FRM Formulas

104
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Transcript of FRM Formulas

Page 1: FRM Formulas

FMP

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Page 2: FRM Formulas

Hedging in a practical world (Basis Risk)

Basis = spot price of asset – futures price contract• Basis = 0 when spot price = futures price

FuturePrice

Spot Price

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Time

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Choice of contracts

• Optimal Hedge Ratio:

Where• σS is the standard deviation of δS, the change in the spot price during the hedging period

• σF is the standard deviation of δF, the change in the futures price during the hedging period

• ρ is the coefficient of correlation between δS and δF

F

h

S

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Optimal number of contracts

The optimal number of contracts (N*) to hedge a portfolio consisting of NA number of units andwhere Qf is the total number of futures being used for hedging

In order to change the beta (β) of the portfolio to (β*), we need to long or short the (N*) numberof contracts depending on the sign of (N*)

APβ*N

f

A

QN*h*N

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APβ*N

AP)-*(*N

Negative sign of (N*) indicates shorting the contracts

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Determination of Forward Price

The price of a forwards contract is given by the equation below:• F0 = S0ert in the case of continuously compounded risk free interest rate, r

• F0 = S0(1+r )t in the case of annual risk free interest rate, r

• Where:– F0: forward price– S0: Spot price– t: time of the contract

Known income from underlying• If the underlying asset on which the forward contract is entered into provides an income with a present

value, I, then the forward contract would be valued as:– F0 = (S0 – I )ert

Known yield from underlying• If the underlying asset on which the forward contract is entered into provides a continuously compounded

yield, q, then the forward contract would be valued as:– F0 = S0e(r-q)t

q: continuously % of return on the asset divided by the total asset price

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The price of a forwards contract is given by the equation below:• F0 = S0ert in the case of continuously compounded risk free interest rate, r

• F0 = S0(1+r )t in the case of annual risk free interest rate, r

• Where:– F0: forward price– S0: Spot price– t: time of the contract

Known income from underlying• If the underlying asset on which the forward contract is entered into provides an income with a present

value, I, then the forward contract would be valued as:– F0 = (S0 – I )ert

Known yield from underlying• If the underlying asset on which the forward contract is entered into provides a continuously compounded

yield, q, then the forward contract would be valued as:– F0 = S0e(r-q)t

q: continuously % of return on the asset divided by the total asset price

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Value of forward contracts

At the time on entering into a forward contract, long or short, the value of the forward is zero

This is because the delivery price (K) of the asset and the forward price today (F0) remains the same

The value of the forward is basically the present value of the difference in the delivery price and the forward price

Value of a long forward, f, is given by the PV of the pay off at time T:• ƒ = (F0 – K )e–rT

K is fixed in the contract, while F0 keeps changing on an everyday basis

For continuous dividend yielding underlying• f = S0e-qt – Ke-rt

For discrete dividend paying stock• f = S0 – I – Ke-rt

Index futures: A stock index can be considered as an asset that pays dividends and the dividends paid are thedividends from the underlying stocks in the index

If q is the dividend yield rate then the futures price is given as:• F0= S0e(r-q)t

Index Arbitrage• When F0 > S0e(r-q)T an arbitrageur buys the stocks underlying the index and sells futures

• When F0 < S0e(r-q)T an arbitrageur buys futures and shorts or sells the stocks underlying the index

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At the time on entering into a forward contract, long or short, the value of the forward is zero

This is because the delivery price (K) of the asset and the forward price today (F0) remains the same

The value of the forward is basically the present value of the difference in the delivery price and the forward price

Value of a long forward, f, is given by the PV of the pay off at time T:• ƒ = (F0 – K )e–rT

K is fixed in the contract, while F0 keeps changing on an everyday basis

For continuous dividend yielding underlying• f = S0e-qt – Ke-rt

For discrete dividend paying stock• f = S0 – I – Ke-rt

Index futures: A stock index can be considered as an asset that pays dividends and the dividends paid are thedividends from the underlying stocks in the index

If q is the dividend yield rate then the futures price is given as:• F0= S0e(r-q)t

Index Arbitrage• When F0 > S0e(r-q)T an arbitrageur buys the stocks underlying the index and sells futures

• When F0 < S0e(r-q)T an arbitrageur buys futures and shorts or sells the stocks underlying the index

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Futures and Forwards on Currencies

Interest rate Parity

Formula to remember:• If Spot rate is given in USD/INR terms then take American Risk-free rate as the first rate• In other words, individual who is interested in USD/INR rates would be an American (Indian will

always think in Rupees not dollars!!!!!), which implies foreign currency (rf) in his case would be rINR

Trr fcbceSF )(00

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Interest rate Parity

Formula to remember:• If Spot rate is given in USD/INR terms then take American Risk-free rate as the first rate• In other words, individual who is interested in USD/INR rates would be an American (Indian will

always think in Rupees not dollars!!!!!), which implies foreign currency (rf) in his case would be rINR

Trr

INRUSD

INRUSD

INRUSDeSF )(

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The Cost of Carry

The cost of carry, c, is the storage cost plus the interest costs less the income earned

For an investment asset F0 = S0ecT

For a consumption asset F0 ≤ S0ecT

The convenience yield on the consumption asset, y, is defined so that: F0 = S0 e(c–y )T

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The cost of carry, c, is the storage cost plus the interest costs less the income earned

For an investment asset F0 = S0ecT

For a consumption asset F0 ≤ S0ecT

The convenience yield on the consumption asset, y, is defined so that: F0 = S0 e(c–y )T

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Calculation of interest rates

Amount compounded annually would be given by:• A = P (1+ r)t

– A terminal amount– P principal amount– r annual rate of interest– t number of years for which the principal is invested

If amount compounded n times a year then:• A = P ( 1+ r/n )nt

When n∞ then we call it continuous compounding:• A = Pert (this formula is derived using limits and continuity)

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Amount compounded annually would be given by:• A = P (1+ r)t

– A terminal amount– P principal amount– r annual rate of interest– t number of years for which the principal is invested

If amount compounded n times a year then:• A = P ( 1+ r/n )nt

When n∞ then we call it continuous compounding:• A = Pert (this formula is derived using limits and continuity)

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Bond pricing

The price of a bond is the present value of all the coupon payment and the final principal payment received at the endof its life

• B the bond price

• C coupon payment

• r zero interest rate at time t

• P bond principal

• T time to maturity

The yield of a bond is the discount rate (applied to all future cash flows) at which the present value of the bond isequal to its market price• Yield to Maturity = Investor’s Required Rate of Return

The par yield is the coupon rate at which the present value of the cash flows equal to the par value (principal value) ofthe bond

If we are looking at a semi-annual 5 year coupon bond with a par value of $100 then the coupon payment would besolved using the following equation:

YTM)(11F

YTMYTM)(111

IB n

n

T

t

rTrt PeCeB1

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The price of a bond is the present value of all the coupon payment and the final principal payment received at the endof its life

• B the bond price

• C coupon payment

• r zero interest rate at time t

• P bond principal

• T time to maturity

The yield of a bond is the discount rate (applied to all future cash flows) at which the present value of the bond isequal to its market price• Yield to Maturity = Investor’s Required Rate of Return

