According to the National Bureau of Economic Research $($NBER$),$ since $1945$ to $2009($extremes included$)$ there has been approximately $11$ business cycles. Assume that on average contractions

$($c$)$ last $10\mathrm{.}24$ months while expansions last $53\mathrm{.}76$ months and define normal times $($n$)$ as $60\%$ of the expansion months, characterizing the residual months as booms $($b$).$ Furthermore, suppose the expected return on the US market conditional on contractions to be $17\%,$ i$.$e$.$M $($c$)=0\mathrm{.}17,$ during normal times to be $1\%,$ and overall to be $8\%.$

Characterize both analytically and graphically:

A The probability distribution function $($pdf$)$ of the market return r$$M

B The cumulative distribution function $($cdf$)$ of the market return r$$M

$$ hint: first compute the probabilities of the states of the world, then check if you have all the associated expected returns $($if not calculate the missing one$($s$)),$ finally refer back to the notes where we introduced the pdf and cdf for discrete random returns

Let$$s now assume the market return to be a continuous random variable, normally distributed with expected value $$M $=0\mathrm{.}08$ and variance $\backslash$sigma $2$

C Compute the variance $\backslash$sigma $2$

assuming is the same as the one from the discrete distribution

D For the following realizations of r$$M $\{$rM $($i$)$M $+$ i$\backslash$sigma M $\}$i$=4,3,...,3,4$

$1.$ compute, using the excel function NORMDIST $($rM $($i$),$M $,\backslash$sigma M $,0),$ the normal pdf $($rM $($i$))$ and draw its graph

$2.$ compute, using the excel function NORMDIST $($rM $($i$),$M $,\backslash$sigma M $,1),$ the normal cdf $($rM $($i$))$ and draw its graph

$$ hint: I am not asking you to draw the graphs with Excel, you can if you know how, you can simply use a pen and a paper.

E Compute the following probabilities:

$1.$ Prob$($M $2\backslash$sigma M $<=$ rM $($i$)<=$M $+2\backslash$sigma M $)$

$2.$ Prob$($rM $($i$)<=0\mathrm{.}5)$