ECON 251 Financial Theory- Notes H1

ECON 251: Financial Theory Lecture 1-2

For any questions please contact: [email protected] 1 / 28

ECON 251: Financial Theory – H1 Professor: John Geanakoplos

All videos and materials are available at http://oyc.yale.edu/economics/econ-251

The note is only for help and this is the first half, enjoy!

1. Why Finance?

2. Utilities, Endowments, and Equilibrium

Course:

1. Methods for pricing financial assets and making optimal financial decisions.

2. Reexamine the logic of laissez-faire1 vs. regulation of financial markets.

3. Explaining the mortgage market and the recent crisis—the leverage cycle.

4. Social security

Model in Economies – Economic equilibrium

Distinguish Exogenous variables e2 from Endogenous variables x3.

Equilibrium conditions: F (e,x)=0;

Equilibrium X(e) conditions: F (e, X(e))=0

Comparative statics =counterfactual reasoning

Double auction4

John’s explanation：People yell and scream at each other and the whole thing would be over in a few

minutes.

Or “Gyration of the market caused by too much borrowing and speculation ”

Other example like New York stock exchange.

Equilibrium price

Demand=supply -> market equilibrium

So that demand price = supply price -> equilibrium price

(PS: The same goes for equilibrium quantity~)

General equilibrium

=> welfare maximize

Total utility function:

U (x, m) = u (x) + m m: money

Exogenous Variables

Agent i, i∈I, Utility Ui, Endowment Ei, Goods x, y∈C

1 Laissez-faire (or sometimes laisser-faire) is an economic environment in which transactions between private parties are free from

government restrictions, tariffs, and subsidies, with only enough regulations to protect property rights. 2 Independent variable that affects a model without being affected by it, and whose qualitative characteristics and method of

generation are not specified by the model builder. 3 A factor in a causal model or causal system whose value is determined by the states of other variables in the system; contrasted with

an exogenous variable. 4 A double auction is a sales proceeding where buyers and sellers submit bid and ask prices to an auctioneer simultaneously, and this

party determines a clearing price for the sale.

U

X

http://oyc.yale.edu/economics/econ-251



Wi (x, y) = Ui(x) + Ui(y),

Ei = (EIx, EIy) initial Endowment

Marginalist’ idea:

It’s part of human nature that the more you get of sth, the less extra advantage it brings you.

Diminishing Marginal Utility

Utility function:

Ui= 100x - 12 x2 +y => M Ui(x) = 100-x differentiate x

Uj= 13 logx+

23 logy

Endogenous Variables

=> price P, trades/final consumption

Budget Set5: EIx +EIy

∵ budget set is constant =>Indifference Curve6

∴ Px(xi - EIx) = Py(EIy -yi )

Equilibrium is defined by (Px, Py), (xi, y

i)

Final consumptions = Final endowments

Equilibrium => price taking, agent optimization, rational

expectations and market clearing

1. ∑ xi = ∑ EIx ∑ yi = ∑ EIy

2. Px(xi - EIx) + Py(yi -EIy )=0

Agent i has no other income or expenses, he needs to sell sth to buy sth.

3. Utility function

If MuixMuiy

= PxPy , get maximized utility

Px, Py, EIx ,EIy are known, slop = PxPy . Every point i on the line

meets the budget set.

5 A budget set or opportunity set includes all possible consumption bundles that someone can afford given the prices of goods and the

person's income level. 6 In microeconomic theory, an indifference curve is a graph showing different bundles of goods between which a consumer is indifferent.

That is, at each point on the curve, the consumer has no preference for one bundle over another. One can equivalently refer to each point on the indifference curve as rendering the same level of utility (satisfaction) for the consumer. Utility is then a device to represent preferences rather than something from which preferences come.[1] The main use of indifference curves is in the representation of potentially observable demand patterns for individual consumers over commodity bundles.

UI(x)

Indifference Curve

X

Y

Budget Set I=Px*x+ Py*y

Slope MuixMuiy

= PxPy

ECON 251: Financial Theory Lecture 3


3. Computing Equilibrium

Review and Introduction

- Financial theory don’t have psychology

- Free markets work best

- Utilitarian view of economics

In equilibrium the final allocation maximizes total welfare

Example 1 –-A, B

Two goods: x, y Two agents: A, B

Welfare function: WA(x, y) =100x - 12 x2 + y WB(x, y) = 30x -

12 x2 + y

Endowment: EA = ( ExA ,EyA) = (4,5000) EB = ( ExB ,EyB) = (80,1000)

Example 2 –-C, D

Welfare function: WC(x,y) = 34 logx +

14 logy WD(x,y) =

23 logx +

13 logy

Endowment: EC= ( ExC ,EyC) = (2,1) ED= ( ExD ,EyD) = (1,2)

Equilibrium is always defined by turning things into equations.

- Endogenous variables: (Px, Py, xA, yA , xB ,yB)

- Exogenous variables: (80, 1000, 100x…)

A maximize WA: Si -> Px* xA +Py* yA = Px* ExA+Py* EyA

B maximize WB: Si -> Px* xB +Py* yB = Px* ExB+Py* EyB

Demands = Supply

xA + xB = ExA + ExB = 4+80 = 84

yA + yB = EyA + EyB = 5000+1000 = 6000

Budget set

Px* xA + Py* yA = 4Px + 5000Py

Px* xB + Py* yB = 80Px + 1000Py

TRICK

MuXA(xA, yA)/Px = (100- xA)/Px = MuY

A(xA, yA)/Py = 1/Py ○1

MuXB(xB, yB)/Px = (30- xB)/Px = MuY

B(xB, yB)/Py = 1/Py

Economy meaning：

In equilibrium, the last 1 dollar spending on no matter which kind of goods should have the same utility,

otherwise Agent i could trade goods with low marginal utility for the high one.

○1 means, the utility by choosing x = the utility by choosing y, Indifference Curve

PS: 1. MuXA(xA, yA) = dWA/dxA, by definition of marginal utility

2. MuXA(xA, yA)/Px: marginal utiliy/price, how much utility A can get by spending one unit of money on x

PS：Beautiful mind of Nash

How to explain Equilibrium: interaction or rational interaction

How to solve:

Take Py=1 => (100 - xA)/Px =1 ∴ xA = 100 - Px, similarly, xB = 30 - Px,

∵ xA + xB = 4+80 = 84 ∴ xA = 77, xB =7, Px = 23

Py can be any other number, it’s just relative price.