The par yield is the coupon rate at which the present value of the cash flows equal to the par value (principal value) ofthe bond

If we are looking at a semi-annual 5 year coupon bond with a par value of $100 then the coupon payment would besolved using the following equation:

9

5

1

5100)2/(100t

rrt eeC

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Forward rate agreements (FRAs)

In general:

Payment to the long at settlement = Notional Principal X (Rate at settlement – FRA Rate) (days/360)

----------------------------------------------------------1 + (Rate at settlement) (days / 360)

12

1122t2t1, TT

TRTRF

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Duration

Macaulay’s duration: is the weighted average of the times when the payments are made. And theweights are a ratio of the coupon paid at time t to the present bond price

Where:• t = Respective time period

• C = Periodic coupon payment

• y = Periodic yield

• n = Total no of periods

• M = Maturity value

pricebondCurrentyMn

yCt

DurationMacaluayn

n

tt )1(

*)1(

*1

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Macaulay’s duration: is the weighted average of the times when the payments are made. And theweights are a ratio of the coupon paid at time t to the present bond price

Where:• t = Respective time period

• C = Periodic coupon payment

• y = Periodic yield

• n = Total no of periods

• M = Maturity value

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Duration contd…

A bond’s interest rate risk is affected by:

• Yield to maturity• Term to maturity• Size of coupon

From Macaulay’s equation we get a key relationship:

In the case of a continuously compounded yield the duration used is modified duration given as:

YDBB

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A bond’s interest rate risk is affected by:

• Yield to maturity• Term to maturity• Size of coupon

From Macaulay’s equation we get a key relationship:

In the case of a continuously compounded yield the duration used is modified duration given as:

nr1

DurationMacaulayD*

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Convexity

Convexity is a measure of the curvature of the price / yield relationship

2

2

dyBd

B1C

Note that this is the second partial derivative of the bond valuation equation w.r.t. the yield

Hence, convexity is the rate of change of duration with respect to the change in yield

Bond price ($)

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YieldY*

P* Actual bond price

Tangent

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…Convexity

The convexity of the price / YTM graph reveals two important insights:• The price rise due to a fall in YTM is greater than the price decline due to a rise in YTM, given an

identical change in the YTM• For a given change in YTM, bond prices will change more when interest rates are low than when they

are high

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Page 16: FRM Formulas

Calculating Bond Price Changes

We can approximate the change in a bond’s price for a given change in yield by usingduration and convexity:

V D i V C V iB M od B B 0 5 2.

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Theories of the Term Structure Three theories are used to explain the

shape of the term structure

Expectations theory

The long rate is the geometric mean ofexpected future short interest rates

Liquidity preference theory

Investors must be paid a “liquiditypremium” to hold less liquid, long-termdebt

Market segmentation theory

Investors decide in advance whether theywant to invest in short term or the longterm

Distinct markets exist for securities ofshort term bonds and long term bonds

Supply demand conditions decide theprices

Where rpn is the risk premium associatedwith an n year bond

)1)...(1)(1()1( 21 yearnst

yearst

yearst

nlt iiii

)1)...(1)(1()1( 21 yearnst

yearst

yearstn

nlt iiirpi

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Three theories are used to explain theshape of the term structure

Expectations theory

The long rate is the geometric mean ofexpected future short interest rates

Liquidity preference theory

Investors must be paid a “liquiditypremium” to hold less liquid, long-termdebt

Market segmentation theory

Investors decide in advance whether theywant to invest in short term or the longterm

Distinct markets exist for securities ofshort term bonds and long term bonds

Supply demand conditions decide theprices

Where rpn is the risk premium associatedwith an n year bond

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Day count conventions

Day count defines the way in which interest is accrued over time. Day count conventions normallyused in US are:• Actual / actual treasury bonds

• 30 / 360 corporate bonds

• Actual/360money market instruments

The interest earned between two dates

(Number of days between dates)*(Interest earned in reference period)

(Number of days in reference period)=

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(Number of days in reference period)=

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Cheapest to deliver bond

The party with the short position can chose to deliver the cheapest bond when it comes todelivery, hence he would chose the cheapest to deliver bond

Net pay out for delivery ( he has to buy a bond and deliver it):• Quoted bond price – (settlement price * conversion factor)

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DV01 – Application to hedging

Hedge ratio is calculated using DV01 with the help of following relation

)instrumenthedgingof100$(01)ositioninitialof100$(1

perDVpperDVO

HR

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Duration based hedging strategies

Number of contracts to hedge is given by the equation:

• FC Contract price for interest rate futures• DF Duration of asset underlying futures at maturity• P Value of portfolio being hedged• DP Duration of portfolio at hedge maturity

FC

P

DFPDN *

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Number of contracts to hedge is given by the equation:

• FC Contract price for interest rate futures• DF Duration of asset underlying futures at maturity• P Value of portfolio being hedged• DP Duration of portfolio at hedge maturity

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Key Rate ‘01 and Key Rate Durations

Key Rate ‘01 measures the dollar change in the value of the bond for every basis point shiftin the key rate• Key Rate ‘01 = (-1/10,000) * (Change in Bond Value/0.01%)

Key rate duration provides the approximate percentage change in the value of the bond• Key Rate Duration = (-1/BV) * (Change in Bond Value/Change in Key rate)

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Put Call parity

Expressed as:• Value of call + Present value of strike price = value of put + share price

Put-call parity relationship, assumes that the options are not exercised before expiration day, i.e. itfollows European options

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Page 24: FRM Formulas

Bounds and Option Values

Option Minimum Value Maximum Value

European call (c) ct ≥ Max(0,St-(X/(1+RFR)t) St

American Call (C) Ct ≥ Max(0, St-(X/(1+RFR)t) St

European put (p) pt ≥Max(0,(X/(1+RFR)t)-St) X/(1+RFR)t

American put (P) Pt ≥ Max(0, (X-St)) X

Where t is the time to expiration

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Where t is the time to expiration

Page 25: FRM Formulas

Binomial Method

• Assuming the price of the underlying asset can take only two values in any given interval of time– Risk Neutral Method

S0

Su

Su2

Sud

IV1 = Max[(Su2-X), 0]

IV2

p

p

1 - p

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S0Sud

Su

Sd2

IV2

IV3

1 - p

1 - p

p

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Black and Scholes Model

Black and Scholes formula allows for infinitesimally small intervals as well as the need to reviseleverage for European options on Non Dividend paying stocks

The formula is:

• Where,

Log is the natural log with base e• N (d) = cumulative normal probability density function• X = exercise price option;• T = number of periods to exercise date• P =present price of stock• σ = standard deviation per period of (continuously compounded) rate of return on stock

Value of Put =

TddT

TRXP

df

12

)]5.0([]ln[1

2

])2([])1([ TR feXdNPdN

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Black and Scholes formula allows for infinitesimally small intervals as well as the need to reviseleverage for European options on Non Dividend paying stocks

The formula is:

• Where,

Log is the natural log with base e• N (d) = cumulative normal probability density function• X = exercise price option;• T = number of periods to exercise date• P =present price of stock• σ = standard deviation per period of (continuously compounded) rate of return on stock

Value of Put =

25

TddT

TRXP

df

12

)]5.0([]ln[1

2

])}1(1[{}]2(1{[ PdNdNeX TR f

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Delta (cont.) The delta of a portfolio of derivatives (such as options) with the same underlying asset, can

be found out if the deltas of each of these derivatives are known

i

n

iiportfolio W

1

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Page 28: FRM Formulas

Theta (cont.)

