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Introduction to Finance

Why two risky assets can be safer than one

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Combine two risky assets and watch the risk shrink, find the least risky mix and the efficient ones, then add T-bills and find the one risky mix that investors with the same forecasts share.

Part 1 · Why diversify · Step 1 of 13

House and full insurance: together safe

houseinsurance policyboth together

Part 1 · Why diversify · Step 1 of 13

House and full insurance: together safe

You own a house worth $100,000 and an insurance policy on it. In a disaster (probability 10%) the house is worth nothing and the policy pays out. Otherwise the house keeps its value and the policy pays nothing. On its own, each asset is risky. Held together, they pay $100,000 whatever happens.

Try it: Cut the cover to 50%. The pair is risky again, but its SD is still far below the two SDs added together.

Probability of a disaster10%

The policy pays this share of the house's value100%

no coverfull cover

	Expected payoff	SD
House	$90,000	$30,000
Insurance policy	$10,000	$30,000
Both together	$100,000	$0

Expected payoffs add: $90,000 + $10,000 = $100,000. The SDs sum to $60,000; the pair’s SD is $0. Positive cover reduces payoff risk. The payoffs have perfect negative correlation; only full cover cancels risk completely.

E[house] = 0.1 × $0 + 0.9 × $100,000 = $90,000

Each payoff times its probability.

SD[house] = √(0.1 × (0 − $90,000)² + 0.9 × ($100,000 − $90,000)²) = $30,000

Square each payoff's gap from the mean, weight by probability, add, then take the square root.

Together the payoff is $100,000 in both states, so SD = $0: no risk at all.

Check yourself

True, False or Uncertain? Expected dollar payoffs add, percentage returns are weighted averages, and portfolio SD depends on covariance.

Question 1 of 5Challenge

Two stocks that always move in opposite directions

The returns of two fictional stocks are perfectly negatively correlated (ρ = −1). Stikine Gold (G) has an expected return of 22% and an SD of 40%. Lakeshore Power (U) has an expected return of 7% and an SD of 15%. Short selling is allowed, and the weights add up to 1 (w_G + w_U = 1).

Find every pair of weights that gives a portfolio SD of 18%. Which pair would a sensible investor choose, and why?
What is the expected return of each of those portfolios?
Find the weights and the expected return of the zero-variance portfolio.
If there is no arbitrage, what must the risk-free rate be in this economy?
Suppose T-bills actually pay 4%, and you can borrow or lend $1,000,000.00 at that rate for one year. Is there a profitable riskless trade? If so, show the trade and its profit.

Given

The two stocks

	Gold (G)	Power (U)
Expected return	22%	7%
Standard deviation	40%	15%

The questions

Target SD of the portfolio18%

T-bill rate actually paid4%

Amount borrowed or lent$1,000,000

Solution

mixes with both weights between 0 and 1mixes that need a short saleSD = 18%

Read the question and the givens first. Try the first part yourself, then press Start solving and open the working to check.

a. First pair, G / U

a. Second pair, G / U

b. Return, first pair

b. Return, second pair

a. The sensible choice, G / U

c. Zero-variance weights, G / U

c. Zero-variance return

d. Risk-free rate

e. Sure profit after one year

Final coming up and the tangency portfolio still feels slippery?

Bring your problem sets. We work through them together until every type feels routine.

Book a session Next toolNext: Systematic risk and the CAPM

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Why two risky assets can be safer than one

House and full insurance: together safe

House and full insurance: together safe

Two stocks that always move in opposite directions

Final coming up and the tangency portfolio still feels slippery?

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