Demands = Supply

XC + xD = ExC + ExD = 2+1 = 3 ○6

yC + yD = EyC + EyD = 1+2 = 3

Budget set

Px* xC + Py* yC = 2Px +Py ○4

Px* xD + Py* yD = Px + 2Py ○5

TRICK

MuXC(xC, yC)/Px = (

34 *

1 xC )/Px =

34 *

1 xC*Px ○2

=> xC*Px, MxC ,how much money C spend on x

MuYC(xC, yC)/Py = (

1y *

14 )/Py =

14 *

1 yC*Py ○3

=> yC*Py, MyC, how much money C spend on y

○2 = ○3 ∴ for Agent C, MxC/MyC = 3/1, MxC = 3/4 C’s Income

MuXD(xD, yD)/Px = (

23 *

1 xD )/Px = MuY

D(xD, yD)/Py = (13 *

1yD )/Py

Similarly for Agent D, MxD/MyD = 2/1, MxD = 2/3 D’s Income

Take Py=1,

Agent C’s Income = Px* xC + Py* yC =2Px +Py ○4

xC = MxC/Px = 2 +1/Px, similarly xD= MxD/Px = 1+2/Px

from ○6 , XC + xD = 3

∴ Px = 5/2, xC = 9/5, xD = 6/5 similarly yC = yD = 3/2

Explanation

- Pareto Efficiency7

- Cobb – Douglas Utility Function

- Marginal Utility

Mu = dTudQ that’s Total utility/ Total quantity

- Diminishing marginal utility

If Δ=Mu, Δ =0,Tu get max

7 PS: There are three conditions for Pareto efficiency.

1. Allocative efficiency. It concerns how a given stock of consumption commodities is allocated to different consumers. This condition requires that the marginal rate of substitution between any pair of goods be equalized across different consumers. 2. Technical efficiency. It concerns how factors of production are transformed into consumption goods. This condition requires that the technical rate of substitution between any pair of productive factors be equalized across different producers. 3. Production-allocative efficiency. It concerns whether the right mix of consumption goods are produced. Production-allocative efficiency requires that the marginal rate of transformation between any pair of goods be equal to their marginal rate of substitution.

PPS: Marginal rate of substitution, the rate at which an individual must give up "good A" in order to obtain one more unit of "good B", while keeping their overall utility (satisfaction) constant. The marginal rate of substitution is calculated between two goods placed on an indifference curve, which displays a frontier of equal utility for each combination of "good A" and "good B". As such, the marginal rate of substitution is always changing for a given point on the indifference curve, and mathematically represents the slope of the curve at that point.

Mu



4. Efficiency, Assets, and Time

Q: Total Welfare max?

CHECK Ⅰ

Example 1 – A, B

Total Welfare W = WA + WB = 100xA - 12 xA2 + yA + 30xB -

12 xB2 + yB

∵ xA + xB = 4+80 = 84, yA + yB = 5000 + 1000 = 6000

∴ W = 100xA - 12 xA2 + 30xB -

12 xB2 + 6000

xB = f(xA) = 84 - xA, ∴ d xB/d xA = -1

Care xA’s allocation to maximize W

dW/dxA = 100 - xA + (30 - xB)*d xB/d xA ∵ xA = 77, xB = 7, ∴ d xB/d xA = -1

dW/dxA = 23-23 = 0, dW = 0, ∴ xA’s allocation do maximize W

Because W is diminishing marginal utility, it is concave. When dW = 0, W gets its max.

Example 2 – C, D

Welfare WC = 34 logxC +

14 logyC WD =

23 logxD +

13 logyD

( ExC ,EyC) = (2,1) ( ExD ,EyD) = (1,2) (xc, yc) = (95,

32) (xD, yD) = (

65,

32)

C 对商品 X 的 marginal utility：MuXC =

34 logxC =

34 *

59 =

512

D 对商品 X 的 marginal utility：MuXD =

23 logxD =

23 *

56 =

59

MuXC ≠ MuX

D, then you could trade goods with small marginal utility for the big one to get more total utility.

∴ Final equilibrium allocation doesn’t maximize the sum of utilities.

Free market is a false argument?

No. It rests on a premise that’s indefensible. There’s constant marginal utility which everyone can measure.

So we need other way of capturing the mathematical idea that the invisible hand is a good thing.

Pareto Efficiency

If you cannot make anybody better off without hurting

somebody else, you are at Pareto efficient point.

Start Economy E (Wi, Exi ,Eyi) goes to Equilibrium E’

E’ is a price vector (Px, Py, (Xi, Yi)),

such that ∑ Xii=I = ∑ Exii=I , ∑ Yii=I = ∑ Eyii=I

to maximize Wi(x, y),

such that Budget set: Px* xC +Py* yC = Px*ExC + Py*ExD

Everyone individually optimizes looking at his own budgets

set, choose Xi, Yi, Supply =Demand -> final allocation.

Pareto criterion allocation (x, y) for all i, Wi (x, y) Wi(x, y).

Theorem:

Point i: initial endowment

Point f: final endowment

By switching C for D, Point O makes WC+W

D

better than i and f, but WC get smaller.

i

WC

f

O

WD



If an feasible allocation (x, y) is an equilibrium, or to say Pareto Optimal, for the economy E,

such that ∑ Xii=I = ∑ Exii=I , ∑ Yii=I = ∑ Eyii=I , then no other feasible allocation (x,y)

such that,

∑ Xii=I ∑ Exii=I for all i

and ∑ Yii=I ∑ Eyii=I for some i.

Pareto optimality, then, is the allocation at which no

individual can increase his utility without

reducing the utility of others

Or to say,

if start with a competitive equilibrium, no way to

make EVERYONE better off than competitive

equilibrium f(95,

32) from view of C.

Proof Ⅱ

First fundamental theorem of welfare economics.

Agent rationality: Wi (x, y) Wi(x, y) for all i,

Px* x +Py*y Px*Ex + Py*Ey, so it will

Px*∑x +Py*∑y Px*∑Ex + Py*∑Ey → constriction!

- Pareto is about efficiency.

- What’s narrow: Externalities not include.

Last thought today

New Economy:

Utility Wi, Endowment (Exi, Eyi), Payoffs (Dxα, Dyα), ownership of the assets θiα

- Q: What means PxPy ?

- PxPy = 1+ real interest rate

It means how much an apple is worth today relative to how much it is worth tomorrow.

Eg. Px= (1+20%) Py means by giving up an apple today you can get 1.2 apples tomorrow.

(3,3)

Total endowment YC

XC

YD

XD

i (2,1)

f (95,

32)

Budget set of C



5. Present Value Prices and the Real Rate of Interest

General Equilibrium: Fisher’s conclusion

1. Market functioning by itself without interference from outside. A situation of Laissez-faire leads to allocation

that are Pareto efficient, total welfare maximized.

2. Free market achieves Pareto efficient at least if there’s no externalities and no monopoly.

3. Price is determined by marginal utility not bu total utility. Just price is not exist.

On interest

- Aristotle: interest is unnatural.

- The Bible says interest is terrible.

- Christianity, Judaism and Islam frowns on interest.

Economy E

- Two Agents A vs. B, two goods (x1, x2)

- Utility of A: logx1 + 12logx2 ① discount factor δ =

12

Utility of B: log x1 + logx2 ②

- Endowment: (E1A, E2

A) = (1, 1) (E1B, E2

B) = (1, 0)

- Stocks α vs. β, which produce sth in the future.

- D2α =1, D2

β = 2.

D is anticipated dividend, that is, output of stock in period 2.

D2α =1 means stock α produce 1 apple in period 2.

Find financial equilibrium ---- (q1, q2, (x1A, x2

A), (x1B, x2

B), πα, πβ, θαA, θαB, θβA, θβB)

- q1 means contemporary price today, q2 is contemporary price tomorrow

- consumption (x1A, x2A), (x1B, x2B)

- πα, πβ means stock α and β’s price in period 1

- (θα’A, θβ’A ) = (1, 0.5), (θα’B , θβ’B) = (0, 0.5)

θα’A = 1 means original ownership by A of stock α is 1.

Define a financial equilibrium

- You buy shares in period 1 and shares get dividend in period 2.

- Budget set for i∈(A, B) in period 1

q1x1 +πα*θα’ + πβ*θβ’ = q1E1 + πα*θα + πβ*θβ

- Budget set for i∈(A, B) in period 2

q2x2 = q2E2 + θα* D2αq2 + θβ* D2

βq2, income in the future

choose x1, x2, θα, θβ to maximize U subject to this budget set.

- Period 1

x1A + x1

B = E1A + E1

B

stock α: θαA + θαB = θα’A + θα’B

stock β: θβA + θβB =θβ’A + θβ’B

- Period 2

X2A + x2

B = E2A + E2

B + (θα’A + θα’B)*D2α + (θβ’A + θβ’B)* D2

β

Define inflation

Inflation = q2

q1 =

price tomorrow price today



On Abitrage

Suppose after finding the equilibrium you add a third asset stock πϥ, which paid 1 dollar in period 2.

price in period 2: 1

price in period 1: πϥ = 1 + i, that is, πϥ = 1

1+i i: nominal interest

The idea of present value prices

Suppose p1 = q1, p1 is the price of an apple today

p2 means the price today of an apple tomorrow. We call p2 the present value price

if πβ = 0.25,

p2

πβ = 1

D2β = 2 ∴ p2 = 0.125

According to definition, by giving up 0.25 today you get 2 dividend tomorrow, so to get 1 devidend tomorrow

you need give up 0.125 today.

Stocks effectively add to the endowments of goods.

Economy E’

Utility: U’A(x1, x2) = UA(x1, x2), U’B(x1, x2) = UB(x1, x2)

Endowment:

E’2A = E2

A + θα’A *D2α + θβ’A * D2

β = 1 + 1 + 0.5*2 = 3, Similarly E’2B = 1

∴ (E’1A, E’2

A) = (1, 3), (E’1B, E’2

B) = (1, 1)

Solve for equilibrium

Take P1 = 1

According to ① and ②, A spends 2/3 of his income on x, B spends 1/2 on x.

A’s money spending on xx's price today P1

+ B's money spending on x

x's price today P1 = E1

A + E1B = 2

2/3(P1* E’1

A + P2*E’2A)

P1 +

1/2(P1* E’1B + P2*E’2

B)P1

= 2

2/3 + 2P2 + 1/2 + 1/2P2 = 2

∴ P2 = 1/3 ∴ P1 = 1, P2 = 1/3

∴ x1A = 4/3, x1

B = 2/3, x2A = 2 , x2

B = 2

Real interest rate

P1

P2 = 1 + r = 3, ∴ r = 200%

Stock price

Take q1 = 1, p1 = 1. How much need you to give up today to get one apple tomorrow.

P1

P2 = 3,

For stock β:

1 apple today --- 2 apple tomorrow --- 2 apple tomorrow each worth 1/3 today’s price --- worth 2/3 apple’s price

today

∴ πα = 13, πβ =

23

Fisher’s theory doesn’t explain:

- Inflation, because q2 is unknown

- Nominal interest rate i, but explain real interest r.



6. Irving Fisher's Impatience Theory of Interest

Financial economy is defined by

Beginning of the economy

- Agent A vs. B in economy and their utilities:

Utility of A: logx1 + 12logx2 Utility of B: log x1 + logx2

- Endowment: (E1A, E2

A) = (1, 1) (E1B, E2

B) = (1, 0)

- Ownership of shares: (θα’A, θβ’A )= (1, 0.5), (θα’B , θβ’B) = (0, 0.5)

- Dividends: D2α =1, D2

β = 2

Find equilibrium (q1, q2, (x1A, x2

A), (x1B, x2

B), πα, πβ, θαA, θαB, θβA, θβB)

q1x1 +πα*θα’ +πβ*θβ’ ≤ q1E1 +πα*θα +πβ*θβ

Define

- Period 1: x1A + x1

B = E1A + E1

B

- Period 2: X2A + x2

B = E2A + E2

B + (θα’A + θα’B)*D2α + (θβ’A + θβ’B)* D2

β

Financial equilibrium Economic equilibrium General equilibrium

General equilibrium (UA, UB, (E1’A, E2’A ),( E1’B , E2’B))

Define

- E1’A = E1A = 1, E1’B= E1

B = 1

E1’A = E1A +θα’A *D2

α + θβ’A * D2β=1 + 1 + 0.5*2 = 3, Similarly E’2

B = 2

- Price: q1 = q2 = 1, πα = 13,πβ =

23

REMARKS: Fisher has no theory for contemporaneous prices. It’s all relative prices.

