Financial Risk Management of Insurance Enterprises Forward Interest Rates

Overview What are forward rates of interest? How do you develop forward rates? What is the relationship between forward rates and spot rates? What is meant by the term structure of interest rates? What theories exist to explain the term structure shape?

Forward Rates - A Definition “Market consensus” of future interest rates Also known as implied forward rates - they are derived from the Treasury yield curve Any future period has a forward rate End of forward period must have a spot rate Typically, forward rates are calculated on bond maturity dates e.g., we used semi-annual maturities last time

Notation

Example An insurer has issued a two year CD and has two investment alternatives 1. Buy a one-year, 5% zero coupon Treasury and reinvest all the proceeds at maturity 2. Buy a two-year, 6% zero coupon Treasury Which investment should the insurer choose?

Example (p.2) Insurer is indifferent if:

Example - Enter Forward Rates From the Treasury curve (above), the current spot rates imply that the market consensus of the future rate of interest is 7.0095% If insurer thinks that the spot rate one year from now will be less than 7.0095%, it will use the two-year, zero coupon strategy In the first strategy, the insurer is exposed to the risk of a decrease in the interest rate

In General Use first principles of interest theory to find any forward rate For example, to determine the 5-year forward rate, 2 years from now:

Relationship Between Spot Rates and Short-Term Forward Rates Extending the previous discussion to short-term forward rates:

Consequences Yield rates, spot rates, and short-term forward rates are all different ways to convey interest rate information Discounting cash flows is most easily performed using either spot rates or short-term forward rates

Are Forward Rates Really a “Market Consensus” Studies show that forward rates do not predict future interest rates well Help investors measure expectations relative to the market Our insurer can speculate, if desired We will see in a future class that there are financial instruments based on forward rates Insurer may be able to hedge interest rate risk Forward rates are really hedgeable rates

Term Structure Four historical shapes: Relationship of interest rates and maturity Four historical shapes: Normal (upward slope) Inverted Humped Flat What explains the shape of the term structure?

Pure Expectations Theory Assumes forward rates are expectations of future interest rates What does this mean for inverted yield curve? No other systematic explanations predicting shape If investors believe interest rates will rise, they prefer short-term bonds, selling long-term bonds

Biased Expectations Theory Forward rates are expected future rates plus some other systematic factor Liquidity Theory Investors prefer short bonds and require a “liquidity premium” for holding longer bonds Preferred Habitat Borrowers/lenders want to be long/short Each requires premium to shift maturity

Market Segmentation Similar to preferred habitat due to maturity preferences Banks are main users of short-term funds Life insurers and pension funds are users of long-term funds Supply and demand among suppliers/users determines market rates

Modeling Interest Rates Interest rates move randomly Need statistical representation of stochastic process One-factor model describes short-term movements All other rates related to short term rate Two factor model has additional variable short term and long term movements short term level and short term volatility

Next time... Forward contracts Foreign exchange forwards FRAs - forwards on interest rates Applications of forwards in financial risk management