You manage the portfolio of assets and liabilities of a boutique insurance company. The assets consist entirely of bonds, and therefore the value of the firm' assets depends on the short$-$term interest rate, x$.$ At the same time, the value of the firm's liabilities also depends on the short$-$term interest rate, x$.$ Your job is to measure how the value of the assets and the value of the liabilities change when the interest rate x changes.

Suppose the value of the firm's assets depends on the interest rate x as follows:

Suppose also that the value of the firm's liabilities depend on the interest rate x as follows:

Finally, suppose that interest rates are currently at x $=2\%.$

Question $1$: By how much would the value of the assets, A$,$ change if interest rates were to increase from $2\%$ to $3\%?$ Please show your work. Please use marginal analysis to answer this question.

Answer:

Choose one of the following answers:

A $=3\mathrm{.}3$

B $=1\mathrm{.}8$

C $=-0\mathrm{.}1$

D $=-2\mathrm{.}4$

E $=-4\mathrm{.}7$

Question $2$: By how much would the value of the firm's liabilities, L$,$ change if interest rates were to increase from $2\%$ to $3\%?$ Please show your work. Please use marginal analysis to answer this question.

Answer:

Choose one of the following answers:

A $=3\mathrm{.}3$

B $=1\mathrm{.}8$

C $=-0\mathrm{.}1$

D $=-2\mathrm{.}4$

E $=-4\mathrm{.}7$

Question $3$: The value of the equity in the firm can be viewed as the difference between the value of the assets, A$,$ and the value of the liabilities, L$.$ If the value of equity falls below $0,$ the firm is officially bankrupt. How much of an interest rates change $($from the current level of x $=2\%)$ is enough to put your firm into bankruptcy? Please show your work.

Answer:

Choose one of the following answers:

A $=-0\mathrm{.}741\%$

B $=-0\mathrm{.}356\%$

C $=-0\mathrm{.}128\%$

D $=0\mathrm{.}215\%$

E $=0\mathrm{.}518$Suppose the value of the firm's assets depends on the interest rate $x$ as follows:

$A(x)=(-10x+5{)}^{2\mathrm{.}5}{e}^{0\mathrm{.}5x-0\mathrm{.}1x^2}$

Suppose also that the value of the firm's liabilities depend on the interest rate $x$ as follows:

$L(x)=\frac{100-0\mathrm{.}1x}{2+0\mathrm{.}4x}$

Finally, suppose that interest rates are currently at $x=2\%.$