- Consumptions: x1A = 4/3, x1

B = 2/3, x2A =2 , x2

B = 2, P1 = 1, P28 = 1/3

Fisher’s principle of no arbitrage

If πα = 13, then πβ =

23. Because D2

α =1, D2β = 2, otherwise it’ll be arbitrage.

Application of the principle of no arbitrage --- looking through the veil

1. Nominal interest rate

If introduce a nominal bond with payoff 1 dollar in period 2, set q1 = q2 = 1

By definition, price of the bond = 1

1 + i, where i is nominal interest rate

1 dollar today 3 units of stock α 3 units of x2 3 dollar dividends

1 + i = 3, so i = 200%

2. Real interest rate

Defined as number of goods today goes into number of goods tomorrow:

1 + r = q1

q2

1 unit of x1 today 1 dollar 3 units of α 3 units of x2

1 + r = 3, so r = 200%

3. Inflation

Suppose q1 = 1, q2 = 2, inflation 1 + g = q2

q1 = 2, so g = 100%

8 p2 means the price today of an apple tomorrow. We call p2 the present value price.



But πα = 13 still, why?

What is the price of stock

1) πα = D2

α

1 + r

- The value of a stock today is the real dividends tomorrow. It’s paying in the future and discounted by real

interest rate.

- If you turn cash next year into cash this year,

πα = D2

α

1 + r*q2

2) πα = p2* D2α

p2 means the price today of an apple tomorrow.

You see nothing is changed, so stock price remains the same.

Suppose q1 = 1, q2 = 2, inflation g = 100%, how about i?

1 dollar today 3 α 3 units of x2 times p2 = 6 dollars

1 + i = 6, so i = 500%

Fisher equation --- the relationship between i and r

1 + i1 + g = 1 + r, so i = 500%

Bank interest

1 + r = 1 + i1 + g = 1.01/1.02

So r = -1%

Why negative? Federal Reserve wants to give money away to the banks

4. Fundamental theorem of asset pricing

The price of assets

- How do you trade off 1 dollar today for 1 dollar tomorrow, you take the nominate interest rate times the

money that’s being produced.

πα D2

α* q2 =

11 + i

- Or you take the real goods in the future discounted by the real interest rate

πα D2

α = 1

1 + r

Remarks

Price today *( 1 + i) = price tomorrow, i is about price

Numbers today *(1 + r) = numbers tomorrow, r is about apples

Question 1

- Assume q1 = q2 = 1, if China lend you money at 0% interest, would people rush to it?

- Of course! If you borrow money from this economy, you need to pay 200% interest but from China you

pay nothing!

Question 2

- If there’s a new technology, which can turn 1 apple today into 2 apples tomorrow, could this technology be

used to make a Pareto improvement?

- Everyone better off? Make himself a profit? No! It loses money! You give up 1 dollar today to get



Today ------------ Tomorrow

Before 3 -------------- 1

New tech 2 -------------- 1

discounted 2/3 dollar tomorrow…

Proof why not Pareto improvement

- If in end it led to an allocation (x1’A, x2’A, x1’B, x2’B), that make everyone better off.

Then, p1 x1’A + p2 x2’A > p1 E1A + p2E2

A, p1 x1’B + p2 x2’B > p1 E1B + p2E2

B

Then, total consumption value > total endowment value, ∴ contradiction.

The total endowment in Fisher’s economy is an equilibrium, but the new tech’s endowment is even less

than Fisher’s, so it’ll be worse.

- Q: new tech but still old price, flawed logic?

No. The unchanged price is the key to this proof. At old prices everybody spend more than the value of

their endowment. So it comes the contradiction.

Now, we add one more step, the new tech changes the old endowments too. So it makes the value even

less than before!

- Basic idea

Financial Equilibrium General Equilibrium

Question 3

- What changes the real interest

- Fisher:

Fisher’s three examples

Example 1

- If Utility of A: logx1 + 12logx2, discount factor δ =

12 →

13,

what about r?

- 12 ->

13,

∴ more impatience ,care less about future.

- Real interest rate ↑ Why?

Formal proof: Cobb-Douglas economy

Set p2 = 1

x1A =

11 + δ*

p1 x1A + p2 x2

A

p1 =

11 + δ*( E1

A + E1

A

p1)

p1↓, demand x1A↑

δ↓ at old price x1A↑, so demand line goes horizontally to the

right.

To clear the market, need P1↑. 1 + r = p1

p2, so r↑

Example 2

- If more optimistic about E2A, how about real interest rate?

Use Pareto Efficiency

Use Marginal Utility

a

b

c

P1

X1

Demand

Supply

1. Impatience theory of interest

Poor imagination: apple today > apple tomorrow

2. Mortality

People might die between today and tomorrow

3.



a

b

c

P1

X1

Demand

Supply

a

b

P1

X1

Demand

Supply

- It’ll goes up.

- Intuitive answer

Better future, more apple ---- need get them to eat all that

extra stuff tomorrow (give them incentive to eat to eat) ---

raise the rate

Proof 2:

At old price, you are going to be rich tomorrow

--- you’ll consume more today --- x1↑

--- but endowment today don’t change

--- how to clear the market --- need raise p1 relative to p2 ,

that to say, r↑

Example 3

- If transfer money from poor to rich, how about r? Define rich

people become rich because they are patient.

- Rich guy (patient guy) are going to consume more in the

future

--- the economy is going to be more in the hands of patient

people, mixture changes.

