$1)$ Two independent loan prospects exist, call them A and B$.$ Both will pay the agreed$-$upon $$12$ return $($on $$100)$ with probability $3/4.$ With probability $1/4,$ on the other hand, they default and pay only $$4.$

a$.$ For an individual saver investing in one or the other, what is the average return $($expected return$)?$ What is the variance?

The average rate of return is $$10.$ The variance is $$6$ since you paid the $$4.$

b$.$ Suppose that an intermediary forms by merging the funds of the $2$ savers and investing in both projects, splitting the total return equally between the $2$ savers. What is the average return to the individual saver? What is the variance?

c$.$ Compare a$.$ and b$.$ Does your result confirm the Law of Large Numbers? Explain

d$.$ Suppose now that there are not just $2$ but huge numbers of independent loan prospects, all identical in their return characteristics to prospects A and B above. An intermediary forms combining funds from large numbers of savers. Each saver derives Happiness$($H$)$ from Income$($I$)$ according to the function: