Analytics of Risk Management I: Sensitivity and Derivative Based Measures of Risk Risk Management Lecturer : Mr. Frank Lee Session 2

Overview  Quantitative measures of risk - 3 main types  Sensitivity and Derivative based measures of risk  Sensitivity analysis  Differentiation  Gap analysis  Duration  Convexity  The ‘Greeks’

Introduction  Risk management relies on quantitative measures of risk.  Various risk measures aim to capture the variation of a given target variable (e.g. earnings, market value or losses due to default) generated by uncertainty.  Three types of quantitative indicators:  Sensitivity  Volatility  Downside measures of risk

Emphasis on Quantitative Measures  When data become available risks are easier to measure - increased use of quantitative measures  Risks can be qualified and ranked even if they cannot be quantified (e.g. ratings agencies)  Regulators’ emphasis and requirements - e.g. banking industry capital requirements.

Sensitivity  Percentage sensitivities are ratios of relative variations of values to the same shock on (variations of) the underlying parameter  E.g. if the sensitivity of a bond price to a unit interest rate variation is 5, 1% interest rate variation generates a relative price variation of a bond of 5 x 1% = 5%.  A value sensitivity is the absolute value of change in value of an instrument for a given change in the underlying parameters  E.g. if the bond price is $1000, its variation is 5% x 1000 = $50

Sensitivity Continued  Return sensitivity - e.g. stock return sensitivity to the index return (Beta).  Market value of an instrument (V) depends on one or several market parameters (m), that can be priced (e.g indexes) or percentages (e.g interest rates)  By definition: s (% change of value) = ( Δ V/V) x Δ m S (value) = ( Δ V/V) x V x Δ m s (% change of value) = ( Δ V/V) x ( Δ m/m) * * % change of parameter

Sensitivity Continued  The higher the sensitivity the higher the risk  The sensitivity quantifies the change  Sensitivity is only an approximation - it provides the change in value for a small variation of the underlying parameter.  It is a ‘local’ measure - it depends on current values of both the asset and the parameter. If they change both s and S do.

Sensitivities and Risk Controlling  Sources of uncertainty are beyond a firms control  random market or environment changes, changes in macroeconomic conditions  It is possible to control exposure or the sensitivities to those exogenous sources of uncertainties  Two ways to control risk:  Through Risk Exposures  Through Sensitivities

Sensitivities and Risk Controlling  Control risk through Risk Exposures - limit the size of the amount ‘at risk’  e.g. banks can cap the exposure to an industry or country  drawback - it limits business volume  Risk control through Sensitivities  e.g. use derivative financial instruments to alter sensitivities  for market risk, hedging exposures help to keep the various sensitivities (the ‘Greeks’) within stated limits

Sensitivities and derivative calculus  Sensitivity is the first derivative of the value (V) with respect to m (parameter)  First derivative measures the rate with which the value changes with changes in an underlying factor  The next order derivative (second derivative) takes care of the change in the first derivative (sensitivity)  Second derivative measures how sensitive is the first derivative measure to changes in the underlying risk factor

Differentiation - A Reminder  Differentiation measures the rate of change  for a function: y = k x n  dy/dx = nk x n-1  e.g. if y=10x 2, then dy/dx = 20x  the original function is not constant, so if e.g. x=2, dy/dx = 10x = 20  for a linear function there is no advantage in using the derivative approach - inspect the equation parameters (e.g. if y=10x, then dy/dx = 10, i.e. is constant)  When looking at a graph plot of y against x, the change can be defined in terms of slope

Derivative Measures of Risk Underlying Risk Factor Value of Financial Obligation a b When looking at a graph plot of y against x, the change can be defined in terms of slope

The Second Derivative - a Reminder  In case of quadratic function (e.g. y=-5x 2 ) we can recognise whether the turning point derived represents a maximum or a minimum.  If we develop business models using higher power expressions we may not be able to do so without looking at a graph  To do this numerically, we need to use the second derivative (the same differentiation rules apply)  If the second derivative is negative - maximum  If the second derivative is positive - minimum

The Second Derivative - a Reminder  E.g. a profit function: Π = x - 5x 2  Its first derivative: d Π /dx = x  The second derivative: d 2 Π /dx 2 = - 10  Since the second derivative is constant and negative - therefore the turning point is a maximum

