FRM Formulas

FMP

© EduPristine For FMP-I (Confidential)

© EduPristine – www.edupristine.com

Hedging in a practical world (Basis Risk)  Basis = spot price of asset – futures price contract • Basis = 0 when spot price = futures price

Future Price Spot Price

Time


1

Choice of contracts • Optimal Hedge Ratio:

h  

 S  F

 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


2

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

h * NA Qf

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

P A

N *  (  * - )

P A

Negative sign of (N*) indicates shorting the contracts © EduPristine For FMP-I (Confidential)

3

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


4

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 the dividends 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


5

Futures and Forwards on Currencies  Interest rate Parity

F0  S 0 e

( rbc  r fc ) T

 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 ( rUSD  rINR )T USD USD INR INR

F

S

e


6

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


7

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)


8

Bond pricing  The price of a bond is the present value of all the coupon payment and the final principal payment received at the end of its life T

B   Ce  rt  Pe  rT t 1

•

B  the bond price

•

C  coupon payment

•

r  zero interest rate at time t

•

P  bond principal

•

T  time to maturity

1  1   (1  YTM) n B  I  YTM  

  1   F (1  YTM) n  

 The yield of a bond is the discount rate (applied to all future cash flows) at which the present value of the bond is equal 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) of the bond  If we are looking at a semi-annual 5 year coupon bond with a par value of $100 then the coupon payment would be solved using the following equation: 5

100   (C / 2)e  rt  100e 5 r t 1


9

Forward rate agreements (FRAs)  In general:

Ft1, t2

R 2 T2  R 1T1  T2  T1

 Payment to the long at settlement = Notional Principal X (Rate at settlement – FRA Rate) (days/360) ---------------------------------------------------------1 + (Rate at settlement) (days / 360)


10

Duration  Macaulay’s duration: is the weighted average of the times when the payments are made. And the weights are a ratio of the coupon paid at time t to the present bond price t *C n*M   t (1  y ) n t 1 (1  y ) Macaluay Duration  Current bond price n

 Where: • t = Respective time period • C = Periodic coupon payment • y = Periodic yield • n = Total no of periods • M = Maturity value


11

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: B   DY B

 In the case of a continuously compounded yield the duration used is modified duration given as: D* 

Macaulay Duration r 1 n


12

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

1 d 2B C B dy 2  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 ($)

P*

Actual bond price Tangent Y*


Yield

13

…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


14

Calculating Bond Price Changes  We can approximate the change in a bond’s price for a given change in yield by using duration and convexity:



 V B    D M o d   i  V B   0 .5  C  V B    i 


2



15

Theories of the Term Structure  Three theories are used to explain the shape of the term structure

(1  ilt ) n  (1  ist

year 1

)(1  ist

year 2

)...(1  ist

yearn

)

yearn

)

 Expectations theory  The long rate is the geometric mean of expected future short interest rates  Liquidity preference theory  Investors must be paid a “liquidity premium” to hold less liquid, long-term debt  Market segmentation theory

(1  ilt ) n  rpn  (1  ist

year 1

)(1  ist

year 2

)...(1  ist

Where rpn is the risk premium associated with an n year bond

 Investors decide in advance whether they want to invest in short term or the long term  Distinct markets exist for securities of short term bonds and long term bonds  Supply demand conditions decide the prices © EduPristine For FMP-I (Confidential)

16

Day count conventions  Day count defines the way in which interest is accrued over time. Day count conventions normally used in US are: • Actual / actual  treasury bonds • 30 / 360  corporate bonds • Actual/360  money market instruments

 The interest earned between two dates =

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


(Number of days in reference period)

17

Cheapest to deliver bond  The party with the short position can chose to deliver the cheapest bond when it comes to delivery, 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)


18

DV01 – Application to hedging  Hedge ratio is calculated using DV01 with the help of following relation

HR 

DVO1( per$100 of initial position) DV 01( per$100 of hedging instrument)


