Lecture 1 - Risk and Return

NPV formula (EC333)

NPV = E(CF) / (1 + r)^T, where E(CF) = expected cash flow, r = discount rate, T = years until the cash flow arrives

E(CF) - meaning

Expected cash flow: the probability-weighted average payout, e.g. 50% x 20,000 + 50% x 10,000 = 15,000

Discount rate - interpretation

The 'exchange rate' between money in the future and money today; equal to the expected return investors require on an investment with a similar risk profile

NPV example - certain cash flow

15,000 in 10 years at a 2% risk-free rate: 15,000 / 1.02^10 = 12,305

NPV example - risky cash flow

Expected 15,000 in 10 years at a 6% discount rate: 15,000 / 1.06^10 = 8,376 - same expected cash flow, lower value because it is risky

Two components of the discount rate

1) Time value of money - money today is worth more than money tomorrow; 2) Risk premium - compensation for bearing risk

Risk premium - definition

Risk premium = discount rate - risk-free rate, e.g. 6% - 2% = 4%

Core question of EC333

The price of risk: where does the risk premium (e.g. the 4%) come from?

Financial assets valued with a risk-free (certain) cash flow

Mortgages, annuities/perpetuities, government bonds

Financial assets with risky (uncertain) cash flows

Corporate bonds, stocks - most financial assets

Common theme of financial economics (corporate finance and asset pricing)

Valuation: what is the fair value of an uncertain future cash flow? Fair = market price; Uncertain = risk; Future = time

EC333 syllabus

Portfolio selection (mean-variance), CAPM, option pricing (Black-Scholes), derivatives in FX markets

Market efficiency - normative vs positive perspective

Normative: how SHOULD risk be priced? (theory). Positive: how IS risk priced? (evidence)

Zoom Technologies (March 2020) - lesson

Shares of tiny China-based Zoom Technologies (ZOOM) surged as investors confused it with Zoom Video (ZM) - prices can move for reasons unrelated to fundamentals, so markets are not perfectly efficient

Newton and the South Sea Company (1720) - lesson

Newton sold at a 100% profit, then bought back in at a higher price and lost £20,000 - even very smart investors get caught in bubbles

Buffett's bet (2008-2017) - lesson

An S&P 500 index fund (+85.4%) beat a hand-picked portfolio of hedge funds over 10 years after fees - markets are very hard to beat consistently (evidence for efficiency)

Can markets be efficient if not everyone is rational?

Yes - market efficiency does not require all individuals to be fully rational; informed traders profit by trading against mistakes, pushing prices back, and random errors tend to cancel out

Earthquake vs fire insurance - why price differently?

Same 1% probability and 1m loss, but fire risks are largely independent (losses average out across 1m houses) while earthquake risks are highly correlated (one quake hits many houses at once) - correlated risk cannot be diversified away, so it should cost more

Key idea from the insurance example

Risk that cannot be diversified away (correlated risk) is the risk that gets priced - the seed of the CAPM

Expected return - formula

r-bar = sum of Pr(r_i) x r_i : the probability-weighted average return

Expected return - example

10% with probability 0.4 and 0% with probability 0.6: 0.4 x 10% + 0.6 x 0% = 4%

Variance - formula

sigma^2 = sum of Pr(r_i) x (r_i - r-bar)^2 : measures how far returns spread around the average; standard deviation sigma is its square root

Variance - example (10% w.p. 0.4, 0% w.p. 0.6)

sigma^2 = 0.4(10% - 4%)^2 + 0.6(0% - 4%)^2 = 0.0024, so sigma = 4.90%

How does EC333 measure risk?

By variance or standard deviation of returns

Five-stock data (Exxon, J&J, Microsoft, GE, JPMorgan, 2010-15) - what it shows

Individual weekly return std. devs of 1.81%-3.54% (average 2.84%), but an equal-weighted portfolio of all five has only 2.08% - diversification reduces risk in real data

Why does a portfolio have lower risk than the average stock?

Stocks are not perfectly correlated, so their ups and downs partly cancel out

Why doesn't diversification eliminate all risk?

Stocks share common market-wide movements; this correlated (systematic) risk remains and is what investors are paid for

Who developed mean-variance portfolio analysis?

Harry Markowitz (1950s-60s; Nobel Prize 1990)

Technical background needed for EC333

Probability (expected values, variance, covariance); calculus incl. constrained maximisation (Lagrange multipliers); linear algebra (matrices, vectors)

[L2 - slides 42-49] Mean-variance frontier graph (slide 42) - what it shows

Each portfolio plotted by risk (std. dev, x-axis) and expected return (y-axis); the curve shows the best risk-return combinations from risky assets; adding a risk-free asset gives a straight line tangent to the curve at the tangency portfolio (highest Sharpe ratio)

[L2] Two stocks A and B - setup

A returns 20% or 0% (50/50), B returns 10% or 0% (50/50), independent. E(rA) = 10%, sigmaA = 10%; E(rB) = 5%, sigmaB = 5%

[L2] Portfolio expected return (weight w in A, 1-w in B)

r_p = w x 0.1 + (1-w) x 0.05 = 0.05 + 0.05w

[L2] If you only care about expected return, which portfolio?

Invest everything in A (w = 1), the asset with the higher expected return

[L2] Variance of a two-asset portfolio - formula

sigma_p^2 = w^2 sigma1^2 + 2w(1-w) cov(r1,r2) + (1-w)^2 sigma2^2, where cov(r1,r2) = rho x sigma1 x sigma2

[L2] Portfolio variance when A and B are independent

Covariance = 0, so sigma_p^2 = w^2(0.1^2) + (1-w)^2(0.05^2)

[L2] If you only care about risk, which portfolio?

Not all in B (the lower-risk asset): minimising variance gives 0.2 in A and 0.8 in B - mixing assets lowers risk because they are not perfectly correlated

[L2] Diversification example - setup (C and D)

C and D are identical and independent: each returns 20% or 0% with equal probability (expected return 10%, sigma 10%)

[L2] Diversification example - 50/50 portfolio of C and D

Returns 20% (prob 0.25), 10% (prob 0.5), 0% (prob 0.25): expected return still 10%, but sigma falls from 10% to 7.07%

[L2] Key lesson of the C/D example

Diversification cuts risk without lowering expected return - a 'free lunch' from combining imperfectly correlated assets