We have theta of call given by:

• Where:

For a put option, theta is given by:

Where:• S0 = Stock price at time 0, i.e. present price of the

stock

• d1 and d2 are as defined in the Black-ScholesPricing formula earlier

• σ = Stock price volatility

• K = Strike price

• T = Time of maturity of the option measured inyears, so that 6 months will be 0.5 years

• r = Risk neutral rate of interest

)(2

)(')( 210 dNrKe

TdNSCall rT

2)('

2/)2^( xexN

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We have theta of call given by:

• Where:

For a put option, theta is given by:

Where:• S0 = Stock price at time 0, i.e. present price of the

stock

• d1 and d2 are as defined in the Black-ScholesPricing formula earlier

• σ = Stock price volatility

• K = Strike price

• T = Time of maturity of the option measured inyears, so that 6 months will be 0.5 years

• r = Risk neutral rate of interest

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)(2

)(')( 210 dNrKe

TdNSPut rT

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Gamma (cont.)

Calculation of Gamma• Gamma for European options can be calculated using the following formula:

• Where symbols have their usual meaning

TSdN

0

)1('

Gamma (ATM) vs. Time0.45

Gamma (Call / Put)0.07

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00.05

0 0.2 0.4 0.6 0.8 1.0 1.2

0.100.150.200.250.300.350.400.45

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49

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Vega

The Vega of a derivative portfolio is the rate of change of the value of the portfolio with the change inthe volatility of the underlying assets. It can be expressed as:• V= , where Π is the value of the portfolio, and σ is the volatility in the price of the underlying.

For European options on a stock that does not pay dividends, Vega can be found by:• V=S0

by:

The Vega of a long position is always positive

A position in the underlying asset has a zero Vega

Thus its behavior is similar to gamma

Vega is maximum for options that are at the money

2)1('

2/)2^1( dedN

16 Vega

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The Vega of a derivative portfolio is the rate of change of the value of the portfolio with the change inthe volatility of the underlying assets. It can be expressed as:• V= , where Π is the value of the portfolio, and σ is the volatility in the price of the underlying.

For European options on a stock that does not pay dividends, Vega can be found by:• V=S0

by:

The Vega of a long position is always positive

A position in the underlying asset has a zero Vega

Thus its behavior is similar to gamma

Vega is maximum for options that are at the money

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2)1('

2/)2^1( dedN

1 4 7 10131619222528313437404346490

468

10121416

2

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Rho

Rho of a portfolio of options is the rate of change of its value with respect to changes in theinterest rate

Rho = , where Π is the value of the portfolio, and r is the rate of interest

For European options on non dividend paying stocks, we have;• Rho (call) = KTe-rTN(d2), where the symbols carry their usual meanings

• Also, Rho (put) = -KTe-rTN(-d2), the symbols carrying their usual meanings

r

30Rho (Call / Put)

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1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49-30

-10

0

10

20

30

-20

Rho (Call)

Rho (Put)

Rho (Call / Put)

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Valuation of swaps

Hence the value of the swap can be given as:• V = Bfix – Bfl

• Where:– Bfix = PV of payments– Bfl = (P+AI)e-rt

• Value of a floating bond is equal to the par value at coupon reset dates and equals to the PresentValue of Par values (P) and Accrued Interest (AI)

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Commodity Forwards

Commodity forward prices can be described using the same formula as used for financialforward prices

For financial assets, is the dividend yield

• For commodities, is the commodity lease rate• The lease rate is the return that makes an investor willing to buy and lend a commodity• Some commodities (metals) have an active leasing market• Lease rates can typically only be estimated by observing forward prices

F 0 , T S 0 e ( r ) T

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Commodity forward prices can be described using the same formula as used for financialforward prices

For financial assets, is the dividend yield

• For commodities, is the commodity lease rate• The lease rate is the return that makes an investor willing to buy and lend a commodity• Some commodities (metals) have an active leasing market• Lease rates can typically only be estimated by observing forward prices

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Futures term structure

The set of prices for different expiration dates for a given commodity is called the forwardcurve (or the forward strip) for that date

If on a given date the forward curve is upward-sloping, then the market is in contango

If the forward curve is downward sloping, the market is in backwardation

Note that forward curves can have portions in backwardation and portions in contango

• Since r is always positive, assets with =0 display upward sloping (contango) futures term structure

• With >0, term structures could be upward or downward sloping

F0,T S0e(r )T

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The set of prices for different expiration dates for a given commodity is called the forwardcurve (or the forward strip) for that date

If on a given date the forward curve is upward-sloping, then the market is in contango

If the forward curve is downward sloping, the market is in backwardation

Note that forward curves can have portions in backwardation and portions in contango

• Since r is always positive, assets with =0 display upward sloping (contango) futures term structure

• With >0, term structures could be upward or downward sloping

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F0,T S0e(r )T

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Commodity loan

With the addition of the lease payment, NPV of loaning the commodity is 0

The lease payment is like the dividend payment that has to be paid by the person whoborroweda stock

Therefore:

Where δ is lease rate

F0 ,T S0e( r )T

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With the addition of the lease payment, NPV of loaning the commodity is 0

The lease payment is like the dividend payment that has to be paid by the person whoborroweda stock

Therefore:

Where δ is lease rate

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Forward Prices and the Lease Rate

The lease rate has to be consistent with the forward price

Therefore, when we observe the forward price, we can infer what the lease rate would haveto be if a lease market existed

The annualized lease rate

The effective annual lease rate

l r 1TIn (F0 ,T / S )

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l r 1TIn (F0 ,T / S )

l (1 r )

(F0 ,T / S )1/T 1

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Storage Costs and Forward Prices

One will only store a commodity if the PV of selling it at time T is at least as great as that ofselling it today

Whether a commodity is stored is peculiar to each commodity

If storage is to occur, the forward price is at least

Where (0,T) is the future value of storage costs for one unit of the commodity from time 0to T

F0 ,T S0erT (0,T )

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F0 ,T S0erT (0,T )

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Storage Costs and Forward Prices (cont’d)

Convenience Yield• Some holders of a commodity receive benefits from physical ownership (e.g., a commercial user)• This benefit is called the commodity’s convenience yield• The convenience yield creates different returns to ownership for different investors, and may or may

not be reflected in the forward price

Convenience and leasing• If someone lends the commodity they save storage costs, but lose the ‘convenience’

– Stated as ( –c)• Therefore, commodity borrower pays a lease rate that covers the lost convenience less the storage

costs:

– = c –

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Convenience Yield• Some holders of a commodity receive benefits from physical ownership (e.g., a commercial user)• This benefit is called the commodity’s convenience yield• The convenience yield creates different returns to ownership for different investors, and may or may

not be reflected in the forward price

Convenience and leasing• If someone lends the commodity they save storage costs, but lose the ‘convenience’

– Stated as ( –c)• Therefore, commodity borrower pays a lease rate that covers the lost convenience less the storage

costs:

– = c –

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Pricing with convenience

So, if:

And if, = c –

Then, F0,T = S0e(r+ -c)T

F0 ,T S0e( r )T

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Page 40: FRM Formulas

No-Arbitrage with Convenience

From the perspective of an arbitrageur, the price range within which there is no arbitrage is:

Where c is the continuously compounded convenience yield

The convenience yield produces a no-arbitrage range rather than a no-arbitrage price. Why?