Partial Differentiation - a Reminder  In case of partial differentiation we only differentiate with respect to one independent variable. Other variables are held constant  For example: if z=2x+3y  Its partial derivative (by x): δ z/ δ x = 2 (since y is constant, 3y is constant - the derivative of a constant is 0)  Its partial derivative (by y): δ z/ δ y = 3

Issues in Relation to Calculus Based Measures  Need to Specify Mathematical Relationship so require a Pricing Model - Bond Valuation, Option Pricing Models  Thus difficult to apply to complicated portfolios of obligations  Applies to Localised Measurement of Risk  An approximation of the function

Sensitivity based measures of risk: Tools and Application

Risk Management Tools  Interest Rate Risk Management:  Gap analysis  Duration  Convexity

Re-pricing Model for Banks  Repricing or funding gap model based on book value.  Contrasts with market value-based maturity and duration models recommended by the Bank for International Settlements (BIS).  Rate sensitivity means time to re-pricing.  Re-pricing gap is the difference between the rate sensitivity of each asset and the rate sensitivity of each liability: RSA - RSL.

Maturity Buckets  Commercial banks must report repricing gaps for assets and liabilities with maturities of:  One day.  More than one day to three months.  More than 3 three months to six months.  More than six months to twelve months.  More than one year to five years.  Over five years.

Repricing Gap Example AssetsLiabilities GapCum. Gap 1-day $ 20 $ 30 $-10 $-10 >1day-3mos >3mos.-6mos >6mos.-12mos >1yr.-5yrs >5 years

Applying the Repricing Model   NII i = (GAP i )  R i = (RSA i - RSL i )  r i Example: In the one day bucket, gap is -$10 million. If rates rise by 1%,  NII i = (-$10 million) ×.01 = -$100,000.

Applying the Repricing Model  Example II: If we consider the cumulative 1-year gap,   NII i = (CGAP i )  R i = (-$15 million)(.01) = -$150,000.

CGAP Ratio  May be useful to express CGAP in ratio form as, CGAP/Assets.  Provides direction of exposure and  Scale of the exposure.  Example:  CGAP/A = $15 million / $270 million = 0.56, or 5.6 percent.

Equal Changes in Rates on RSAs & RSLs  Example: Suppose rates rise 2% for RSAs and RSLs. Expected annual change in NII,  NII = CGAP ×  R = $15 million ×.01 = $150,000 With positive CGAP, rates and NII move in the same direction.

Unequal Changes in Rates  If changes in rates on RSAs and RSLs are not equal, the spread changes. In this case,  NII = (RSA ×  R RSA ) - (RSL ×  R RSL )

Unequal Rate Change Example  Spread effect example: RSA rate rises by 1.2% and RSL rate rises by 1.0%  NII =  interest revenue -  interest expense = ($155 million × 1.2%) - ($155 million × 1.0%) = $310,000

Restructuring Assets and Liabilities  The FI can restructure its assets and liabilities, on or off the balance sheet, to benefit from projected interest rate changes.  Positive gap: increase in rates increases NII  Negative gap: decrease in rates increases NII

Weaknesses of Repricing Model  Weaknesses:  Ignores market value effects and off-balance sheet cash flows  Overaggregative Distribution of assets & liabilities within individual buckets is not considered. Mismatches within buckets can be substantial.  Ignores effects of runoffs Bank continuously originates and retires consumer and mortgage loans. Runoffs may be rate-sensitive.

The Maturity Model  Explicitly incorporates market value effects.  For fixed-income assets and liabilities:  Rise (fall) in interest rates leads to fall (rise) in market price.  The longer the maturity, the greater the effect of interest rate changes on market price.  Fall in value of longer-term securities increases at diminishing rate for given increase in interest rates.

Maturity of Portfolio  Maturity of portfolio of assets (liabilities) equals weighted average of maturities of individual components of the portfolio.  Principles stated on previous slide apply to portfolio as well as to individual assets or liabilities.  Typically, M A - M L > 0 for most banks

Effects of Interest Rate Changes  Size of the gap determines the size of interest rate change that would drive net worth to zero.  Immunization and effect of setting M A - M L = 0.

Maturity Matching and Interest Rate Exposure  If M A - M L = 0, is the FI immunized?  Extreme example: Suppose liabilities consist of 1-year zero coupon bond with face value $100. Assets consist of 1-year loan, which pays back $99.99 shortly after origination, and 1¢ at the end of the year. Both have maturities of 1 year.  Not immunized, although maturities are equal.  Reason: Differences in duration.