--- people are more patient than before, so they consume

less today than before

--- demands today ↓, to clear the market, r needs ↓

Summarize of Fisher’s three examples

- More impatient people, higher interest rate. πα

D2α =

11 + r, r

↑, so πα↓.

- More optimistic about future, higher interest rate.

- Transfer money from poor to rich, lower interest rate.

Interest as a price

- Instead of thinking of money today for more money tomorrow. Fisher thought of goods given up today for

goods tomorrow.

- The real rate of interest is no more or less than the relative price of goods today vs. goods tomorrow.

- Thus the rate of interest is the most important price in the economy.

Real interest rate matters!

Why real interest rate is positive?

Because people are impatient.

Theory of impatience

“Impatience is a fundamental attribute of human nature. As long as people like to have things today rather than

tomorrow, there will be a rate of interest. Interest is, as it were, impatience crystallized into a market rate. ”

---Fisher

Fisher’s advice: all contract and wages should be inflation indexed.

Patience and wealth

- The patient will accumulate wealth

- By waiting they make possible more production. I.e. they allow society to produce wealth.



- Their patience earns them their returns.

Shakespeare anticipated all of Fisher’s Impatience Theory of Interest and went a step further. He took collateral

into consideration.



7. Shakespeare's Merchant of Venice, Collateral.

Present Value and the Vocabulary of Finance

Present value9

πt = price at time of 1 dollar at time t

Pt = price at time of one apple at time t, take out of inflation

Constant nominal interest rates

πt = 1

(1 + i)t for all i = 1…T

Fisher: present value price

Theorem: If a stream of future cash flows (m1, m2, m3, … , mT) can be bought at time 0, and if there is no

arbitrage, then its price must be

PV(m, π) = π1m1 + π2m2 + π3m3 + … +πTmT PV: assets today’s price

PV(m, i) = m1/(1 + i) + m2/(1 + i)2 + m3/(1 + i)3 + … + mT/(1 + i)T

Cash flow: (m1, m2, m3, … , mT)

If the price of the bond is less than PV, buy it and sell it at t1, t2,…, tT, then you make money. If there’s no

arbitrage, then the price of bond = this discounted cash flow PV(m,π)

Doubling time

Q: how many years at interest rate I does it take to double you money?

A: (1 + i)n = 2, nlog(1 + i) = log2 ≈ 0.69, n = 0.69

log(1 + i), according to Taylor’s theorem,

log(1 + i) ≈ 0 + (1 + i) – 1 + 12(- 1)(I + i - 1)2 ≈ 0 + i -

12* i2

For i is very small, i - 12* i2 ≈ i

∴ n = 0.69

log(1 + i) ≈ 0.69

i

For i = 0.07, log (1 + i) = i - 12* i2 = 0.0675 ≈ i, n =

0.690.0675 ≈ 10.2 ≈

0.720.07 ≈

0.72i footnote10

∴ Doubling time ≈0.72

i , if i = 6%, doubling time = 12 years, if i = 8%, it’ll be 9 years.

Power of exponential growth

Indian sold Manhattan for 24 dollar in 1646.

- At 6% interest, Doubling time = 12 years, hence after 360 years get 30 doubling now.

230 = (210)3 = 10243 ≈ 1 billion, 24 dollar → 24 billion now.

- If at 7% interest, Doubling time = 72/7 years, 360/72/7 = 35,235 ≈ 1 billion* 25 = 32 billion,

24 dollar → 32 *24 = 768 billion in 360 years.

Exponential growth sensitive to rate

Q: Suppose you deposit 1 dollar in a fund. 36 years later you withdraw the money. If the fund returns 7% every year,

how much money do you get in the end? Suppose the fund changes a fee of 1% per year, means you get 6%,

9 Present value, also known as present discounted value, is a future amount of money that has been discounted to reflect its current

value, as if it existed today. 10

Why take 0.72? Because if you take i = (6% ~ 10%), accumulated error is smaller than you take 0.70 or 0.69. See Wiki “Rule of 72”. For higher accuracy, see adjustments of rule 72.



after 36 years what fraction of your money is lost?

A: At 7%, Doubling time = 72/7, 236/72/7 = 23 + 0.5 = 23* 20.5≈ 8*1.4 = 11.2

At 6%, Doubling time = 72/6 = 12 years, 236/12 = 23 = 8

(11.2-8)/8 = 28%

1 percent less every year, after 36 years you lose 28%. How huge a difference a percent makes!!

Coupon bond, Coupon rate and Annual coupon

A coupon bond has a face value and a coupon.

Coupon = coupon rate c * face value

Theorem: If a coupon bond pays the same coupon rate c as the constant interest rate i prevailing in the economy,

c = i, then its price must be

PV(m, i) = 100*(c

1 + i + c

(1 + i)2 + c

(1 + i)3 + … + c + 1

(1 + i)T)

= 100*(i

1 + i + i

(1 + i)2 + i

(1 + i)3 + … + i + 1

(1 + i)T)

= 100

if i > c, then PV(m,i) < 100. If i < c, then PV(m, i) > 100.

Perpetuity

Perpetuity pays the same fixed coupon c forever.

Theorem: A perpetuity has present value

PV(m, i) = ci* (1 -

1(1 + i)T) ≈

ci , T is forever, for

1(1 + i)T is very small

Q: If at 6% interest, you get 12 $ every year, what’s the present value?

A: 126% = 200 dollar

Annuity

Annuity gives a fixed payment each year until T, no principal in the end.

0 1 2 3 4 T-1 T

Principal

C

(m0, m1, m2, … , mT ) = (0, 100c, 100c, 100c, … , 100c + 100) at rate c%

0 1 2 3 4 T-1 T T+1 forever

0 1 2 3 4 T-1 T T+1



time 0 T

PT

P

Annuity (m0, m1, m2, … , mT )= (0, c, c, c, … , c)

- Be changed in two ways:

Be protected against inflation

Be timed to last the rest of your life, social security annuity

Annuity present value

An annuity with payment c and maturity T has present value depending on the interest rate.

