In this exercise we consider a life insurance company $$Bronze$$ with the following initial balance sheet:

The value of the liabilities is $85$ NOK. The liabilities consist of one certain cash flow after $40$ years. The

value of $85$ NOK has been determined by discounting this single cash flow with the $40$ years interest rate

according to EIOPA $($i$.$e$.$ the organization that is responsible for Solvency II$).$ See the table in the

appendix. $($This is the curve that was published by EIOPA for December $2016).$ The curve converges to

the Ultimate Forward Rate of $4\mathrm{.}2\%.)$

The value of the bonds is $90$ NOK. In fact Bronze only invests in one government bond. The bond is a so

called zero$-$bond $($i$.$e$.$ there is only one payment at maturity date and no coupon payments in between$).$

The maturity of the zero$-$bond is $10$ years. The value of $90$ NOK has been determined by discounting the

single cash flow with the $10$ years interest rate according to EIOPA. There is no default risk.

We assume the EIOPA curve is applicable both to liabilities as to government bonds.

Questions:

$3)$ What would the balance sheet look like if the EIOPA interest rate curve would drop with $1\%$ the

coming split second $($and the equity investments will stay at the current level$)?$

a$.$ Approximate the new balance sheet using the duration approximation

b$.$ Approximate the new balance sheet using the duration $+$ convexity approximation

c$.$ Calculate the exact balance sheet without using approximations

Suppose that $$Bronze$$ is holding the same assets and liabilities for the coming $10$ years and suppose that

the equity investments stay at the current levels and interest rates do not change during the entire $40$

years. $($Ignore the interest rate change in $3).)$

$4)$ What does the balance sheet look like at T$=10$ one split second before the zero$-$bond matures? What

is the value of the Net Assets?

At T$=10$ the zero bond expires and is reinvested in a new zero$-$coupon bond with again a maturity of $10$

years.

$5)$ What is the size of the new zero$-$bond cash flow at T$=20?$

$6)$ What does the balance sheet look like at T$=20$ one split second before the new zero$-$bond expires?

What is the value of the Net Assets?

At T$=20$ the expiring zero$-$bond expires and is reinvested again in a new zero$-$coupon bond with again a

maturity of $10$ years.

$7)$ What does the balance sheet look like at T$=30$ one split second before the new zero$-$bond expires?

What is the value of the Net Assets?

At T$=30$ the expiring zero$-$bond expires and is reinvested again in a new zero$-$coupon bond with a maturity

of $40$ years.

$8)$ What does the balance sheet look like at T$=40$ one split second before the new zero$-$bond expires?

What is the value of the Net Assets?

$9)$ Describe the change in the value of the Net Assets between t$=0$ and T$=40.$ Explain what causes the

changes you see.

$10)$ Do you think the balance sheet as shown above gives a fair picture of the Net Assets $/$ the buffer the

company has at T$=0?$ Please comment.