Financial Risk Management of Insurance Enterprises

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Financial Risk Management of Insurance Enterprises

• Introduction to Asset/Liability Management,
  Duration Convexity

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Review...

• For the first third of the course, we have
  discussed
• The need for financial risk management
• How to value fixed cash flows
• Basic derivative securities
• We will now discuss techniques used to evaluate
  asset and liability risk

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Today

• An introduction to the asset/liability management
  (ALM) process
• What is the goal of ALM?
• The concepts of duration and convexity
• Extremely important for insurance enterprises
• Extensions to duration
• Partial duration or key rate duration

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Asset/Liability Management

• As its name suggests, ALM involves the process of
  analyzing the interaction of assets and
  liabilities
• In its broadest meaning, ALM refers to the
  process of maximizing risk-adjusted return
• Risk refers to the variance (or standard
  deviation) of earnings
• More risk in the surplus position (assets minus
  liabilities) requires extra capital for
  protection of policyholders

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The ALM Process

• Firms forecast earnings and surplus based on
  best estimate or most probable assumptions
  with respect to
• Sales or market share
• The future level of interest rates or the
  business activity
• Lapse rates
• Loss development
• ALM tests the sensitivity of results for changes
  in these variables

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ALM of Insurers

• For insurance enterprises, ALM has come to mean
  equating the interest rate sensitivity of assets
  and liabilities
• As interest rates change, the surplus of the
  insurer is unaffected
• ALM can incorporate more risk types than interest
  rate risk (e.g., business, liquidity, credit,
  catastrophes, etc.)
• We will start with the insurers view of ALM

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The Goal of ALM

• If the liabilities of the insurer are fixed,
  investing in zero coupon bonds with payoffs
  identical to the liabilities will have no risk
• This is called cash flow matching
• Liabilities of insurance enterprises are not
  fixed
• Policyholders can withdraw cash
• Hurricane frequency and severity cannot be
  predicted
• Payments to pension beneficiaries are affected by
  death, retirement rates, withdrawal
• Loss development patterns change

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The Goal of ALM (p.2)

• If assets can be purchased to replicate the
  liabilities in every potential future state of
  nature, there would be no risk
• The goal of ALM is to analyze how assets and
  liabilities move to changes in interest rates and
  other variables
• We will need tools to quantify the risk in the
  assets AND liabilities

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Price/Yield Relationship

• Recall that bond prices move inversely with
  interest rates
• As interest rates increase, present value of
  fixed cash flows decrease
• For option-free bonds, this curve is not linear
  but convex

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Simplifications

• Fixed income, non-callable bonds
• Flat yield curve
• Parallel shifts in the yield curve

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Examining Interest Rate Sensitivity

• Start with two 1000 face value zero coupon bonds
• One 5 year bond and one 10 year bond
• Assume current interest rates are 8

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Price Changes on Two Zero Coupon BondsInitial
Interest Rate 8
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Price Volatility Characteristics of Option-Free
Bonds

• Properties
• 1 All prices move in opposite direction of change
  in yield, but the change differs by bond
• 23 The percentage price change is not the same
  for increases and decreases in yields
• 4 Percentage price increases are greater than
  decreases for a given change in basis points
• Characteristics
• 1 For a given term to maturity and initial yield,
  the lower the coupon rate the greater the price
  volatility
• 2 For a given coupon rate and intitial yield, the
  longer the term to maturity, the greater the
  price volatility

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Macaulay Duration

• Developed in 1938 to measure price sensitivity of
  bonds to interest rate changes
• Macaulay used the weighted average
  term-to-maturity as a measure of interest
  sensitivity
• As we will see, this is related to interest rate
  sensitivity

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Macaulay Duration (p.2)
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Applying Macaulay Duration

• For a zero coupon bond, the Macaulay duration is
  equal to maturity
• For coupon bonds, the duration is less than its
  maturity
• For two bonds with the same maturity, the bond
  with the lower coupon has higher duration

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Modified Duration

• Another measure of price sensitivity is
  determined by the slope of the price/yield curve
• When we divide the slope by the current price, we
  get a duration measure called modified duration
• The formula for the predicted price change of a
  bond using Macaulay duration is based on the
  first derivative of price with respect to yield
  (or interest rate)

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Modified Duration and Macaulay Duration

• i yield CF Cash flow
• P price

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An Example

• Calculate
• What is the modified duration of a 3-year, 3
  bond if interest rates are 5?

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Solution to Example
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Example Continued

• What is the predicted price change of the 3 year,
  3 coupon bond if interest rates increase to 6?

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Example Continued

• What is the predicted price change of the 3 year,
  3 coupon bond if interest rates increase to 6?