There may be no way for an average investor to earn the convenience yield when engagingin arbitrage

S0e( r c )T F0 ,T S0e

( r )T

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From the perspective of an arbitrageur, the price range within which there is no arbitrage is:

Where c is the continuously compounded convenience yield

The convenience yield produces a no-arbitrage range rather than a no-arbitrage price. Why?

There may be no way for an average investor to earn the convenience yield when engagingin arbitrage

39

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Interest rate parity

Interest Rate Parity (IRP)

Where; rDC = Domestic currency rate

rFC = Foreign currency rate

T

FC

DC

rrSpotForward

11

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Where; rDC = Domestic currency rate

rFC = Foreign currency rate

Page 42: FRM Formulas

Default rates

Issuer default rate =Number of issuers that default

Total number of issuers at the beginning of issue

Dollar default rate =Cumulative dollar value of all defaulted bonds

Cumulative $ value of all issuance *Weighted Avg. number of years outstanding

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Cumulative $ value of all issuance *Weighted Avg. number of years outstanding

Cumulative annual default rate =Cumulative dollar value of all defaulted bonds

Cumulative dollar value of issue

Page 43: FRM Formulas

Foundation of Risk Management

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Foundation of Risk Management

Page 44: FRM Formulas

Expected Return and Standard Deviation of Portfolio

Return of Portfolio

Standard Deviation of Portfolio

Nto1kRWR kkp

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ikN;to1iN;to1kPσσWWσW kiikik2

kk p

Page 45: FRM Formulas

Portfolio Variance for two asset portfolio

For two-asset portfolio• Var(wAkA+ wBkB) = wA

2 σA2 + wB

2 σB2 + 2 wA wB σA σB ρAB

Where ρ is correlation coefficient between A and B

wA ,wB are weights of the asset A and B• If ρ =1– Var(wAkA + wBkB) = (wAσA + wBσB)2

• If ρ <1– Var(wAkA+ wBkB) < (wAσA+ wBσB)2

So there is a risk reduction from holding a portfolio of assets if assets do not move inperfect unison

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For two-asset portfolio• Var(wAkA+ wBkB) = wA

2 σA2 + wB

2 σB2 + 2 wA wB σA σB ρAB

Where ρ is correlation coefficient between A and B

wA ,wB are weights of the asset A and B• If ρ =1– Var(wAkA + wBkB) = (wAσA + wBσB)2

• If ρ <1– Var(wAkA+ wBkB) < (wAσA+ wBσB)2

So there is a risk reduction from holding a portfolio of assets if assets do not move inperfect unison

44

Page 46: FRM Formulas

Correlation and Portfolio Diversification

Perfect Positive Correlation• ρ =1 & Var (wAkA+ wBkB)= (wAσA + wBσB)2

Perfect Negative Correlation• ρ =-1 & Var (wAkA + wBkB) = (wAσA - wBσB)2

Zero Correlation• Correlation between two assets is zero

Moderate Positive Correlation• Correlation between two assets lies between 0 and 1

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Perfect Positive Correlation• ρ =1 & Var (wAkA+ wBkB)= (wAσA + wBσB)2

Perfect Negative Correlation• ρ =-1 & Var (wAkA + wBkB) = (wAσA - wBσB)2

Zero Correlation• Correlation between two assets is zero

Moderate Positive Correlation• Correlation between two assets lies between 0 and 1

45

Page 47: FRM Formulas

Capital Market Line

Capital Market Line: A line used in the capital asset pricing model to illustrate the rates of returnfor efficient portfolios depending on the risk-free rate of return and the level of risk(standard deviation) for a particular portfolio

Represents all possible combinations of the market portfolio (P) and risk free asset

p

sffs

σσR)E(RR)E(R p

CML

Risk Free Asset Introduced

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Rf

Efficient Frontier

CML

Return

Volatility

Pe

Page 48: FRM Formulas

Capital Asset Pricing Model (CAPM)

As per CAPM, stock’s required rate of return = risk-free rate of return + market risk premium

Rm- Rf: Risk Premium

β: Quantity of Risk

fmfs RRβRR

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m

mii RVar

R,Rcovβ

Page 49: FRM Formulas

Relaxing Assumptions of CAPM

CAPM equation is adjusted to include dividend yield on the market portfolio and the stock

factor taxTistockforyielddividend

portfoliomarketofyielddividend

)()())(()E(R p

i

M

FiFMFMF RRRRER

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Page 50: FRM Formulas

Beta

Sensitivity of the return of the asset to the market return is known as Beta

Beta is calculated as follows:-

m

mii RVar

R,Rcovβ

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Portfolio Beta

Beta can also be calculated for portfolio

Portfolio Beta is the weighted average of the betas of individual assets in the portfolio

Page 51: FRM Formulas

Beta

Sharpe ratio:

Sharpe ratio• Rp = portfolio return, Rf = risk free return

• The higher the Sharpe measure, the better the portfolio

p

fp

σRR

Treynor ratio:

Treynor ratio• Rp = portfolio return, Rf = risk free return

• The higher the Treynor measure, the better the portfolio

• However, this measure should be used only for well-diversified portfolio

Beta

RR fp

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Treynor ratio:

Treynor ratio• Rp = portfolio return, Rf = risk free return

• The higher the Treynor measure, the better the portfolio

• However, this measure should be used only for well-diversified portfolio

Beta

RR fp

Jenson’s alpha:

Jenson’s alpha• Rp = portfolio return, Rc = return predicted by CAPM

• Positive alpha (portfolio with positive excess return) is always preferred over negative alpha

cp RRα

Page 52: FRM Formulas

Measures of performance

Tracking Error (TE):

(Std. dev. of portfolio’s excess return over Benchmark index)

• Where Ep = RP – RB

• RP = portfolio return, RB = benchmark return• Lower the tracking error lesser the risk differential between portfolio and the benchmark index

Information Ratio (IR):• Measure of risk-adjusted return for a portfolio, defined as expected active return per unit of tracking error

• Higher IR indicates higher active return of portfolio at a given risk level

Sortino Ratio (SR):

• MAR is Minimum Accepted Return. SSD is standard deviation of returns below MAR. (Or) SSD is the Semi Standard Deviationfrom MAR where Rp<MAR

• Higher the Sortino Ratio, lower is the risk of large losses

PETE

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Tracking Error (TE):

(Std. dev. of portfolio’s excess return over Benchmark index)

• Where Ep = RP – RB

• RP = portfolio return, RB = benchmark return• Lower the tracking error lesser the risk differential between portfolio and the benchmark index

Information Ratio (IR):• Measure of risk-adjusted return for a portfolio, defined as expected active return per unit of tracking error

• Higher IR indicates higher active return of portfolio at a given risk level

Sortino Ratio (SR):

• MAR is Minimum Accepted Return. SSD is standard deviation of returns below MAR. (Or) SSD is the Semi Standard Deviationfrom MAR where Rp<MAR

• Higher the Sortino Ratio, lower is the risk of large losses

51

TE

RRIR bp

SSD

MARRSR p

,MARR1/tSSD 2p

Page 53: FRM Formulas

Quantitative Analysis

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Quantitative Analysis

Page 54: FRM Formulas

Counting Principle

Number of ways of selecting r objects out of n objects

nCr

n!/(r!)*(n-r)!