19

Duration based hedging strategies  Number of contracts to hedge is given by the equation:

N* 

PDP FC DF

• 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


20

Key Rate ‘01 and Key Rate Durations  Key Rate ‘01 measures the dollar change in the value of the bond for every basis point shift in 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)


21

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. it follows European options


22

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


23

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

p

Su p


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

1-p

S0 1-p

Su 2

p Su

1-p

Sud

Sd 2

IV2

IV3

24

Black and Scholes Model  Black and Scholes formula allows for infinitesimally small intervals as well as the need to revise leverage for European options on Non Dividend paying stocks  The formula is:

[ N ( d 1)  P ]  [ N ( d 2)  X  e

 R f T

• Where,

]

P ln[ ]  [Rf  (0.5 2 )]T d1  X  T d 2  d1  T

 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 = [ X  e

R f T


 {1  N (d 2}]  [{1  N (d1)} P]

25

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




portfolio



n



i1

W i

i

26

Theta (cont.)  We have theta of call given by:  S 0 N ' ( d 1 )  rKe 2 T

 ( Call ) 

 Where:  rT

N (d 2 )

• K = Strike price • T = Time of maturity of the option measured in years, so that 6 months will be 0.5 years

• Where:

 rT

• d1 and d2 are as defined in the Black-Scholes Pricing formula earlier • σ = Stock price volatility

e (x^2)/2 N '(x)  2

 S 0 N ' ( d 1 )  ( Put )   rKe 2 T

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

N ( d 2 )

• r = Risk neutral rate of interest

 For a put option, theta is given by:


27

Gamma (cont.)  Calculation of Gamma • Gamma for European options can be calculated using the following formula:

 

N ' ( d 1) S 0 T

• Where symbols have their usual meaning

Gamma (ATM) vs. Time 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0

Gamma (Call / Put) 0.07 0.06 0.05 0.04 0.03 0.02 0.01

0

0.2

0.4

0.6


0.8

1.0

1.2

0

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

28

Vega  The Vega of a derivative portfolio is the rate of change of the value of the portfolio with the change in the 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:

e  (d 1^ 2 ) / 2 N ' ( d 1)  2  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


16 14 12 10 8 6 4 2 0

Vega

1 4 7 1013161922252831343740434649

29

Rho  Rho of a portfolio of options is the rate of change of its value with respect to changes in the interest rate  

 Rho = 

r

, 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

Rho (Call / Put)

30 20 10

Rho (Call)

0

Rho (Put)

-10 -20 -30

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


30

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 Present Value of Par values (P) and Accrued Interest (AI)


31

Commodity Forwards  Commodity forward prices can be described using the same formula as used for financial forward prices

F

0 , T



S

0

e

( r 



) T

 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


32

Futures term structure  The set of prices for different expiration dates for a given commodity is called the forward curve (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

(r )T

F0,T  S0e

• 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


33

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 who borrowed a stock  Therefore:

F0 ,T  S 0 e ( r   )T  Where δ is lease rate


34

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 have to be if a lease market existed  The annualized lease rate  The effective annual lease rate

1  l  r  In (F0 ,T / S ) T

(1  r ) l  1 1/T (F0 ,T / S )


35

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 of selling 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 0 to T

F0 ,T  S 0 e rT   (0,T )


36

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–


37

Pricing with convenience  So, if:

F0 ,T  S 0 e ( r   )T  And if,  = c –   Then, F0,T = S0e(r+  -c)T


38

No-Arbitrage with Convenience  From the perspective of an arbitrageur, the price range within which there is no arbitrage is:

S0 e

( r    c )T

 F0 ,T  S 0 e

( r   )T

 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 engaging in arbitrage


39

Interest rate parity  Interest Rate Parity (IRP)