PV(m,r) = c*(1

(1 + r) + 1

(1 + r)2 + 1

(1 + r)3 + … + 1

(1 + r)T )

= cr (1 -

1(1 + r)T )

Annuities are often inflation corrected, r for the real rate of interest

T- period Annuity with coupon c = Perpetuity with coupon c – Perpetuity of coupon c contracted from T (or

beginnes at time T+1)

T-period Annuity = ci -

ci *

1(1 + r)T

Example

Q: At 6% rate a 12 dollar perpetuity is worth 200 dollar today ci =

126% = 200, a 24 dollar 36 year annuity is

worth?

A: At 6% interest, doubling time = 12 years, so in 24 year you double it twice.

PV(m,r) = ci (1 -

1(1 + i)T ) = 200*(1-

14) = 150

Suppose i = 8%, a 100 dollar 30 year annuity is worth?

A: At 8%, doubling time is 9 years, you double it 3 times in 27 years.

(1 + i) = 1.0830 = 1.083 * 23 ≈ 1.25*8 = 10

100 = c

0.08*(1 - 1

10), now you get c… if you buy it with 100,000 dollar, you get 8888 dollar every year.

Mortgage11

Mortgage is an annuity, defined by a principal, a mortgage coupon rate and a maturity.

In the last example, a mortgage at 8% for 30 years on a 100,000 dollar principal get payment 8888 dollar per

year. Which means 8888 dollar every year discounted is 100,000 dollar today at 8% interest rate.

But mortgage have monthly payments.

Monthly rate = coupon rate

12

8%12 ≈ 0.67%, at monthly rate you pay slightly more than

888812 month every month.

Mortgage is nominal fixed payments.

11

A debt instrument that is secured by the collateral of specified real estate property and that the borrower is obliged to pay back with a predetermined set of payments. Mortgages are used by individuals and businesses to make large purchases of real estate without paying the entire value of the purchase up front.



Practical problem

Suppose your salary at Yale is 115,000 dollar per year and it goes up at 3% inflation every year. Nominal interest

rate = 5.3%, inflation = 3%. You work for 30 years and retired for 30 years. How much should you spend every

year? Let’s say you want level real consumption.

A: Fisher said don’t care about inflation, what you should care is real interest rate.

Real interest rate: 1 + r = i + i

1 + g = 1.0531.03 = 1.023, r = 2.3%

Present value: 72/2.3=31.3 doubling time is about 30 years.

PV(m,r) = ci (1 -

1(1 + i)T ) =

1150000.023 *(1 -

11.02330 ) =

1150000.023 *(1 -

12 ) = 2.5 million

2.5million = c

0.023 *(1 - 1

1.02360 ) = c

0.023 * 34 , c ≈ 76000

The question is just like a 115,000 dollar at 2.3% rate 30 years perpetuity is worth how much today, and with

these money how much will you get every year if it is a 60 years perpetuity at the same rate.



8. How a Long-Lived Institution Figures an Annual Budget

Yale’s maintenance

Suppose the budget of Yale is 100 million every year for 10 years, how to cut out the budget for maintenance?

The real interest rate is 5%.

A: Suppose no inflation.

PV0 = 1005% (1 -

1(1 + 5%)10 ) = 772

So the annual deficit should have been around

PV1 = c/i, c = 5% * 772 = 38.6

If take a 4% expected inflation into account, now the present value including real interest and inflation

Real interest rate = 1- 1.05*1.04 = 0.09

PV0’ =

1009% (1 -

1(1 + 9%)10 ) = 641

The annual deficit should be

C = 5% * 641 = 32 million

Internal rate of return12 (in term s of a hedge fund)

Yield is to figure out a number that summarizes how good a bond is.

Simple coupon bond

A simple coupon bond pays the same coupon every year and pays the principal and the coupon at its maturity.

Suppose 10 years, payments (0, m1, m2, … , mT ) ≥ (0, 0, … , 0) but not all zero

PV(m, r) = 0

105 = 7

1+y + 7

(1+y)2 + … + 107

(1+y)10

If it’s 100 instead of 105, you’ll see y =7%, but it is 105, so y < 7%

coupon rateprice =

7105 = 6.7% called the current yield

If the actual interest rate is 6%, then price of the bond would be more than 100.

Q: If sb tells current yield 7%PV > 6% (market actual interest rate), should you buy it?

Premium bond: price > face

Discount bond: price < face

Par bond: price = face

12

IRR is a discount rate, that is often used in capital budgeting that makes the net present value of all cash flows from a particular project equal to zero.

7 = rate

105 = price

0 1 2 3 4 T-1 T

100 = face value



Theorem: if market price = present value at a going interest rate, then current yield on a premium bond > the

interest rate.

But why?

To keep this simpler, let’s suppose it was 10 and the interest rate went down to 5%, now the present value of

this bond is going to be less than 200. Because if it were 200 at 5% it would give you 10, 10,… , 210, but this

gives you 10, 10, … , 110, so obviously the present value is less than 200. Therefore 10 over something less than

200 is going to be more than 5%.

If you have a coupon bond that’s a premium bond then the current yield is always above the interest rate.



9. Yield Curve Arbitrage

Yield curve

Yield is an attempt to look at an investment and without paying any attention to the market or anything outside the

investment.

Cash flow: P0 + c1

1 + y + c2

(1 + y)2 + … + cT

(1 + y)T = 0, P0 is the price negative

EXAMPEL --- Sometimes yield to maturity is misleading

Cash flow: 1,-4, 3

Yield to maturity: 1 + -4

1 + y + 3

(1+ y)2 = 0, y = 0 or y = 200%, so it is ambiguous.

- On the other hand, the yield will depend too sensitively on stuff early rather than the stuff late. And so again,

you get into troubles yielding just yield to maturity.

Fisher: Use market interest rate to figure out what the present value of all the cash flows is.

EXAMPLE

To keep it simple let’s say the treasury has issued 5 different bonds over 5 different years on the same day. They

set the coupon was 1 dollar for the 1-year bond, 2 dollars for the 2-year bond, … 5 dollars for the 5-year bond.

The face value is always 100. The prices turn out to be 100.1, 100.2, 100.3, 100.4 and 100.5. 100.5 is the price

the market’s paying for the 5-year bond.