Number of ways of giving r objects to n people, such that repetition is allowed

Nr

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Page 55: FRM Formulas

Some definitions and properties of Probability

Definitions• Mutually Exclusive: If one event occurs, then other cannot occur

• Exhaustive: All exhaustive events taken together form the complete sample space (Sum of probability = 1)

• Independent Events: One event occurring has no effect on the other event

The probability of any event A:

If the probability of happening of event A is P(A), then the probability of A not happening is(1-P(A))

For example, if the probability of a company going bankrupt within one year period is 20%, thenthe probability of company surviving within next one year period is 80%

]1,0[)( AP

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Definitions• Mutually Exclusive: If one event occurs, then other cannot occur

• Exhaustive: All exhaustive events taken together form the complete sample space (Sum of probability = 1)

• Independent Events: One event occurring has no effect on the other event

The probability of any event A:

If the probability of happening of event A is P(A), then the probability of A not happening is(1-P(A))

For example, if the probability of a company going bankrupt within one year period is 20%, thenthe probability of company surviving within next one year period is 80%

54

)(1)( APAP

Page 56: FRM Formulas

Sum Rule & Bayes’ Theorem

The unconditional probability of event B is equal to the sum of joint probabilities of event (A,B)and the probability of event (A’,B). Here A’ is the probability of not happening of A• The joint probability of events A and B is the product of conditional probability of B, given A has occurred

and the unconditional probability of event A

• We know that P(AB) = P(B/A) * P(A)

• Also P(BA)= P(A/B) * P(B)

• Now equating both P(AB) and P(BA) we get:

• P(B) can be further broken down using sum rule defined above:

)()/()()/()()()( ccc APABPAPABPBAPBAPBP

)()(*)/()/(

BPAPABPBAP

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The unconditional probability of event B is equal to the sum of joint probabilities of event (A,B)and the probability of event (A’,B). Here A’ is the probability of not happening of A• The joint probability of events A and B is the product of conditional probability of B, given A has occurred

and the unconditional probability of event A

• We know that P(AB) = P(B/A) * P(A)

• Also P(BA)= P(A/B) * P(B)

• Now equating both P(AB) and P(BA) we get:

• P(B) can be further broken down using sum rule defined above:

55

)()(*)/()/(

BPAPABPBAP

)()/()()/()()/()/( cc APABPAPABPAPABPBAP

Page 57: FRM Formulas

Mean

The expected value(Mean) measures the central tendency, or the center of gravity of thepopulation

It is given by:

The graph shows the mean of normal distributions

N

xXE

n

ii

1)(

0.450.400.35

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Standard Normal Distribution

= 0, = 2

= 1, = 1

0 2 4-4 -2

0.400.350.300.250.200.150.100.05

0

Page 58: FRM Formulas

Geometric Mean

Geometric Mean: is calculated as:

• Where there are n observations and each observation is Xi

• Compound Annual Growth Rate(CAGR): It’s the geometric mean of the returns

nnXXXXG ...321

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Page 59: FRM Formulas

Properties of Expectation

E(cX) = E(X) x c

E(X+Y) = E(X) + E(Y)

E(X2) ≠ [E(X)]2

E(XY) = E(X) x E(Y) [if X and Y are independent]

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E(cX) = E(X) x c

E(X+Y) = E(X) + E(Y)

E(X2) ≠ [E(X)]2

E(XY) = E(X) x E(Y) [if X and Y are independent]

58

Page 60: FRM Formulas

Variance & Standard deviation

Variance is the squared dispersion around the mean.

The standard deviation, which is the square root of the Variance, is more convenient to use,as it has the same units as the original variable X

• SD(X) =

N

xVAR

n

ii

1

2)(

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Variance is the squared dispersion around the mean.

The standard deviation, which is the square root of the Variance, is more convenient to use,as it has the same units as the original variable X

• SD(X) =

59

)(xVARN

xn

ii

1

2)(

Page 61: FRM Formulas

Covariance & correlation

Covariance describes the co-movement between 2 random numbers, given as:• Cov(X1, X2) = σ12

Correlation coefficient is a unit-less number, which gives a measure of linear dependencebetween two random variables.• ρ(X1, X2) = Cov(X1, X2) / σ1σ2

Correlation coefficient always lies in the range of +1 to -1

A correlation of 1 means that the two variables always move in the same directionA correlation of -1 means that the two variables always move in opposite direction

If the variables are independent, covariance and correlation are zero, but vice versais not true

YX

YX

XYEYXCovYXEYXCov

)(),()])([(),(

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Covariance describes the co-movement between 2 random numbers, given as:• Cov(X1, X2) = σ12

Correlation coefficient is a unit-less number, which gives a measure of linear dependencebetween two random variables.• ρ(X1, X2) = Cov(X1, X2) / σ1σ2

Correlation coefficient always lies in the range of +1 to -1

A correlation of 1 means that the two variables always move in the same directionA correlation of -1 means that the two variables always move in opposite direction

If the variables are independent, covariance and correlation are zero, but vice versais not true

60

Page 62: FRM Formulas

Some Properties of Variance

Variance of a constant = 0

Covariance between same variables is also their variance

For independent or uncorrelated variables,• covariance or correlation = 0

)()( 2 XVarabaXVar

n

ii

n

ii XVarXVar

11)()(

n

i

n

jji

n

ii XXCovXVar

1 11),()(

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Variance of a constant = 0

Covariance between same variables is also their variance

For independent or uncorrelated variables,• covariance or correlation = 0

61

n

ii

n

ii XVarXVar

11)()(

),(2)()()( 22 YXabCovYVarbXVarabYaXVar

Page 63: FRM Formulas

Skewness

Skewness describes departures from symmetry

Skewness can be negative or positive.

Negative skewness indicates that the distributionhas a long left tail, which indicates a high probabilityof observing large negative values.

If this represents the distribution of profits andlosses for a portfolio, this is a dangerous situation.

31

3)(

n

ii

k

xS

Symmetric Distribution

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Skewness describes departures from symmetry

Skewness can be negative or positive.

Negative skewness indicates that the distributionhas a long left tail, which indicates a high probabilityof observing large negative values.

If this represents the distribution of profits andlosses for a portfolio, this is a dangerous situation.

