723: Capital Markets

Lecture 4

1
Weiling Liu

Ec1723

1
Slides adapted from Prof. Xavier Gabaix.
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Announcements

First homework has been posted (and will be submitted)

virtually using Gradescope.
This homework is due next Tuesday 2/11 by 1:30pm (see
syllabus for strict homework policy).
If you have any questions on the homework or class material that
wasn’t covered in section, please attend office hours (posted on
Ed). They will be available on Friday, Sunday, and Monday.

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Key questions

Return measurement.
How do we measure expected rates of return ? When do we use
the arithmetic average or the geometric average?
How do we measure the rate of return of projects ?
What is the economic reason for why different types of assets
have different expected rates of return ?
examples we will consider: bills, bonds, stock, and gold

What is the stochastic discount factor (SDF) and how does it

price assets?

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Roadmap

1 Arithmetic and Geometric Average Rates of Return

2 Evaluating Projects: Internal Rates of Return

3 A statistical view of financial history

4 Optimality and equilibrium

5 The stochastic discount factor

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Arithmetic average return

The arithmetic average return is the sample mean of the return.

If we have historical observations for periods 1, .., T , then the
arithmetic average A is given by:
T T
1 X 1 X
1+A= (1 + Rt ) , or A = Rt .
T t=1 T t=1

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Geometric average return
An alternative measure of average return is the cumulative
return over the period from 1 to T , annualized by taking the
T th root (i.e., by raising to the power 1/T ).
This is called the geometric average return:
"T #1/T
Y
1+G = (1 + Rt ) .
t=1

Intuition: G is the average return per year if returns are

reinvested:
T
Y
T
(1 + G ) = (1 + Rt ) .
t=1

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Arithmetic average is greater than the geometric
average

Learning Catalytics: Consider two returns -10%, 60%. What

is A? What is G ? How do they compare?
One can prove generally that A ≥ G .
A good rule of thumb (accurate for the lognormal distribution)
is that:
1
A = G + (Return variance) .
2
So the difference is important for volatile assets or a long
holding period.

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Arithmetic and geometric averages of stock and
bond returns

BKM 9th ed. Table 5.3: History of nominal (not inflation adjusted)
portfolio returns in percentage points, 1926-2009.

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Log gross returns

Consider the log gross return log(1 + Rt ).

This is very close to Rt for small Rt (and smaller than Rt for
larger Rt ).
Log returns are analytically convenient for a few reasons:
1 Returns over longer horizons are additive in logs:
 

log  (1 + R1 ) (1 + R2 )  = log (1 + R1 ) + log (1 + R2 ) .

 
| {z }
gross return over two periods

2 For most asset returns, log gross returns are approximately

normal over periods of a month or more (because of the Central
Limit Theorem).

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Lognormality and the median
Why do log gross returns tend to be normal?
Monthly gross returns are generated by multiplying daily gross
returns within the month.
Monthly log gross returns are generated by adding daily log
gross returns within the month.
Daily returns are approximately independent.
The Central Limit Theorem says that sums of independent
random variables approach the normal distribution under mild
assumptions.
DANGER: This logic is not bulletproof (eg Black Swans).

When log gross returns are normal, G is also the median of the
simple return, Rt .

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The summary of A, G, and lognormality

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Which average is better? A or G?

It depends.
G is a measure of average returns per year when returns are
reinvested (which they usually are in a brokerage account).
Under the reasonable assumption that log returns are normal, G
gives the median of the annual return distribution. Half the time
you will do better, half the time worse than G .
But A is the expected return if you hold the asset for a single
randomly selected period.

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Roadmap

1 Arithmetic and Geometric Average Rates of Return

2 Evaluating Projects: Internal Rates of Return

3 A statistical view of financial history

4 Optimality and equilibrium

5 The stochastic discount factor

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Quick Review: Present value
Suppose that interest rates are constant at R
The price at time 0 of a bond paying Ct at time t is:
Ct
P0 =
(1 + R)t

More generally, the price at time 0 of a bond paying Ct at time t

(t = 1, ..., T ) is:
T
C1 C2 CT X Ct
P0 = + + ... + =
1 + R (1 + R) 2
(1 + R)T
t=1
(1 + R)t

“The price is the present value of future cash-flows”

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Internal Rate of Return (IRR)
In previous slide, return R was given, and we derived P0 from it
You can also ask: what’s the R such that price = present value
(PV) of cash-flows?
This is the ”Internal rate of return”. It’s the number IRR such
that:
C1 CT
P0 = 1 + ... +
(1 + IRR) (1 + IRR)T
In sum, IRR gives a way to estimate rate of return for assets
that don’t have market prices. Or for the rate of return earned
on a project.
How to solve? Define C0 := −P0 , and solve for unknown IRR in
C1 CT
0 = C0 + 1 + ... +
(1 + IRR) (1 + IRR)T