PV prices 1-year coupon 2-year 3-year 4-year 5-year

Paid year 1 1 101 2 3 4 5

Paid y 2 1 0 102 3 4 5

Paid y 3 1 0 0 103 4 5

Paid y 4 1 0 0 0 104 5

Paid y 5 1 0 0 0 0 105

Bond price 100.1 100.2 100.3 100.4 100.5

For the 4-year bond, 100.4 = 4

1 + y + 4

(1 + y)2 + 4

(1 + y)3 + 104

(1 + y)4, you’ll find the unique y.

Price of the bond:( Π1, Π2, Π3, Π4, Π5) = (100.1, 100.2, 100.3, 100.4, 100.5)

Today’s money price for 1 dollar at time T: π1, π2, π3, π4, π5 --- the price of zero coupon bond

Fisher: Find the interest and the price of Zero coupon bonds13

- The right price of the bond P0 = C1π1 + C2π2 + C3π3 + C4π4 + C5π5

- If some guy is offering you the investment opportunity at a lower price than P0, you should buy it. You will lock

in a profit for sure. So if you knew the πs, you would know for sure how to value any project.

- How to deduce the πs are from the data?

π1 is the price you would pay today to buy 1 dollar tomorrow.

π1 = Π1

101 = 100.1101 = 0.991

π2 = Π2

102 - 2π1

102 = 0.962

13

A zero-coupon bond (also called a discount bond or deep discount bond) is a bond bought at a price lower than its face value, with the face value repaid at the time of maturity. The price of a zero-coupon bond can be calculated by using the following formula: Price = Face/(1 + y)

T



π3 = Π3

103 - 3π1

103 - 3π2

103 = 0.917

The principle of duality

100.1 = 101π1

100.2 = 102π2 + 2π1

100.3 = 103π3 + 3π2 + 3π1

100.4 = 104π4 + 4π3 + 4π2 + 4π1

100.5 = 105π5 + 5π4 + 5π3 + 5π2 + 5π1

Forward interest rate

I promise to give you 1 dollar in year 2 and how much money will you give me in year 3?

1 + it means the number of dollars at t+1 in exchange for 1 dollar at time t.

1 + it = πT

πT+1

- Forward rate changes faster than the yields, because yields are average numbers.

- If everyone had a perfect forecast of what was going to happen in the future. Then of course the forward rates

in the market today would have to be exactly equal to the forward interest rate.

- Could you tell me what the price of the 5-year coupon bond was going to be next year? Suppose people don’t

have any doubt about what’s going to happen in the future.

There’s a formula waiting for you to find out ^_^ .

Forward interest rate

T + 1 Today T



10. Dynamic Present Value

Present value of the assets is just the discount of future dividends.

Suppose π0 = 1, π1 = π1

π0 =

11 + i0

, π2 = π1

π0 *

π2

π1 =

1(1 + i0)(1 + i1)

PV0 = C1π1 + C2π2 + C3π3 + … + CTπT

= c1

1 + i0 +

c2

(1 + i0)(1 + i1) + … + cT

(1 + i0)(1 + i1) … (1 + iT)

PV1 = c2

1 + i1 +

c3

(1 + i1)(1 + i2) + … + cT

(1 + i1)(1 + i2) … (1 + iT)

∴ PV0 = c1 + PV1

1 + i0

Mark to market

- How to define profit at time 1?

Rate of return = c1 + PV1 - PV0

PV0 = i0, which is what you should expect to get…

EXAMPLE – Premium bond

Let’s say you have PV0 = 8

1.06 + 8

1.062 + … + 108

1.06T = 108.4, the forwards rate are 10% for all t.

The rate of return on the first year = 8/108.4 > 6%, the market rate of interest, it seems very good. But the fund is

losing value, PV1 < PV0. It is not good.

Carry trade14

- Suppose you have a 2-year bond which pays 2 and 102, and another 5-year bond which pays 4, 4, 4, 4 and 104.

i0 = i1 = 2%, it = 5% for all t ≥ 3. Would it possible that the present value of 2-year bond equals the present value

of the 5-year bond?

It’s possible. Because you are discounting the first two payments by 2% a year but after that you are discounting

these payments by 5%. They are getting discounted by a lot.

- So what is the carry trade? The carry trade is you buy the long bond and sell short the short bond. Which means,

in this case, you are going to make money at the beginning but lose it back later.

How mortgages work

- By this formula PV0 = c1 + PV1

1 + i0 we can use backward induction to figure out the PV0.

I don’t know PV1 but i know at the end of time the present value of the bond is 0, so i can find PVT-1.

FOR EXAMPLE

A 30 - year mortgage with payment 8 for every year at an interest rate 7%. So what is the present value of this

bond?

It pays nothing at time 30, so it is 0.

The present value of what’s left at time 29 is 8

1+ 7% = 7.47

The present value of what’s left at time 28 is 8 + 7.471+ 7% = 14.4

And so on.

14

Carry trade, without further modification, refers to currency carry trade: investors borrow low-yielding currencies and lend (invest in) high-yielding currencies to make profits.



- It used to be in the old days that mortgages were coupon bonds. They pay 8, 8, 8, … , 108. But just before the

108 payment everyone would default. So the bank said we should make the payment be constant and in that

way there’s no reason for the guy to default right at the end.

But of course if it’s constant that means the present value of what’s left is going down all the time. So that’s why

it’s called an Amortizing mortgage. And bankers wanted that to happen because that way their risk is going

down every year.

The purpose of a mortgage is you take out a loan using the house as collateral. But if in year 5 just after you are

making your payment of 8.05 dollars you decide to move, you say to the bank, you want to cancel the mortgage,

you just need to pay the remaining balance, 93.91. so that’s why this remaining balance is an important

number.