Negatively Skewed Distribution

Positively Skewed Distribution

Page 64: FRM Formulas

Kurtosis

Kurtosis describes the degree of “flatness” of a distribution, or width of its tails

Because of the fourth power, large observations in the tail will have a large weight and hencecreate large kurtosis. Such a distribution is called leptokurtic, or fat tailed

A kurtosis of 3 is considered average

High kurtosis indicates a higher probabilityof extreme movements

41

4)(

n

iix

K

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Kurtosis describes the degree of “flatness” of a distribution, or width of its tails

Because of the fourth power, large observations in the tail will have a large weight and hencecreate large kurtosis. Such a distribution is called leptokurtic, or fat tailed

A kurtosis of 3 is considered average

High kurtosis indicates a higher probabilityof extreme movements

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

-4 -3 -2 -1 0 1 2 3 4

PlatykurticK<3

LeptokurticK>3

MesokurticK=3

Page 65: FRM Formulas

Errors in estimation

Type I and Type II Errors• Type I error occurs if the null hypothesis is rejected

when it is true

• Type II error occurs if the null hypothesis is not rejectedwhen it is false

Significance Level• -> Significance level– the upper-bound probability of a Type I error

• 1 - ->confidence level– the complement of significance level

Actual

InferenceH0 is True H0 is False

H0 is TrueCorrect DecisionConfidenceLevel = 1-α

Type-II ErrorP(Type-II Error)= β

H0 is FalseType-I ErrorSignificanceLevel = α

Power=1-β

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Type I and Type II Errors• Type I error occurs if the null hypothesis is rejected

when it is true

• Type II error occurs if the null hypothesis is not rejectedwhen it is false

Significance Level• -> Significance level– the upper-bound probability of a Type I error

• 1 - ->confidence level– the complement of significance level

Page 66: FRM Formulas

Hypothesis tests for variances

Hypothesis Test of VariancesHypothesis Test of Variances

Test forSingle Population Variance

Test forSingle Population Variance

Test forTwo Population Variances

Test forTwo Population Variances

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Example HypothesisExample Hypothesis

H0: σ2 = σ02

HA: σ2 ≠ σ02

Chi-Square TestStatistic

Chi-Square TestStatistic

20

22

)1(,)1(

snn

Example HypothesisExample Hypothesis

H0: σ12 – σ2

2 = 0HA: σ1

2 – σ22 ≠ 0

F-test StatisticF-test Statistic

22

21

,, ssF ddfndf

Page 67: FRM Formulas

Test for single population variance

Single population variance test has 3different forms:• Two Tailed Test:

• Lower Tail test:

• Upper Tail Test

H0: σ2 = σ02

HA: σ2 ≠ σ02

H0: σ2 σ02

HA: σ2 < σ02

/2

/2

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Single population variance test has 3different forms:• Two Tailed Test:

• Lower Tail test:

• Upper Tail Test

H0: σ2 σ02

HA: σ2 < σ02

H0: σ2 ≤ σ02

HA: σ2 > σ02

Page 68: FRM Formulas

Chi-square test statistic

The chi-squared test statistic for aSingle Population Variance is:

Where

2 = standardized chi-square variable

n = sample size

s2 = sample variance

σ2 = hypothesized variance

2

22

σ1)s(n

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The chi-squared test statistic for aSingle Population Variance is:

Where

2 = standardized chi-square variable

n = sample size

s2 = sample variance

σ2 = hypothesized variance

Page 69: FRM Formulas

Finding the critical value

The critical value, 2 , is found from the chi-square table:

H0: σ2 ≤ σ02

HA: σ2 > σ02

Upper tail test:

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2

Do not reject H0 Reject H0

2

Page 70: FRM Formulas

Lower tail or two tailed Chi-square tests

H0: σ2 = σ02

HA: σ2 ≠ σ02

H0: σ2 σ02

HA: σ2 < σ02

/2

/2

Lower tail test: Two tail test:

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Do not reject H0Reject

21-

2

2/2

Do notreject H0

Reject

21-/2

2

Reject

Page 71: FRM Formulas

F-test for difference in two population variances

Two population variance test has 3different forms:• Two Tailed Test:

• Lower Tail test:

• Upper Tail Test

H0: σ12 – σ2

2 = 0HA: σ1

2 – σ22 ≠ 0

H0: σ12 – σ2

2 0HA: σ1

2 – σ22 < 0

/2

/2

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Two population variance test has 3different forms:• Two Tailed Test:

• Lower Tail test:

• Upper Tail Test

H0: σ12 – σ2

2 0HA: σ1

2 – σ22 < 0

H0: σ12 – σ2

2 ≤ 0HA: σ1

2 – σ22 > 0

Page 72: FRM Formulas

F-test for difference in two population variances (cont.)

The F test statistic is:

= Variance of Sample 1

(n1 – 1) = numerator degrees of freedom

= Variance of Sample 2

(n2 – 1) = denominator degrees of freedom

21s

22

21

ssF

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The F test statistic is:

= Variance of Sample 1

(n1 – 1) = numerator degrees of freedom

= Variance of Sample 2

(n2 – 1) = denominator degrees of freedom

21s

22s

Page 73: FRM Formulas

Chebyshev’s inequality

Chebyshev's inequality says that at least 1 - 1/k2 of the distribution's values are within kstandard deviations of the mean.

Where k is any positive real number greater than 1

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Page 74: FRM Formulas

Population linear regression

Random Error forthis x value

Y

Observed Value ofY for Xi

Predicted Value ofY for Xi

Slope = β1

uXY 10

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Mean marks for hours of study

Individual person’s marks

Random Error forthis x value

Predicted Value ofY for Xi

xi

Intercept = β0

ui

x

Page 75: FRM Formulas

Population regression function

Populationy intercept

PopulationSlope

Coefficient

RandomError

term, orresidual

DependentVariable

IndependentVariable

uXββY 10

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But can we actually get this equation?If yes what all information we will need?

Linear component Random Errorcomponent

uXββY 10

Page 76: FRM Formulas

Sample regression function

Random Error forthis x value

Y

Observed Value ofY for Xi

Predicted Value ofY for Xi

Slope = β1

exbby 10

ei

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Random Error forthis x value

Predicted Value ofY for Xi

xi

Intercept = β0

x

Page 77: FRM Formulas

Sample regression function

Estimate of theregressionintercept Error term

Estimated(or predicted)

y value

Estimate of theregression

slope

Independentvariable

exbby 10i

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Notice the similarity with the Population Regression FunctionCan we do something of the error term?

exbby 10i

Page 78: FRM Formulas

One method to find b0 and b1

Method of Ordinary Least Squares (OLS)

b0 and b1 are obtained by finding the values of b0 and b1 that minimize the sum of thesquared residuals

210

22

x))b(b(y

)y(ye

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210

22

x))b(b(y

)y(ye

Page 79: FRM Formulas

OLS regression properties

The sum of the residuals from the least squares regression line is 0

The sum of the squared residuals is a minimumMinimize ( )

The simple regression line always passes through the mean of the y variable and the meanof the x variable

The least squares coefficients are unbiased estimates of β0 and β1

0)ˆ( yy

2)ˆ( yy

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The sum of the residuals from the least squares regression line is 0

The sum of the squared residuals is a minimumMinimize ( )

The simple regression line always passes through the mean of the y variable and the meanof the x variable

The least squares coefficients are unbiased estimates of β0 and β1

78

Page 80: FRM Formulas

The least squares equation

The formulas for b1 and b0 are:

