AP

You see utility of the form u(W) = −exp(−aW) and returns are normally distributed. What do you do?

This is CARA utility with risk aversion coefficient a. By the MGF of the normal distribution, maximising E[u(W)] is equivalent to: max μ_p − (a/2)σ_p²

Two conditions required:

1. CARA utility → u(W) = −exp(−aW)

2. Normally distributed returns

The coefficient a is whatever number sits in the exponent. In an exam if you see u(W) = −exp(−20W), then a = 20.

What is the Capital Market Line (CML) formula and what does it describe?

The CML gives the expected return of any efficient portfolio combining the risk-free asset and the market portfolio:

μ_p = r_f + [(μ_M − r_f) / σ_M] × σ_p

Where:

- r_f = risk-free rate

- μ_M = expected return on market portfolio

- σ_M = standard deviation of market portfolio

- σ_p = standard deviation of your portfolio

The slope (μ_M − r_f) / σ_M is the Sharpe ratio of the market — the "price of risk".

Key points:

1. Only holds for EFFICIENT portfolios (on the frontier)

2. x-axis is σ_p, y-axis is μ_p (mean-std dev space)

3. The market portfolio M is the tangency point between the CML and the risky asset frontier

What is the Espinoza-Tsomocos core argument about why the yield curve slopes upward?

The puzzle they're solving

The yield curve has historically been upward sloping even during long periods when interest rates barely moved. Standard theories (expectations, risk premiums) struggle to explain this — if rates are stable, why should long-term bonds always yield more?

Their key insight

It's not about risk or expectations — it's about liquidity and money. People need cash to complete transactions. In any future state of the world where interest rates are higher, money is tighter and more valuable. That scarcity makes those states more 'expensive' — they carry a higher state price. This mechanically pushes long-term yields up.

The intuition in one sentence

Higher future interest rates = tighter money = higher state price for that future = long-term bonds must yield more to compensate.

How does Espinoza-Tsomocos explain the flattening yield curve — and what role does default play?

Why default matters

Default is what gives money its value in the model. If no one could ever default, there'd be no real cost to borrowing and money would be essentially free. Default risk creates a genuine financing cost which feeds directly into how assets are priced across different future states.

Why the curve has been flattening

As markets develop and function better — lower inflation volatility, pension fund reforms, high savings from emerging economies — the liquidity constraint relaxes. When it's easier to access cash when you need it, high-interest-rate states become less 'scary', their state prices fall, and the premium for long-term bonds shrinks. The curve flattens — not because rates changed, but because liquidity improved.

The bottom line

Liquidity is the root cause of both the upward slope and its flattening over time. The model unifies both puzzles under one framework — something expectations theory and risk premium theory alone cannot do.

What is the fundamental theorem of finance?

1. Security prices exclude arbitrage iff there exist strictly positive state prices qs>0 ∀ s∈S 

2. Security prices exclude strong arbitrage iff there exist positive state prices qs≥0 ∀ s∈S but at least one qs=0, so can make profit on weak arbitrage
   

What is the formula for put-call parity?

C + K/(1+rf) = P + S₀

C - Call price

P - Put price

K - Strike price

S₀ - Current stock price

What are the differences between European and American calls?

What are the differences between European and American puts?

Calls: An American call on a non-dividend paying stock is worth the same as a European call. Early exercise is never optimal because you always throw away time value and insurance value — it's better to sell the option than exercise it early. Dividends change this, as you may want to exercise just before the ex-dividend date.

Puts: An American put is always worth more than a European put. Early exercise can be optimal — if the stock falls deep enough in the money, it's worth exercising early to receive K now and earn interest on it. This benefit can outweigh the lost time and insurance value.

What is the formula for the risk-neutral probability of the up state?

q = (Rf - d) / (u - d)

Note: Risk neutral probabilities require discounting unlike state prices