1  rDC  Forward  Spot   1  rFC 

T

 Where; rDC = Domestic currency rate rFC = Foreign currency rate


40

Default rates

Issuer default rate =

Dollar default rate =

Cumulative annual default rate =


Number of issuers that default Total number of issuers at the beginning of issue

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

Cumulative dollar value of all defaulted bonds Cumulative dollar value of issue

41

Foundation of Risk Management



Expected Return and Standard Deviation of Portfolio  Return of Portfolio

k  1 to N

R p   Wk R k

 Standard Deviation of Portfolio

p 

Wσ  WWσ σ P 2

k k


k

i k i ki

k 1toN;i 1toN;k i

43

Portfolio Variance for two asset portfolio  For two-asset portfolio • Var(wAkA+ wBkB) = wA2 σA2 + wB2 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 in perfect unison


44

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


45

Capital Market Line  Capital Market Line: A line used in the capital asset pricing model to illustrate the rates of return for 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 E(R s )  R f   E(R p )  R f  

σs σp

Risk Free Asset Introduced CML Return Pe

Efficient Frontier

Rf

Volatility © EduPristine For FMP-I (Confidential)

46

Capital Asset Pricing Model (CAPM)  As per CAPM, stock’s required rate of return = risk-free rate of return + market risk premium

R s  R f  βR m  R f   Rm- Rf: Risk Premium  β: Quantity of Risk

covR i , R m  βi  VarR m 


47

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

E(R

p

)  R F  ( E ( R M )  R F )   (  M  R F )   (  i  R F ) 

 M  dividend yield of market portfolio  i  dividend yield for stock i T  tax factor


48

Beta  Sensitivity of the return of the asset to the market return is known as Beta  Beta is calculated as follows:-

cov R i , R m  βi  Var R m 

Portfolio Beta  Beta can also be calculated for portfolio  Portfolio Beta is the weighted average of the betas of individual assets in the portfolio


49

Beta  Sharpe ratio: Sharpe ratio 

R R  p

f

σp • Rp = portfolio return, Rf = risk free return • The higher the Sharpe measure, the better the portfolio

 Treynor ratio: Treynor ratio 

R

p

 Rf



Beta • 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

 Jenson’s alpha: Jenson’s alpha α  R p  R c • Rp = portfolio return, Rc = return predicted by CAPM • Positive alpha (portfolio with positive excess return) is always preferred over negative alpha


50

Measures of performance  Tracking Error (TE): TE   E P (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

IR 

R

p

 Rb



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

 Sortino Ratio (SR): SR 

SSD 

R

p

 MAR



SSD

1/t  R p  MAR 2 ,

• MAR is Minimum Accepted Return. SSD is standard deviation of returns below MAR. (Or) SSD is the Semi Standard Deviation from MAR where Rp
51

Quantitative Analysis



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


53

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:

P( A)  [0,1]

 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%, then the probability of company surviving within next one year period is 80%

P( A)  1  P( A)


54

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

P( B)  P( A  B)  P( Ac  B)  P( B / A) P( A)  P( B / Ac ) P( Ac ) • 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 / A) * P( A) P( A / B)  P( B)

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

P( A / B) 

P( B / A) P( A) P( B / A) P( A)  P( B / Ac ) P( Ac )


55

Mean  The expected value(Mean) measures the central tendency, or the center of gravity of the n population  It is given by:

  E(X ) 

x i 1

i

N

 The graph shows the mean of normal distributions 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0

Standard Normal Distribution  = 0,  = 2  = 1,  = 1

-4

-2


0

2

4

56

Geometric Mean  Geometric Mean: is calculated as:

G  n X 1  X 2  X 3  ... X n • Where there are n observations and each observation is Xi • Compound Annual Growth Rate(CAGR): It’s the geometric mean of the returns


57

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]


58

Variance & Standard deviation  Variance is the squared dispersion around the mean. n

VAR





i1

(xi   )2 N

 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 n

• SD(X) =

VAR ( x )

  




i1

(xi   )2 N

59

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

Cov( X , Y )  E[( X   X )(Y  Y )] Cov( X , Y )  E ( XY )   X Y  Correlation coefficient is a unit-less number, which gives a measure of linear dependence between 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 direction A 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 versa is not true