21 )())((

xxyyxx

b

Algebraic equivalent:

nx

x

nyx

xyb 2

21 )(

And

xbyb 10

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nx

x

nyx

xyb 2

21 )(

xbyb 10

Page 81: FRM Formulas

Interpretation of the Slope and the Intercept

b0 is the estimated average value of y when the value of x is zero. More often than not itdoes not have a physical interpretation

b1 is the estimated change in the average value of y as a result of a one-unit change in x

y

XbbY 10

slope of the line(b1)

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x

b0

slope of the line(b1)

Page 82: FRM Formulas

Explained and unexplained variation

yi

y

y

y

•RSS = Residual sum of squares

_

_

TSS = Total sumof squares

RSS = (yi - yi )2

TSS = (yi - y)2

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Xix

y_

y_y

ESS = (yi - y)2 _

•ESS = Explained Sum of squares

Page 83: FRM Formulas

Explained and unexplained variation

Total variation is made up of two parts:

ESSRSSTSS Total sum of SquaresTotal sum of Squares Sum of Squares Regression /

Explained Sum of SquaresSum of Squares Regression /

Explained Sum of SquaresSum of Squares Error /

Residual Sum of SquaresSum of Squares Error /

Residual Sum of Squares

2)( yyTSS 2)ˆ( yyRSS 2)ˆ( yyESS

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2)( yyTSS 2)ˆ( yyRSS 2)ˆ( yyESS

Where:• = Average value of the dependent variable• y = Observed values of the dependent variable• = Estimated value of y for the given x valuey

y

Page 84: FRM Formulas

Coefficient of determination, R2

The coefficient of determination is the portion of the total variation in the dependentvariable that is explained by variation in the independent variable The coefficient of determination is also called R-squared and is denoted as R2

SSTSSRR 2

1R0 2 where

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Page 85: FRM Formulas

Coefficient of determination, R2

Coefficient of determination

Note: In the single independent variable case, the coefficient of determination is

squaresofsumtotalregressionbyexplainedsquaresofsum

SSTSSRR 2

22 rR

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22 rR

Where:• R2 = Coefficient of determination• r = Simple correlation coefficient

Page 86: FRM Formulas

Calculating the correlation coefficient

Sample correlation coefficient:

])yy(][)xx([

)yy)(xx(r

22

or the algebraic equivalent:

])y()y(n][)x()x(n[

yxxynr

2222

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])y()y(n][)x()x(n[

yxxynr

2222

Where:• r = Sample correlation coefficient• n = Sample size• x = Value of the independent variable• y = Value of the dependent variable

Page 87: FRM Formulas

Standard Error of “Estimate”

The standard deviation of the variation of observations around the regression line isestimated by:

1

knRSSsu

Where:• RSS = Residual Sum of Squares (summation of e2)• n = Sample size• k = number of independent variables in the model

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Where:• RSS = Residual Sum of Squares (summation of e2)• n = Sample size• k = number of independent variables in the model

Standard Error of Estimate (SEE) is another name of Standard Error of regression

Page 88: FRM Formulas

The Standard Deviation of the intercept

2

u2

ib )x(x

sXs

o n

nx)(

x

s

)x(x

ss

22

u

2

ub1

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Where:• = Estimate of the standard error of the least squares slope

• = Sample standard error of the estimate

nx)(

x

s

)x(x

ss

22

u

2

ub1

1bs

2nRSSs u

Page 89: FRM Formulas

Multiple Regression

Using more than one explanatory variable in a regression model• Y = b0 + b1X1 + b2X2 + b3X3 + uI

Omitted variable bias• The biasness incurred due to omission of one or more explanatory variable from the model.

Omitted variable bias occurs when two conditions are met:• Omitted variables are correlated with the independent variable• Variables that are not accounted for in the model but affect the dependent variable

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Using more than one explanatory variable in a regression model• Y = b0 + b1X1 + b2X2 + b3X3 + uI

Omitted variable bias• The biasness incurred due to omission of one or more explanatory variable from the model.

Omitted variable bias occurs when two conditions are met:• Omitted variables are correlated with the independent variable• Variables that are not accounted for in the model but affect the dependent variable

88

Page 90: FRM Formulas

Multiple Regression Basics

General Multiple Linear Regression model take the following form:

ikikiii XbXbXbbY .........22110

Where:• Yi = ith observation of dependent variable Y• Xki = ith observation of kth independent variable X• b0 = intercept term• bk = slope coefficient of kth independent variable• εi = error term of ith observation• n = number of observations• k = total number of independent variables

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Where:• Yi = ith observation of dependent variable Y• Xki = ith observation of kth independent variable X• b0 = intercept term• bk = slope coefficient of kth independent variable• εi = error term of ith observation• n = number of observations• k = total number of independent variables

Page 91: FRM Formulas

Estimated Regression Equation

As we calculated the intercept and the slope coefficient in case of simple linear regressionby minimizing the sum of squared errors, similarly we estimate the intercept and slopecoefficient in multiple linear regression

• Sum of Squared Errors is minimized and the slope coefficient is estimated.

The resultant estimated equation becomes:

Now the error in the ith observation can be written as:

n

ii

1

2

kikiii XbXbXbbY

.........22110

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As we calculated the intercept and the slope coefficient in case of simple linear regressionby minimizing the sum of squared errors, similarly we estimate the intercept and slopecoefficient in multiple linear regression

• Sum of Squared Errors is minimized and the slope coefficient is estimated.

The resultant estimated equation becomes:

Now the error in the ith observation can be written as:

90

kikiii XbXbXbbY

.........22110

kikiiiiii XbXbXbbYYY .........22110

Page 92: FRM Formulas

Estimation of Volatility

Let xi be the continuously compounded return during day i (between the end of day“i-1” and end of day “I”)

Let σn be the volatility of the return on day n as estimated at the end of day n-1

Variance estimate for next day is usually calculated as:• variance = average squared deviation from average return over last ‘n’ days

Mean of returns (x-bar) is usually zero, especially if returns are over short-time period(say, daily returns). In that case, variance estimate for next day is nothing but simple average (equallyweighted average) of previous ‘n’ days’ squared returns

1n

xxVariance

n

1i

2

i

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Let xi be the continuously compounded return during day i (between the end of day“i-1” and end of day “I”)

Let σn be the volatility of the return on day n as estimated at the end of day n-1

Variance estimate for next day is usually calculated as:• variance = average squared deviation from average return over last ‘n’ days

Mean of returns (x-bar) is usually zero, especially if returns are over short-time period(say, daily returns). In that case, variance estimate for next day is nothing but simple average (equallyweighted average) of previous ‘n’ days’ squared returns

91

1n

xxVariance

n

1i

2

i

11

2

n

xVariance

n

ii

What if the volatility is dependent on the values of volatility observed in the recent past?What if they also depend on the latest returns?

