$1.$ Some Insurance Company issued a $$90$ million, one$-$year note at $8$ percent annual interest

$($paying one coupon at the end of the year$)$ with an $8$ percent yield. The proceeds, plus $$10$ million in

equity, were used to fund a $$100$ million, two$-$year commercial loan with a $10$ percent coupon rate

and a $10$ percent yield. Immediately after these transactions were simultaneously closed, all market

interest rates increased $1\mathrm{.}5\%(150$ basis points$).$

a$.$ What was the duration of the commercial loan when it was first issued?

b$.$ What is the duration of the liability, which is a note, at the time of issuance?

c$.$ What is the leverage adjusted duration gap of Some Insurance Company after the issuance of

the note?

d$.$ What is the predicted change in equity value that forecasted by the duration gap when the

interest rate increases by $1\mathrm{.}5\%?$

e$.$ What is the market value of the loan investment and the liability after the change in interest

rate? What is actual impact on the market value of equity when the interest rate increased by

$1\mathrm{.}5\%?$

f$.$ Using the duration rule, what is the expected change in the value of the loan if the interest rate

is expected to increase to $11\mathrm{.}5\%$ from the initial $10\%?$