Duration and equity issuance [30 points]

You are hired as a consultant by an insurance company. The company has the following balance

sheet:

Assets Liabilities, Equity

Cash (D = 0) $150 Liabilities toward policy holders (D = 1:5) $1,039

Medium-term bonds (D = 5) $272 Liabilities toward policy holders (D = 2) $351

Short-term bonds (D = 1:2) $1,148 Equity $180

TOTAL $1,570 TOTAL $1,570

where "D" denotes the duration of each item, in years.

1. What is the duration of assets? (Use the formula to compute the duration of a portfolio when

all assets have the same YTM. The balance-sheet numbers refer to current market values; for

instance, PB = $272 for the bonds with D = 5) [4 points]

2. What is the duration of liabilities? (Follow the same approached you've used in the previous

question.) [4 points]

3. The answers to your previous questions should highlight a mismatch between the duration

of assets (i.e., DA) and the duration of liabilities (i.e., DL). The insurance company wants to

immunize the balance sheet to obtain DA = DL, and is considering multiple options.

(a) The first option is to issue new equity in exchange for cash, so that equity and cash

in the balance sheet increase by the same amount. Compute the dollar value of new

equity (= new cash) to obtain DA = DL. (Round your answer to the second decimal.)

[5.5 points]

(b) The second option is to renegotiate the liabilities toward the policy holders. To keep

things simple, let's assume that

Liabilities can have only duration of 1.5 or 2 years;

The total value of liabilities has to remain the same.

What is the value of liabilities with duration 1:5 years that achieves immunization?

And what is the value of liabilities with duration 2 years? [5.5 points]

(c) The third option is to issue additional liabilities with duration 1.5 years, in exchange

for cash. That is, for every dollar of new liability, the insurance company gets $1 of

cash. What is the dollar value of new liabilities that needs to be issued? (Round your

answer to the second decimal.) [5.5 points]

(d) The last option is to use the cash to buy bonds with duration > 0. Without doing

any calculation, answer and explain briefly: can this option achieve the immunization

objective? [5.5 points]