60

Some Properties of Variance

Var ( aX  b )  a 2Var ( X ) n

n

 Variance of a constant = 0

n

Var ( X i )   Cov ( X i , X j )  Covariance between same variables is also their variance i 1

i 1 j 1

n

n

i 1

i 1

Var ( X i )   Var ( X i )

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

Var (aX  bY )  a 2Var ( X )  b 2Var (Y )  2abCov( X , Y )


61

Skewness  Skewness describes departures from symmetry n

Sk 

3 ( x   )  i i 1

3

 Skewness can be negative or positive.

Symmetric Distribution

 Negative skewness indicates that the distribution has a long left tail, which indicates a high probability of observing large negative values. Negatively Skewed Distribution

 If this represents the distribution of profits and losses for a portfolio, this is a dangerous situation.

Positively Skewed Distribution


62

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

K 

 (x i 1

i

  )4

4

 Because of the fourth power, large observations in the tail will have a large weight and hence create large kurtosis. Such a distribution is called leptokurtic, or fat tailed  A kurtosis of 3 is considered average

Platykurtic K<3

0.45

 High kurtosis indicates a higher probability of extreme movements

0.4 0.35

Leptokurtic K>3

0.3 0.25 0.2

Mesokurtic K=3

0.15 0.1 0.05 0 -4


-3

-2

-1

0

1

2

3

4

63

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 rejected when 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 Inference

H0 is True

H0 is False

H0 is True

Correct Decision Confidence Level = 1-α

Type-II Error P(Type-II Error) =β

H0 is False

Type-I Error Significance Level = α

Power=1-β

Hypothesis tests for variances Hypothesis Test of Variances

Test for Single Population Variance

Example Hypothesis

H0: σ2 = σ02 HA: σ2 ≠ σ02


Test for Two Population Variances

Chi-Square Test Statistic

 ,( n 1)  2

(n  1) s

 02

Example Hypothesis 2

H0: σ1 – σ2 = 0 HA: σ12 – σ22 ≠ 0 2

2

F-test Statistic

F ,ndf ,ddf

s12  2 s2

Test for single population variance  Single population variance test has 3 different forms:

/2 /2

• Two Tailed Test:

H0: σ2 = σ02 HA: σ2 ≠ σ02 • Lower Tail test:



H0: σ2  σ02 HA: σ2 < σ02

• Upper Tail Test

H0: σ2 ≤ σ02 HA: σ2 > σ02




Chi-square test statistic  The chi-squared test statistic for a Single Population Variance is: 2 (n  1)s 2  σ2

Where

2 = standardized chi-square variable n = sample size s2 = sample variance σ2 = hypothesized variance


Finding the critical value  The critical value, 2 , is found from the chi-square table:

Upper tail test:

H0: σ2 ≤ σ02 HA: σ2 > σ02



2 Do not reject H0

Reject H0 2


Lower tail or two tailed Chi-square tests Lower tail test:

H0: σ2  σ02 HA: σ2 < σ02

Two tail test:

H0: σ2 = σ02 HA: σ2 ≠ σ02

/2



/2

2 Reject

Do not reject H0

21-


2 Reject 21-/2

Do not reject H0

Reject 2/2

F-test for difference in two population variances  Two population variance test has 3 different forms:

/2 /2

• Two Tailed Test:

H0: σ12 – σ22 = 0 HA: σ12 – σ22 ≠ 0 • Lower Tail test:



H0: σ12 – σ22  0 HA: σ12 – σ22 < 0

• Upper Tail Test

H0: σ12 – σ22 ≤ 0 HA: σ12 – σ22 > 0




F-test for difference in two population variances (cont.)  The F test statistic is:

s12 F  2 s2 s 12

= Variance of Sample 1 (n1 – 1) = numerator degrees of freedom

s 22

= Variance of Sample 2 (n2 – 1) = denominator degrees of freedom


Chebyshev’s inequality  Chebyshev's inequality says that at least 1 - 1/k2 of the distribution's values are within k standard deviations of the mean.  Where k is any positive real number greater than 1