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IRR example
Suppose you’re a small healthcare startup considering purchasing
a new ultrasound machine, expected to generate the following
cash-flows.
What is the IRR on this investment ?
Initial investment of machine: $100K
Sales in year 1: $50K
Sales in year 2: $50K
Sales in year 3: $30K, machine is depreciated
Solve:
50 50 30
0 = −100 + 1 + 2 +
(1 + IRR) (1 + IRR) (1 + IRR)3
In Excel, this is the IRR function.
Solution: IRR = 16.65%
Intuitively, it’s as if “the average geometric return was 16.65%”
Weiling Liu (Ec1723) Lecture 4 16 / 46
Roadmap

1 Arithmetic and Geometric Average Rates of Return

2 Evaluating Projects: Internal Rates of Return

3 A statistical view of financial history

4 Optimality and equilibrium

5 The stochastic discount factor

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Frequency and probability

If we repeat an experiment, the frequency of an event is the

fraction of times it occurs.
As we increase the number of experiments, the frequency of an
event approaches its probability.
A probability distribution is the set of possible outcomes along
with their associated probabilities.
A histogram shows events on the horizontal axis, and frequencies
or probabilities on the vertical axis.

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History as a succession of experiments

Economists view history as the record of a succession of

experiments.
Historical evidence gives us event frequencies, which tell us
about event probabilities.
Very different from the narrative approach to history.
DANGER: We cannot assume that probabilities always equal
frequencies. Things can happen that haven’t happened yet!

Next, let’s look at some distribution of asset returns

estimated from history.

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Yearly real returns (inflation adjusted) for T-bills

How can the real return of T-bills be negative?

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Yearly returns for treasury bonds

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Yearly returns for stocks

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Stocks have done relatively well
The arithmetic average for real returns over 1928-2023 for
different asset classes:
T-bills: 0.32%.
Bonds: 1.86%.
US stocks: 8.40%.
The average stock return for the longer 1802-2023 period is
8.41%.
RStocks − RT −bills over 1928-2023 averages 8.08% (8.32%
nominal)
It seems that stocks have done relatively well...
Source: Aswath Damodaran’s website NYU Stern. Link:
https://pages.stern.nyu.edu/~adamodar/?_ga=2.112484871.
2029911818.1674882599-427745302.1611509454

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Stocks, bonds, and bills: Cumulative returns

Warning: Perhaps the historical US data overstate the premium for

stocks if the US stock market happens to have overachieved in this
period of history (frequencies are not always probabilities).
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Stocks, bonds, and bills

It looks as if stocks are more volatile than bonds, which are

more volatile than bills.
It also looks as if stock returns are higher on average than bond
returns, which are higher on average than bill returns.
This all seems sensible until we look at another asset. . .

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Yearly returns for gold

Standard deviation roughly the same as stocks.

Lower mean (3.08% from 1928-2023).
Source: https://www.measuringworth.com/datasets/gold/
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Stocks, bonds, bills, and gold

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To understand gold prices, we need a more formal
analysis

Looking at returns and risks of individual asset classes could only

get us so far: Why would anyone hold gold?
For a better understanding, we need to analyze risks at the
portfolio level.
Next: Portfolio choice (optimality) and equilibrium.

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Roadmap

1 Arithmetic and Geometric Average Rates of Return

2 Evaluating Projects: Internal Rates of Return

3 A statistical view of financial history

4 Optimality and equilibrium

5 The stochastic discount factor

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Consider the portfolio choice problem in our
discrete states framework
Consider a 2-state example with 2 Arrow-Debreu securities.
Investor buys W1 shares of Arrow-Debreu security for state 1,
and W2 shares of Arrow-Debreu security for state 2.
Remember the price of Arrow-Debreu securities (state prices)
are denoted by q1 and q2 .
Investor takes these prices as given, and solves:

max E [U (W )] = π 1 U (W1 ) + π 2 U (W2 ) ,

W1 ,W2

subject to the budget constraint:

W0 = W1 q1 + W2 q2 .

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Deriving the first order conditions for optimality

The budget constraint implies:

W0 − W1 q1
W2 = .
q2
Substitute this to write the problem on the previous slide as:
 
W0 − W1 q1
max π 1 U (W1 ) + π 2 U .
W1 q2

The first-order condition for optimality is:

 
′ ′ q1
π 1 U (W1 ) + π 2 U (W2 ) − = 0.
q2

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Optimality: Investors care about state prices and
probabilities

The first order condition can be rewritten as:

π 1 U ′ (W1 ) π 2 U ′ (W2 )
=
q1 q2
Intuition: if an investment plan is optimal, then,
Expected marginal utility of wealth from investing $1 in
Arrow-Debreu security 1 = Expected marginal utility of wealth
from investing $1 in Arrow-Debreu security 2.
The FOC implies you economize on wealth in states that are
expensive relative to their probability of occurring.

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From optimality to equilibrium: Lucas’ thought
experiment
Investor takes the prices q1 and q2 as given, and chooses an
optimal allocation of wealth, W1 and W2 .
But what determines q1 and q2 ?
For this, we need to specify the supply of Arrow-Debreu
securities and consider market clearing (equilibrium).