11. Social Security

12. Overlapping Generations Models of the Economy

Overlapping generation model

Land produce 1 1 1 1 1

1

3 1

3 1

3 1

3 1

Time Time 1 Time 2 Time 3 Time 4 …

Every generation Gt has endowments: (Ett, Et+1

t ) = (3, 1)

U0 (x1) = logx1, G0 owns the land

Ut (xt, xt+1) = logxt + logxt+1,

Find equilibrium

Describe equilibrium: (qt, Πt, (xt, xt+1), x1, θ)

qt means the apple price of every period

Πt is the price of land in each period

Θ is how much land they hold

Budget set for t ≥1, (young y, old z, θ)

Young: qt*y + Πt,* θ ≤ qt* Ett

Old: qt+1*z ≤ qt+1*1 + Πt+1,* θ + qt+1*1*θ qt+1*1*θ: dividend from the land

G0 z: q1*z ≤ q1*1 + Π1,* 1 + q1*1

Equilibrium

(young y, old z, θ) = (xt, xt+1, θ) is best for generation t in budget set t

Z = x1 is best for generation 0

When people are young, they have to think about the price of land next period when they are old.

How to solve

Take qt = 1. We assume there is no inflation, so we just renormalize all nominal prices in terms of apples. We

measure the price of the land at time t in terms of apples in time t.

Fisher:

- Forget about the assets by putting their dividends into the endowments.

- Look at present value prices.

Suppose pT+1

pT = P, P is a constant number, pT is the price of an apple at time t.

Πt+1 + 1Πt

= 1 + r = 1P , P is like the interest rate between time t and time t + 1, means if I had one apple at time t

how many apples could I get at time t + 1, 1P .

Equilibrium for t ≥ 2

12(3 + 1p)

p +

12(3 + 1p)

1 = old apple 1 + young apple 3 + land produces 1 = 5



The old spend half of his income (3 + 1p) for apples

P = 3 ± √6, take p = 3 - √6 ≈ 0.55

The young eat apples = 12 (3 + 1p) = 1.775

The old eat apples = 5 – 1.775 = 3.225

The young spent 1.775 on apples and the left is for lands.

So the price of the land = 3 – 1.775 = 1.225 (we’ve supposed that θ = 1)

OR you can figure it out in Fisher’s way:

- Fisher means the price is the present value of all the payments. The land pays 1 apple every period. So it’s price

should be:

Π = 1p + 1p2 + 1p3 + … + pt = 1r , this is like perpetuity.

r = 1P - 1 = 0.81, Π =

1r = 1.225, it is looked at from the point view of time 1.

Πt+1 + 1Πt

= 2.2251.225 = 1.81 = 1 + r

I’m trading off consumption today for consumption when I’m old at 81%.

- Above proves the market would be cleared when t ≥ 2

At time 0, the old guy get the dividend of land because he’s owned the land.

1 endowment + 1 land produce + sells the land for 1.225 = 3.225, add the young’s 1.775, it will clear the market

at time 0.

- So the market is going to clear in every single period. We solve it for period 2 and onwards which were all

symmetric.

Fisher: The price of every asset is the present value of its dividends

Use that to figure out the price of land.

If you buy the lands today, the rate of return is 81%.

CHANGE – if (Ett, Et+1

t ) = (2, 2)


2

2 2

2 2

2 2

2 2


12(2 + 2p)

p +

12(2 + 2p)


P = (3 ± √5)/2, take p = 0.38

r = 1P - 1 = 1.61, r goes up.81% ----161%, which means Gt lose more. There’s a substantial loss for ever one’s

utility except for the first generation.

Experiment 1



Suppose we had more and more children ever generation and every 30 years the population doubles.


1

3 1

6 2

12 4

24 8

Time Time 1 Time

2

Time 3 Time

4

…

You need solve for the new equilibrium.

You’ll find the social security isn’t solved. The old get one apple from 2 youngs, but the interest rate goes up.

(‧_‧？),how could it be?!



13. Demography and Asset Pricing:

Will the Stock Market Decline when the Baby Boomers Retire?

Overlapping generation model


1

3 1

3 1

3 1

3 1


Principle here:

Fisher: Price of asset = discounted dividends

Endowments of G0: E0 = 2, 1, 1, 1, 1,…

PV of the land = 1, p, p2, p3, p4, p5,…

Utility (y, z) = 12 logy +

12 logz

If it is not symmetric, the function should be:

12(3pt-1 + 1pt)

pt +

12(3pt + 1pt+1)

pt = 1 + 3 + 1 = 5

PV of the land = p + p2 + p3 + p4 + p5 …

= 1

1+r + 1

(1+r)2 + 1

(1+r)3 + …

If the utility function changes, Utility (y, z) = 12 logy +

13 logz, means people become more impatience, so the

interest rate will go up.

Question 1:

- Suppose we have growth in the economy, what will happen?

Let’s say growth rate = 1 + g, the population of the young = (1 + g) the old

- Market cleaning equation at Period t:

12(3pt-1 + 1pt)

pt (1 + g)t-1 +

12(3pt + 1pt+1)

pt ( 1 + g)t = 1*(1 + g)t-1 + 3*(1 + g)t + (1 + g)t

Question 2:

- Let’s suppose that the model is an alternation between big generations and small generations.

By symmetry, you can guess that P is different between the small generation and the big one, let’s just say the

relative price for the small generation is ps and for the big generation is pb. And it’s going to keep repeating itself

over and over again. It seems just as follows.

Put dividends into endowments Financial Equilibrium General Equilibrium




1

3 1

6 2

3 1

6 2


- When the small generation is young:

12*2(3 + 1pb)

pb +

12(3 + 1ps)


When the big generation is young:

12(3 + 1ps)

ps +

12*2(3 + 1pb)


PV of the land at time 1 = 1

1 + rs +

1(1 + rs)(1 + rb) +

1(1 + rs)

2(1 + rb) + …

PV of the land at time 2 = 1

1 + rb +

1(1 + rb)(1 + rs)

+ 1

(1 + rb)2(1 + rs) + …

Question 3:

- Would you rather be in the big generation or in the small generation?

- In the small one.

Because if you’re making money when you are young and you’ve got a high interest rate, which means a high

return. But if you were in the big generation, it goes just by contraries.

TO BE CONTINUED ~

A big PS:

This entire staff is written on my own. Ideally these would be full thoughts of the content of the audio and video, but

since sometimes the board was not able to be seen, some ideas were beyond my abilities to capture, there might be

mistakes. All comments and questions are welcome. Please contact [email protected], let’s make it better.

Thank you.

Shareal

mailto:[email protected]

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ECON 251 Financial Theory- Notes H1