Population linear regression Y

Y   0  1 X  u

Observed Value of Y for Xi

Slope = β1

Predicted Value of Y for Xi Intercept = β0

Random Error for this x value ui xi

x Mean marks for hours of study Individual person’s marks


73

Population regression function

Dependent Variable

Population y intercept

Population Slope Coefficient

Independent Variable

Random Error term, or residual

Y  β 0  β1X  u Linear component

Random Error component

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


74

Sample regression function Y

y  b 0  b1x  e

Observed Value of Y for Xi

ei

Predicted Value of Y for Xi

Random Error for this x value

Slope = β1

Intercept = β0 xi


x

75

Sample regression function Estimate of the regression intercept

Estimated (or predicted) y value

y

i

 b

Estimate of the regression slope

0

Independent variable

Error term

 b 1x  e

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


76

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 the squared residuals 2 e  



2 ˆ (y  y ) 

2 (y  (b  b x))  0 1


77

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

 ( y  yˆ )  0  The sum of the squared residuals is a minimum Minimize (  ( y  yˆ ) 2 )  The simple regression line always passes through the mean of the y variable and the mean of the x variable  The least squares coefficients are unbiased estimates of β0 and β1


78

The least squares equation  The formulas for b1 and b0 are:

b1

( x  x )( y  y )    (x  x) 2

 Algebraic equivalent:

b1 

x y   xy 



n 2 ( x ) x2   n


 And

b 0  y  b1 x

79

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 it does 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 

Y

 b

0

 b

1

X

slope of the line(b1)

b0 x


80

Explained and unexplained variation

y yi

•RSS = Residual sum of squares  RSS = (yi - yi )2

TSS = Total sum of squares _ TSS = (yi - y)2  y

 _2 ESS = (yi - y)

_ y

•ESS = Explained Sum of squares

Xi


 y

_ y

x

81

Explained and unexplained variation  Total variation is made up of two parts:

TSS  Total sum of Squares

TSS 

2 ( y  y ) 

RSS  Sum of Squares Error / Residual Sum of Squares

RSS 

2 ˆ ( y  y ) 

ESS Sum of Squares Regression / Explained Sum of Squares

ESS 

2 ˆ ( y  y ) 

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


82

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

SSR R  SST 2


where

2

0  R 1

83

Coefficient of determination, R2  Coefficient of determination

SSR sum of squares explained by regression R   SST total sum of squares 2

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

R2  r2  Where: • R2 = Coefficient of determination • r = Simple correlation coefficient


84

Calculating the correlation coefficient  Sample correlation coefficient:

r

 ( x  x )( y  y ) [  ( x  x ) ][  ( y  y ) 2

2

]

 or the algebraic equivalent:

r

n  xy   x  y

[n(  x 2 )  (  x ) 2 ][ n(  y 2 )  (  y ) 2 ]

 Where: • r = Sample correlation coefficient • n = Sample size • x = Value of the independent variable • y = Value of the dependent variable © EduPristine For FMP-I (Confidential)

85

Standard Error of “Estimate”  The standard deviation of the variation of observations around the regression line is estimated by:

su 

RSS n  k 1

 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 © EduPristine For FMP-I (Confidential)

86

The Standard Deviation of the intercept

 Xi su 2

s bo  s b1 

n  (x  x ) 2

su

 (x  x )

2

su



2 x  

(  x) 2 n

 Where: s b = Estimate of the standard error of the least squares slope • 1

•

su 

RSS n2

= Sample standard error of the estimate


87

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


88

Multiple Regression Basics  General Multiple Linear Regression model take the following form:

Yi  b0  b1 X 1i  b2 X 2 i  .........  bk X ki   i  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


89

Estimated Regression Equation  As we calculated the intercept and the slope coefficient in case of simple linear regression by minimizing the sum of squared errors, similarly we estimate the intercept and slope coefficient in multiple linear regression n

• Sum of Squared Errors  i i 1

2

is minimized and the slope coefficient is estimated.