Thought experiment by Robert Lucas:

What if W1 and W2 are fixed, e.g., by technological constraints?
Example:
State 1 is recession with low real resources.
In addition, there is no way to change investment plans today to
reduce W2 and increase W1 (e.g., because real investment
opportunities are fixed).
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Equilibrium: Asset prices depend on probabilities
and marginal utilities

Investors try but cannot reallocate wealth between states!

Investors’ desire to reallocate pushes up or down state prices,
until investors are indifferent.
Equilibrium state prices are characterized by the same
equation written in different way:

q1 π 1 U ′ (W1 )
= (1)
q2 π 2 U ′ (W2 )
In equilibrium, state prices are determined by probabilities and
investors’ marginal utilities in those states.

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Equilibrium with risk-neutral investors

When investors are risk neutral, marginal utility, U ′ , is constant.

In this case, Eq. (1) implies:
q1 π1
= ,
q2 π2

so state prices must equal probabilities (up to a scale factor).

Concept check: What would happen if qq21 < ππ12 ?

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Equilibrium with risk-averse investors

But in general, with risk averse investors we have the condition

in (1):
q1 π 1 U ′ (W1 )
= .
q2 π 2 U ′ (W2 )

Payments to be received in probable and unpleasant states

are expensive.

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What happens when investment opportunities are
not completely fixed?

Lucas’ thought experiment is unrealistic but insightful.

More generally, investors are able to reallocate wealth between
states to some extent by changing real investment plans.
The allocations (W1 and W2 ) and prices (q1 and q2 ) are jointly
determined.
Nonetheless, equation (1) still holds when the dust settles.
Because the first order condition holds for any pair of prices, q1
and q2 , including the equilibrium prices.
Thus, the implications for state prices apply more generally.

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Going from state prices to asset prices

We have so far characterized the state prices.

But remember the basic relationship between asset prices and
the state prices:
pn = q1 Xn1 + q2 Xn2 . (2)
Thus, the same insights also apply for more general assets (not
just Arrow-Debreu securities).

Assets that pay off in unpleasant states of the world are

expensive because they provide insurance. Thus, they have a
low expected return.

Assets that pay off in good states of the world are less
valuable, and must offer a high expected return.

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Another look at stocks, bonds, bills, and gold

The return history fits the basic story of equilibrium with risk
aversion.
Stocks do well when the economy does well, and have the
highest average return.
Gold does well when the economy does poorly, and has a lower
average return.
Bonds and bills are intermediate.

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Stocks, bonds, bills, and gold

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Roadmap

1 Arithmetic and Geometric Average Rates of Return

2 Evaluating Projects: Internal Rates of Return

3 A statistical view of financial history

4 Optimality and equilibrium

5 The stochastic discount factor

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The stochastic discount factor (SDF)

The same basic equations can be written in yet another way.

Recall the price decomposition equation in (2).
Rewrite this as:
   
q1 q2
pn = π 1 Xn1 + π 2 Xn2 ,
π1 π2
= π 1 M1 Xn1 + π 2 M2 Xn2 ,
= E [MXn ] .

Here,
qs
Ms = .
πs
is called the “stochastic discount factor” for state s.

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SDF approach to asset pricing
The stochastic discount factor is a random variable that differs
across states, but is the same for all assets.
The price of any asset is the expected product of the SDF
and the payoff.
In recent years, this has become an important paradigm of
academic finance.
For reviews of finance from this perspective (but much harder
than Ec 1723), see:
John Cochrane, Asset Pricing, Princeton University Press, 2005
John Campbell, “Financial Decisions and Markets: A Course in
Asset Pricing ”, Princeton University Press, 2017
The other paradigms are: behavioral (psychology matters); and
demand-based (frictions in demand / supply matter). We’ll see
them later in the class.
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Some takeaways

Is the arithmetic mean or the geometric mean the right way to

measure an investor’s typical return experience? How do they
compare?
Depends on the question. Both are useful.
The arithmetic mean is higher than the geometric mean.
If an asset pays off in states where marginal utility is high it will
be more valuable, other things equal. Why is this?
Investors’ desire to allocate wealth to states with high marginal
utility pushes up the prices for those states.
Assets that pay off in those states command higher prices.

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Some takeaways

Does the history of bills, bonds, stock, and gold returns make
sense in this context?
Stocks pay off when times are good and gold pays off when
times are bad, consistent with their average returns.
How does the stochastic discount factor (SDF) approach price
assets?
The price of any asset is the average of its payoffs, weighted by
state probabilities and marginal utilities.
Equivalently, it is the expected product of the SDF and the
payoff, where the SDF is proportional to marginal utility.

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For next time

Next time we will study portfolio optimization in more detail, by

assuming that investors care about the mean and variance of
return on their wealth portfolio.
Initially, we will consider the case with one risky and one
risk-free asset.
Prepare by viewing the online lecture module on the insurance
premium.
Also read BKM Chapter 5, section 5.5, and Chapter 6, sections
6.1-6.5.

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