Consider an economy in which $($only$)$ two states, State $1$ and State

$2,$ are deemed to be possible in one year from today. There are two securities whose

cash$-$flows, expressed in Swedish Crowns, are reported in Table $1.$Today$$s price of

each security is equal to SEK $100.$

Cash$-$flow in State $1$ Cash$-$flow in State $2$

Security $1750$

Security $0250$

Table $1.$ The cash$-$flows of securities $$ and $$ in the two states.

$2\mathrm{.}2\mathrm{.}1.$ Which sums should we invest in Security $$ and in Security $$ to create a

portfolio that pays out SEK $100,$ regardless of the state, in one year?

$2\mathrm{.}2\mathrm{.}2.$ What are the amount and the sign of the cash$-$flow that we would face today

if we constructed the portfolio in $2\mathrm{.}2\mathrm{.}1?$

$2\mathrm{.}2\mathrm{.}3.$ What must be the risk$-$free one year$-$rate of interest in this economy, if the

prices do not admit arbitrage opportunities?

$2\mathrm{.}2\mathrm{.}4.$ Suppose that we are told that in this economy, all individuals are risk$-$neutral.

What must be the probabilities attached to the two states?

Hint. Relatively to $2\mathrm{.}2\mathrm{.}4,$ a risk$-$neutral investor is indifferent between a risk$-$free cashflow of SEK $$ due on any given date and a cash$-$flow $$ paid out on the same date with

probability $,$ provided $=.$ You may conveniently choose $=100(1+),$ where $$

is the one year, risk$-$free rate of interest as the $$benchmark$.$