 The resultant estimated equation becomes: 









Yi  b0  b1 X 1i  b 2 X 2 i  .........  b k X ki  Now the error in the ith observation can be written as:       i  Yi  Yi  Yi   b0  b1 X 1i  b2 X 2 i  .........  bk X ki    


90

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

 x n

Variance



i 1

i

x



2

n 1

 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 (equally weighted average) of previous ‘n’ days’ squared returns n

Variance



 x  i 1

i

2

n 1

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? © EduPristine For FMP-I (Confidential)

91

EWMA Model  In an exponentially weighted moving average model, the weights assigned to the u 2 decline exponentially as we move back through time  This leads to:  2n    2n  1  (1   ) u n2  1  Apply the recursive relationship:

 n2     n2 2  (1   ) u n2 2   (1   ) u n21  n2  (1   ) u n21   u n2 2    2 n2 2

 Hence we have



2 n

 m i 1 2   (1   )    u n  i    m   i 1 

2 nm

• Variance estimate for next day (n) is given by (1-λ) weight to recent squared return and λ weight to the previous variance estimate • Risk-metrics (by JP Morgan) assumes a Lambda of 0.94


92

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:  t21  (1  0.94)  t2  0.94 t2  In this equation, variance for time ‘t’ was also an estimate. So we can substitute for it as follows:

 t21  (1  0.94)  t2  0.94(1  0.94)  t21  0.94 t21 

 t21  0.06 * t2  0.94 * 0.06 * t21  (0.94 * 0.94 t21 )  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 (impact of older data is allowed to persist). Lower λ gives higher weight to recent data (i.e. previous data impacts 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

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 to a long-term Volatility rate over time.



2 t1

 V

L

 

2 t

 

2 t

Long-term average Volatility


94

GARCH (1,1)



2 t 1

      2 t

2 t

 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:

VL

  1   


95

Simulating a Price Path

S    t   S Drift

 S is the stock price,

t

 μ is the expected return,  σ is the standard deviation of returns,  "t" is time, and

Shock

 ε is the random variable

 The first step in simulating a price path is to choose a random process to model changes in financial assets  Stock prices and exchange rates are modeled by geometric Brownian motion (GBM) shown in the above equation


96

Valuations and Risk Models



Measuring Value-at-Risk (VAR) 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 -4

-2

VAR X % (in %)  Z X % * 

0

2

4

Mean = 0

 ZX% : the normal distribution value for the given probability (x%) (normal distribution has mean as 0 and standard 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:

VAR  VARX % (in %) * Asset Value  This can also be written as:

VAR  Z X % * * Asset Value © EduPristine For FMP-I (Confidential)

98

Measuring Value-at-Risk (VAR)  VAR for n days can be calculated from daily VAR as:

VaR (n days) (in %)  VaR (daily VaR) (in %) * n  This comes from the known fact that the n-period volatility equals 1-period volatility multiplied by the square root of number of periods(n).

VaR (n days) (in %)  ZX% * * Asset Value * n  As the volatility of the portfolio can be calculated from the following expression:

 portfolio  wa2  a2  w 2b  b2  2w a w b *  a *  b *  ab  The above written expression can also be extended to the calculation of VAR:

VaR portfolio (in %)  wa2 (%VARa ) 2  w 2b (%VARb ) 2  2w a w b * (%VARa ) * (%VARb ) *  ab © EduPristine For FMP-I (Confidential)

99

Expected Loss (EL)  EL = (Assured payment at Maturity Time T )* Loss Given Default * (Probability that the default occurs 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


100

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)


101

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


102

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 LGD and EDF terms will come into picture. Though it has been found that they will affect the result only slightly


103

FRM Formulas

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