Problem $1.(6$ points$)$

The process for the prices of a $5-$year maturity zero$-$coupon bond and of a derivative on

the interest rate that matures in three years are decribed by the following trees. The

probablities that an analyst associates with going up and down are $60\%$ and $40\%$ at each

node of the tree, respectively. $($NOTE: These are NOT the risk neutral probabilities.$)$

$5-$year zero coupon bond price

Suppose that you hold a portfolio of $10$ five$-$years zeros and $20$ derivatives. How does

the portfolio payoff evolve over three years? Construct the tree.

How can you change your position in the derivative in order to make the portfolio

riskless between date $t=0$ and $t=1?$

What is the implied interest$-$rate tree up to $t=2?$ For simplicity, use annual

compounding.

What is the price of a zero$-$coupon bond that matures at time $t=2?$ For simplicity,

use annual compounding.

Suppose that at time $t=1$ the interest rate is $5\%.$ How many the $3-$year zero$-$coupon

bonds do you need to hold for each unit of the derivative to obtain a Sharpe ratio of

$0\mathrm{.}75?$ For simplicity, use annual compounding. $($Hint: Recall that the variance

of a security $x$ is ${}_x^2=E[(x-E(x){)}^2]$