Page 93: FRM Formulas

EWMA Model

In an exponentially weighted moving average model, the weights assigned to the u2 declineexponentially as we move back through time

This leads to: 21

21

2 )1( nnn u

Apply the recursive relationship:

Hence we have

• Variance estimate for next day (n) is given by (1-λ) weight to recent squared return and λ weight to the previousvariance estimate

• Risk-metrics (by JP Morgan) assumes a Lambda of 0.94

2

222

22

12

21

22

22

2

)1(

)1()1(

nnnn

nnnn

uuuu

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Apply the recursive relationship:

Hence we have

• Variance estimate for next day (n) is given by (1-λ) weight to recent squared return and λ weight to the previousvariance estimate

• Risk-metrics (by JP Morgan) assumes a Lambda of 0.94

2

222

22

12

21

22

22

2

)1(

)1()1(

nnnn

nnnn

uuuu

22

1

12 )1( mnm

in

m

i

in u

Page 94: FRM Formulas

EWMA Model

Since returns are squared, their direction is not considered. Only the magnitude is considered

In EWMA, we simply need to store 2 data points: latest return & latest volatility estimate

Consider the equation:

In this equation, variance for time ‘t’ was also an estimate. So we can substitute for it as follows:

What are the weights for old returns and variance?

λ is called ‘Persistence factor’ or even “Decay Factor”. Higher λ gives more weight to older data (impactof older data is allowed to persist). Lower λ gives higher weight to recent data (i.e. previous dataimpacts are not allowed to persist)

Higher λ means higher persistence or lower decay

Since, (1- λ) is weight given to latest square return, it is called ‘Reactive factor’

2221 94.0)94.01( ttt

21

21

221 94.0)94.01(94.0)94.01( tttt

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Since returns are squared, their direction is not considered. Only the magnitude is considered

In EWMA, we simply need to store 2 data points: latest return & latest volatility estimate

Consider the equation:

In this equation, variance for time ‘t’ was also an estimate. So we can substitute for it as follows:

What are the weights for old returns and variance?

λ is called ‘Persistence factor’ or even “Decay Factor”. Higher λ gives more weight to older data (impactof older data is allowed to persist). Lower λ gives higher weight to recent data (i.e. previous dataimpacts are not allowed to persist)

Higher λ means higher persistence or lower decay

Since, (1- λ) is weight given to latest square return, it is called ‘Reactive factor’

93

)94.0*94.0(*06.0*94.0*06.0 21

21

221 tttt

Page 95: FRM Formulas

GARCH (1,1)

GARCH stands for Generalized Autoregressive Conditional Heteroscedasticity

Heteroscedasticity means variance is changing with time.

Conditional means variance is changing conditional on latest volatility.

Autoregressive refers to positive correlation between volatility today and volatility yesterday.

(1,1) means that only the latest values of the variables.

GARCH model recognizes that variance tends to show mean – reversion i.e. it gets pulled toa long-term Volatility rate over time.

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GARCH stands for Generalized Autoregressive Conditional Heteroscedasticity

Heteroscedasticity means variance is changing with time.

Conditional means variance is changing conditional on latest volatility.

Autoregressive refers to positive correlation between volatility today and volatility yesterday.

(1,1) means that only the latest values of the variables.

GARCH model recognizes that variance tends to show mean – reversion i.e. it gets pulled toa long-term Volatility rate over time.

94

2221 ttLt V

Long-term average Volatility

Page 96: FRM Formulas

GARCH (1,1)

Generally γ*VL is replaced by ω

Since the sum of all the weights is equal to 1 we get the following equation as well:

2221 ttt

1LV

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1LV

Page 97: FRM Formulas

Simulating a Price Path

S is the stock price,

μ is the expected return,

σ is the standard deviation of returns,

"t" is time, and

ε is the random variable

ttSS

Drift Shock

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The first step in simulating a price path is to choose a random process to model changes infinancial assets

Stock prices and exchange rates are modeled by geometric Brownian motion (GBM) shownin the above equation

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Valuations and Risk Models

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Valuations and Risk Models

Page 99: FRM Formulas

Measuring Value-at-Risk (VAR)

Mean = 0

00.050.10.150.20.250.30.350.40.45

-4 -2 0 2 4

*%)( %% XX ZinVAR

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ZX% : the normal distribution value for the given probability (x%) (normal distribution has mean as 0 andstandard deviation as 1)

σ : standard deviation (volatility) of the asset (or portfolio)

VAR in absolute terms is given as the product of VAR in % and Asset Value:

This can also be written as:

98

Mean = 0

ValueAssetinVARVAR X *%)(%

ValueAssetZVAR X **%

Page 100: FRM Formulas

Measuring Value-at-Risk (VAR)

VAR for n days can be calculated from daily VAR as:

This comes from the known fact that the n-period volatility equals 1-period volatility multiplied bythe square root of number of periods(n).

As the volatility of the portfolio can be calculated from the following expression:

The above written expression can also be extended to the calculation of VAR:

n*%)(inVaR%)(inVaR VaR)(dailydays)(n

n*ValueAsset** Z%)(inVaR X%days)(n

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VAR for n days can be calculated from daily VAR as:

This comes from the known fact that the n-period volatility equals 1-period volatility multiplied bythe square root of number of periods(n).

As the volatility of the portfolio can be calculated from the following expression:

The above written expression can also be extended to the calculation of VAR:

99

***w2w w abbaba2b

2b

2a

2aportfolio w

*)(%VAR*)(%VAR*w2w)(%VAR w)(%VAR%)(inVaR abbaba2

b2b

2a

2aportfolio w

Page 101: FRM Formulas

Expected Loss (EL)

EL = (Assured payment at Maturity Time T )* Loss Given Default * (Probability that the defaultoccurs before maturity Time T)

The term “Assured payment” is more relevant for bonds than loans

For Bank Loans the terms Assured Payment is better replaced by “Exposure”

EL = Exposure * LGD*PD

EL is the amount the bank can lose on an average over a period of time

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Adjusted Exposure

Let the value of bank asset at time T be V

Let the already drawn amount be OS (outstanding)

Let COM be the commitment

Let “d” be the fraction of the commitment which would be drawn before the default

Portion which is not drawn and risk free = COM*(1-d)

Risky portion = OS + d*COM

This Risky Portion is known as Adjusted Exposure also known as Exposure At Default

EL = Adjusted Exposure*LGD*PD

“d” is also known as Usage Given Default (UGD)

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Let the value of bank asset at time T be V

Let the already drawn amount be OS (outstanding)

Let COM be the commitment

Let “d” be the fraction of the commitment which would be drawn before the default

Portion which is not drawn and risk free = COM*(1-d)

Risky portion = OS + d*COM

This Risky Portion is known as Adjusted Exposure also known as Exposure At Default

EL = Adjusted Exposure*LGD*PD

“d” is also known as Usage Given Default (UGD)

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Page 103: FRM Formulas

Causes of Unanticipated Risk

Two Primary Sources

The occurrence of defaults (PD)• PD is never zero for any rating

• PD is calculated using historical data or Analytical methods like Option theory

Unexpected Credit Migration – unanticipated change in credit quality• An obligor undergoes financial crisis which deteriorates the credit quality although it is not a default

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Page 104: FRM Formulas

Unexpected Loss

UL is the estimated volatility of the potential loss in value of the asset around its EL

UL is the standard deviation of the unconditional value of the asset at the time horizon

UL = s.d. of expected asset value

UL = AE*√[EDF* σ2LGD +LGD2* σ2EDF ]

• Underlying assumption that EDF is independent of LGD. In case it is not so then correlation between LGDand EDF terms will come into picture. Though it has been found that they will affect the result only slightly

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