Interest Rate Risk Management

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

Balance Sheet of Bronze

Equity 10	Net Assets 15
Bonds 90	Liabilities 85
Total 100	Total 100

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 4.2%.)

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:

1) What is the size of the liability cash flow at T=40?

2) What is the size of the zero-bond cash flow at T=10?

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.

Appendix:

EIOPA interest rates

Maturity Rate

1 1.005%

2 1.186%

3 1.257%

4 1.358%

5 1.461%

6 1.564%

7 1.658%

8 1.747%

9 1.816%

10 1.874%

11 1.937%

12 2.004%

13 2.072%

14 2.141%

15 2.208%

16 2.273%

17 2.337%

18 2.398%

19 2.457%

20 2.513%

21 2.567%

22 2.618%

23 2.667%

24 2.714%

25 2.758%

26 2.801%

27 2.841%

28 2.880%

29 2.917%

30 2.952%

31 2.986%

32 3.018%

33 3.049%

34 3.078%

35 3.106%

36 3.133%

37 3.159%

38 3.184%

39 3.208%

40 3.230%