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Other Interest Rate Sensitivity Measures

• Instead of expressing duration in percentage
  terms, dollar duration gives the absolute dollar
  change in bond value
• Multiply the modified duration by the yield
  change and the initial price
• Present Value of a Basis Point (PVBP) is the
  dollar duration of a bond for a one basis point
  movement in the interest rate
• This is also known as the dollar value of an 01
  (DV01)

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A Different Methodology

• The Valuation book does not use the formulae
  shown here
• Instead, duration can be computed numerically
• Calculate the price change given an increase in
  interest rates of ?i
• Numerically calculate the derivative using actual
  bond prices

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A Different Methodology (p.2)

• Can improve the results of the numerical
  procedure by repeating the calculation using an
  interest rate change of -?i
• Duration then becomes an average of the two
  calculations

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Error in Price Predictions

• The estimate of the change in bond price is at
  one point
• The estimate is linear
• Because the price/ yield curve is convex, it lies
  above the tangent line
• Our estimate of price is always understated

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Convexity

• Our estimate of percentage changes in price is a
  first order approximation
• If the change in interest rates is very large,
  our price estimate has a larger error
• Duration is only accurate for small changes in
  interest rates
• Convexity provides a second order approximation
  of the bonds sensitivity to changes in the
  interest rate
• Captures the curvature in the price/yield curve

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Computing Convexity

• Take the second derivative of price with respect
  to the interest rate

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Example

• What is the convexity of the 3-year, 3 bond with
  the current yield at 5?

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Predicting Price with Convexity

• By including convexity, we can improve our
  estimates for predicting price

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An Example of Predictions

• Lets see how close our estimates are

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Notes about Convexity

• Again, the Valuation textbook computes
  convexity numerically, not by formula
• Also, Valuation defines convexity differently
• It includes the ½ term used in estimating the
  price change in the definition of convexity

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Convexity is Good

• In our price/yield curve, we can see that as
  interest increases, prices fall
• As interest increases, the slope flattens out
• The rate of price depreciation decreases
• As interest decreases, the slope steepens
• The rate of price appreciation increases
• For a bondholder, this convexity effect is
  desirable

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Applications of Duration

• Remember, ALM evaluates the interaction of asset
  and liability movements
• Insurers attempt to equate interest sensitivity
  of assets and liabilities so that surplus is
  unaffected
• Surplus is immunized against interest rate risk
• Immunization is the technique of matching asset
  duration and liability duration

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Limits to Duration

• Our predictions of the change in bond value was
  based on some assumptions
• The change is interest rates is small
• Convexity can improve results to a point
• Changes in the yield curve are parallel
• All rates move the same amount
• The bond has no options
• Effective duration reflects cash flow
  sensitivity to the level of interest rates

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Assuming Parallel Shifts

• The assumption of parallel shifts in the yield
  curve is not plausible
• In reality, short-term rates move more than
  long-term rates
• Also, it is possible that the yield curve
  twists
• Short-term and long-term rates move in opposite
  directions

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An Illustration

• There are two cash flows, 100 at the end of year
  1 and 100 at the end of the second year
• The interest rate is a flat 5
• Calculating modified duration

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Partial Duration

• Each term in the calculation tells us something
  about interest rate sensitivity
• It is the sensitivity of the cash flow to that
  interest rate
• In this example, define two partial durations
• One for each cash flow period

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Interpreting Partial Duration

• Note that the sum of the partial durations is
  equal to the original duration calculation
• Using partial duration, we can determine the
  interest rate sensitivity to any non-parallel
  shift in the yield curve
• We can use partial duration to predict price
  changes

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Example

• From our two period cash flow, what is the change
  in value if the one year rate goes to 4 and the
  two year rate goes to 6

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Key Rates

• Interest rates of similar maturities move in
  the same fashion
• The 10 year rate and the 10½ year rate move
  similarly
• Therefore, partial durations can be based on a
  few points on the yield curve
• These are called key rates
• Partial durations are sometimes referred to as
  key rate durations

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Typical Key Rates

• Popular key rates are
• 3 month and 6 month rate
• 1 year
• 2 years
• 3 years
• 5 years
• 7 years
• 10 years
• 30 years

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Applications of Key Rate Durations

• Key rate durations are very useful for hedging
  purposes
• Because multiple partial durations provide more
  information than a single duration number,
  insurers can determine their sensitivity to
  interest rates based on various parts of the
  yield curve
• If the insurer is not immunized, it can use
  interest rate derivatives to hedge the risk

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Next time...

• An introduction to stochastic processes
• The use of stochastic movements in modeling
  interest rates