*Term Structure of Interest Rates YTM Time to Maturity YTM

*Unbiased Expectations Theory  Yield curve reflects market’s expectations of future short-term rates.  Long-term rates are geometric average of current and expected short-term rates. _ _ ~ ~ R N = [(1+R 1 )(1+E(r 2 ))…(1+E(r N ))] 1/N - 1

*Liquidity Premium Theory  Allows for future uncertainty.  Premium required to hold long-term. *Market Segmentation Theory  Investors have specific needs in terms of maturity.  Yield curve reflects intersection of demand and supply of individual maturities.

*Market Segmentation Theory  Investors have specific needs in terms of maturity.  Yield curve reflects intersection of demand and supply of individual maturities.

Price Sensitivity and Maturity  In general, the longer the term to maturity, the greater the sensitivity to interest rate changes.  Example: Suppose the zero coupon yield curve is flat at 12%. Bond A pays $ in five years. Bond B pays $ in ten years, and both are currently priced at $1000.  Bond A: P = $1000 = $ /(1.12) 5  Bond B: P = $1000 = $ /(1.12) 10

Example continued...  Now suppose the interest rate increases by 1%.  Bond A: P = $ /(1.13) 5 = $  Bond B: P = $ /(1.13) 10 = $  The longer maturity bond has the greater drop in price because the payment is discounted a greater number of times.

Coupon Effect  Bonds with identical maturities will respond differently to interest rate changes when the coupons differ. This is more readily understood by recognizing that coupon bonds consist of a bundle of “zero-coupon” bonds. With higher coupons, more of the bond’s value is generated by cash flows which take place sooner in time. Consequently, less sensitive to changes in R.

Price Sensitivity of 6% Coupon Bond

Price Sensitivity of 8% Coupon Bond

Remarks on Preceding Slides  The longer maturity bonds experience greater price changes in response to any change in the discount rate.  The range of prices is greater when the coupon is lower.  The 6% bond shows greater changes in price in response to a 2% change than the 8% bond. The first bond is has greater interest rate risk.

Duration  The average life of an asset or liability  The weighted-average time to maturity using present value of the cash flows, relative to the total present value of the asset or liability as weights.

Duration  Duration  Weighted average time to maturity using the relative present values of the cash flows as weights.  Combines the effects of differences in coupon rates and differences in maturity.  Based on elasticity of bond price with respect to interest rate.

Duration  Duration D =  n t=1 [C t t/(1+r) t ]/  n t=1 [C t /(1+r) t ] Where D = duration t = number of periods in the future C t = cash flow to be delivered in t periods n= term-to-maturity & r = yield to maturity (per period basis).

Duration  Since the price of the bond must equal the present value of all its cash flows, we can state the duration formula another way: D =  n t=1 [t  (Present Value of C t /Price)]  Notice that the weights correspond to the relative present values of the cash flows.

Duration of Zero-coupon Bond  For a zero coupon bond, duration equals maturity since 100% of its present value is generated by the payment of the face value, at maturity.  For all other bonds:  duration < maturity

Computing duration  Consider a 2-year, 8% coupon bond, with a face value of $1,000 and yield-to-maturity of 12%. Coupons are paid semi-annually.  Therefore, each coupon payment is $40 and the per period YTM is (1/2) × 12% = 6%.  Present value of each cash flow equals CF t ÷ ( ) t where t is the period number.

Duration of 2-year, 8% bond: Face value = $1,000, YTM = 12%

Special Case  Maturity of a consol: M = .  Duration of a consol: D = 1 + 1/R

Duration Gap  Suppose the bond in the previous example is the only loan asset (L) of an FI, funded by a 2-year certificate of deposit (D).  Maturity gap: M L - M D = 2 -2 = 0  Duration Gap: D L - D D = =  Deposit has greater interest rate sensitivity than the loan, so DGAP is negative.  FI exposed to rising interest rates.

Features of Duration  Duration and maturity:  D increases with M, but at a decreasing rate.  Duration and yield-to-maturity:  D decreases as yield increases.  Duration and coupon interest:  D decreases as coupon increases

Economic Interpretation  Duration is a measure of interest rate sensitivity or elasticity of a liability or asset: [dP/P]  [dR/(1+R)] = -D Or equivalently, dP/P = -D[dR/(1+R)] = -MD × dR where MD is modified duration.

Economic Interpretation  To estimate the change in price, we can rewrite this as: dP = -D[dR/(1+R)]P = -(MD) × (dR) × (P)  Note the direct linear relationship between dP and -D.

