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) = (ΔV/V) x ΔmS (value) = (ΔV/V) x V x Δms (% change of value) = (ΔV/V) x (Δm/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=10x2, 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

UnderlyingRisk Factor

Value of FinancialObligation

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=-5x2) 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= -100 +100x - 5x2

Its first derivative: d Π/dx = 100 - 10x

The second derivative: d 2Π/dx2 = - 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):

δz/δx = 2 (since y is constant, 3y is constant - the derivative

of a constant is 0)

Its partial derivative (by y): δz/δy = 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

Assets Liabilities Gap Cum. Gap1-day $ 20 $ 30 $-10 $-10>1day-3mos. 30 40 -10 -20>3mos.-6mos. 70 85 -15 -35>6mos.-12mos. 90 70 +20 -15>1yr.-5yrs. 40 30 +10 -5>5 years 10 5 +5 0

Applying the Repricing Model

NIIi = (GAPi) Ri = (RSAi - RSLi) ri

Example: In the one day bucket, gap is -$10

million. If rates rise by 1%,

NIIi = (-$10 million) × .01 = -$100,000.

Applying the Repricing Model Example II: If we consider the cumulative 1-year gap,

NIIi = (CGAPi) Ri = (-$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 × RRSA ) - (RSL × RRSL )

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, MA - ML > 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 MA - ML = 0.

Maturity Matching and Interest Rate Exposure If MA - ML = 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.

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 $1762.34 in five years. Bond B pays $3105.85 in ten years, and both are currently priced at $1000. Bond A: P = $1000 = $1762.34/(1.12)5 Bond B: P = $1000 = $3105.84/(1.12)10

Example continued...

Now suppose the interest rate increases by 1%. Bond A: P = $1762.34/(1.13)5 = $956.53 Bond B: P = $3105.84/(1.13)10 = $914.94

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

r 8% 6% 4% Range

n

40 $802 $1,000 $1,273 $471

20 $864 $1,000 $1,163 $299

10 $919 $1,000 $1,089 $170

2 $981 $1,000 $1,019 $37

Price Sensitivity of 8% Coupon Bond

r 10% 8% 6% Range

n

40 $828 $1,000 $1,231 $403

20 $875 $1,000 $1,149 $274

10 $923 $1,000 $1,085 $162

2 $981 $1,000 $1,019 $38

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

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 = nt=1[Ct• t/(1+r)t]/ n

t=1 [Ct/(1+r)t]

WhereD = durationt = number of periods in the future

Ct = 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 = nt=1[t (Present Value of

Ct/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 CFt ÷ (1+ 0.06)t where t is the period number.

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

t years CFt PV(CFt) Weight(x)

x × years

1 0.5 40 37.736 0.041 0.020

2 1.0 40 35.600 0.038 0.038

3 1.5 40 33.585 0.036 0.054

4 2.0 1,040 823.777 0.885 1.770

P = 930.698 1.000 D=1.883(years)

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: ML - MD = 2 -2 = 0 Duration Gap: DL - DD = 1.885 - 2.0 = -

0.115 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 × dRwhere 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 = [-DAA + DLL] R/(1+R) or

DA - DLk]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 DA = 5 years, DL = 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.

DA - DLk]A[R/(1+R)]

= -[5 - 3(90/100)]100[.01/1.1] = - $2.09. Methods of immunizing balance sheet.

Adjust DA , DL 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 108.

*Calculation of CX

Example: convexity of 8% coupon, 8% yield, six-year maturity Eurobond priced at $1,000.

CX = 108[P-/P + P+/P] = 108[(999.53785-1,000)/1,000 +

(1,000.46243-1,000)/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

OC = Ps[N(d1)] - S[N(d2)]e-rt

OC- Call Option Price

Ps - Stock Price

N(d1) - Cumulative normal density function of (d1)

S - Strike or Exercise price

N(d2) - Cumulative normal density function of (d2)

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 ModelBlack-Scholes Option Pricing Model

Value of Option

OC = 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