Immunizing the Balance Sheet of an FI  Duration Gap:  From the balance sheet, E=A-L. Therefore,  E=  A-  L. In the same manner used to determine the change in bond prices, we can find the change in value of equity using duration.   E = [-D A A + D L L]  R/(1+R) or   D A - D L k]A(  R/(1+R))

Duration and Immunizing  The formula shows 3 effects:  Leverage adjusted D-Gap  The size of the FI  The size of the interest rate shock

An example:  Suppose D A = 5 years, D L = 3 years and rates are expected to rise from 10% to 11%. (Rates change by 1%). Also, A = 100, L = 90 and E = 10. Find change in E.   D A - D L k]A[  R/(1+R)] = -[5 - 3(90/100)]100[.01/1.1] = - $2.09.  Methods of immunizing balance sheet. Adjust D A, D L or k.

Limitations of Duration  Immunizing the entire balance sheet need not be costly. Duration can be employed in combination with hedge positions to immunize.  Immunization is a dynamic process since duration depends on instantaneous R.  Large interest rate change effects not accurately captured. Convexity  More complex if nonparallel shift in yield curve.

Convexity  The duration measure is a linear approximation of a non-linear function. If there are large changes in R, the approximation is much less accurate. All fixed-income securities are convex. Convexity is desirable, but greater convexity causes larger errors in the duration-based estimate of price changes.

Convexity  Recall that duration involves only the first derivative of the price function. We can improve on the estimate using a Taylor expansion. In practice, the expansion rarely goes beyond second order (using the second derivative).

*Modified duration   P/P = -D[  R/(1+R)] + (1/2) CX (  R) 2 or  P/P = -MD  R + (1/2) CX (  R) 2  Where MD implies modified duration and CX is a measure of the curvature effect. CX = Scaling factor × [capital loss from 1bp rise in yield + capital gain from 1bp fall in yield]  Commonly used scaling factor is 10 8.

*Calculation of CX  Example: convexity of 8% coupon, 8% yield, six- year maturity Eurobond priced at $1,000. CX = 10 8 [  P - /P +  P + /P] = 10 8 [( ,000)/1,000 + (1, ,000)/1,000)] = 28

Duration Measure: Other Issues  Default risk  Floating-rate loans and bonds  Duration of demand deposits and passbook savings  Mortgage-backed securities and mortgages  Duration relationship affected by call or prepayment provisions.

Contingent Claims  Interest rate changes also affect value of off- balance sheet claims.  Duration gap hedging strategy must include the effects on off-balance sheet items such as futures, options, swaps, caps, and other contingent claims.

More on Sensitivity based measures  Changing the sensitivities to risk factors or keeping the sensitivities within the stated limits by use of derivative financial instruments  Forwards, Futures, Swaps, Options

General idea of hedging Need to look for hedge that has opposite characteristic to underlying price risk Change in price Change in value Underlying risk Hedge position

O C = P s [N(d 1 )] - S[N(d 2 )]e -rt O C - Call Option Price P s - Stock Price N(d 1 ) - Cumulative normal density function of (d 1 ) S - Strike or Exercise price N(d 2 ) - Cumulative normal density function of (d 2 ) r - discount rate (90 day comm paper rate or risk free rate) t - time to maturity of option (as % of year) v - volatility - annualized standard deviation of daily returns Black-Scholes Option Pricing Model

Value of Option O C = f(P,S,v(P),t,r)

More on Sensitivity  The Greek Letters:  Delta ( Δ )  Gamma ( Γ )  Theta ( Θ )  Vega (V)*  Rho (P)  Each Greek letter measures a different dimension of risk in an option position  The aim to manage the Greeks so that all risks are acceptable *Not a Greek letter but considered one of the ‘Greeks’

The Greeks - Risk Measures for Options  Delta: Partial Derivative of the call price (Oc) with respect to underlying asset price (Ps)  Rate of change of the option price with respect to the price of the underlying asset  Gamma: 2nd Partial Derivative of Oc with respect to Ps  Rate of change of the portfolios delta with respect to the price of the underlying asset  Theta: Partial Derivative of Oc with respect to t  Rate of change of the portfolio value with respect to the passage of time when all else remains the same  Vega: Partial Derivative of Oc with respect to v(P)  Rate of change of the value of the portfolio with respect to the volatility of the underlying asset ( σ )  Rho: Partial Derivative of Oc with respect to r  Rate of change of the portfolio value with respect of the interest rate