FRM Formulas

FMP

FMP

Hedging in a practical world (Basis Risk)

• Basis = 0 when spot price = futures price

Future Price Spot Price

Choice of contracts

• Optimal Hedge Ratio:

 Where

• σSis the standard deviation of δS, the change in the spot price during the hedging period

• σFis 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  

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

 _{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*) A P β * N  f A Q N * h * N  A P β * N  A P ) -* ( * N   

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

 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

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 F0keeps 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)Tan arbitrageur buys the stocks underlying the index and sells futures

 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 F0keeps 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}

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 (r_f) in his case would be r_INR

T r r_{bc fc}

e

S

F



 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 (r_f) in his case would be r_INR

T r r INR USD INR

USD

S

e

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

 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

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)}

 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:

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

• 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 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:

YTM) (1 1 F YTM YTM) (1 1 1 I B n _n                   



 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

• 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 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 _{( /2) 100} 5

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)

1 2 1 1 2 2 t2 t1,

R

_T

T

_T

R

T

F







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

 Where:

• t = Respective time period

• C = Periodic coupon payment

• y = Periodic yield • n = Total no of periods • M = Maturity value price bond Current y M n y C t Duration Macaluay n n t t (1 ) * ) 1 ( * 1    



 _{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

 Where:

• t = Respective time period

• C = Periodic coupon payment

• y = Periodic yield

• n = Total no of periods

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: Y

D B

B _{ }



 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:

n r 1 Duration Macaulay D*  

Convexity

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

2 2 dy B d B 1 C 

 _{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 ($)

Yield Y*

P* Actual bond price

…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

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:













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 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

 _{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

Where rp_nis the risk premium associated with an n year bond

) 1 )...( 1 )( 1 ( ) 1 ( 1 2 yearn st year st year st n lt i i i i      ) 1 )...( 1 )( 1 ( ) 1

( ilt n  rpn  istyear1 istyear2 istyearn

 _{Three theories are used to explain the}

shape of the term structure

 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

 _{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

Where rp_nis the risk premium associated with an n year bond

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)

=

(Number of days in reference period) =

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):}

DV01 – Application to hedging

 _{Hedge ratio is calculated using DV01 with the help of following relation}

)

instrument

hedging

of

100

$

(

01

)

osition

initial

of

100

$

(

1

per

DV

p

per

DVO

HR



Duration based hedging strategies

 _{Number of contracts to hedge is given by the equation:}

• F_C Contract price for interest rate futures

• D_F Duration of asset underlying futures at maturity

• P Value of portfolio being hedged

• D_P Duration of portfolio at hedge maturity

F C P

D

F

PD

N

*



 _{Number of contracts to hedge is given by the equation:}

• F_C Contract price for interest rate futures

• D_F Duration of asset underlying futures at maturity

• P Value of portfolio being hedged

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}

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

Bounds and Option Values

Option Minimum Value Maximum Value

European call (c) c_t≥ Max(0,S_t-(X/(1+RFR)t_{) S} t

American Call (C) C_t≥ Max(0, S_t-(X/(1+RFR)t_{) S} t

European put (p) p_t≥Max(0,(X/(1+RFR)t_)-S

t) X/(1+RFR)t

American put (P) P_t≥ Max(0, (X-S_t)) X

Where t is the time to expiration Where t is the time to expiration

Binomial Method

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

S₀ S_u S_u2 S_ud IV₁= Max[(S_u2_{-X), 0]} IV₂ p p 1 - p S₀ Sud S_u S_d2 IV₂ IV₃ 1 - p 1 - p p

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:

• 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 =} T d d T T R XP d f           1 2 )] 5 . 0 ( [ ] ln[ 1 2 ] ) 2 ( [ ] ) 1 ( [N d P  N d  X eRfT

 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:

• 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 =} T d d T T R XP d f           1 2 )] 5 . 0 ( [ ] ln[ 1 2 ] )} 1 ( 1 [{ }] 2 ( 1 { [X eRfT  N d  N d P

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 i i portfolio  W  



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

• d₁and d₂are as defined in the Black-Scholes Pricing formula earlier

• σ = Stock price volatility

• K = Strike price

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

• r = Risk neutral rate of interest ) ( 2 ) (' ) ( 0 1 _{rKe N d 2} T d N S Call _  _ rT  



 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

• d₁and d₂are as defined in the Black-Scholes Pricing formula earlier

• σ = Stock price volatility

• K = Strike price

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

• r = Risk neutral rate of interest ) ( 2 ) (' ) ( 0 1 _{rKe N d 2} T d N S Put _  _ rT _   

Gamma (cont.)

 _{Calculation of Gamma}

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

• Where symbols have their usual meaning

T S d N  0 ) 1 ('  

Gamma (ATM) vs. Time

0.45

Gamma (Call / Put)

0.07 0 0.05 0 0.2 0.4 0.6 0.8 1.0 1.2 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 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

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:

 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 ('d e ( d1^ 2)/2 N   16 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:

 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 ('d e ( d1^ 2)/2 N   1 4 7 1013161922252831343740434649 0 4 6 8 10 12 14 16 2

Rho

 _{Rho of a portfolio of options is the rate of change of its value with respect to changes in the}

interest 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-rT_{N(d2), where the symbols carry their usual meanings}

• Also, Rho (put) = -KTe-rT_{N(-d2), the symbols carrying their usual meanings}

r

  

30 Rho (Call / Put)

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)

Valuation of swaps

 _{Hence the value of the swap can be given as:}

• V = B_fix – B_fl

• Where:

– B_fix= PV of payments – B_fl= (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)

Commodity Forwards

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



_S

_e

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

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

• 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

F

_0,T



_S

₀

_e

(r





)T

 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

• 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

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:

 Where δ is lease rate

F



_S

_e

 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:

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





_r



1

T

In (F

/ S)





_r



1

T

In (F

/ S)





(1



r)

(F

/ S)



1

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

F



_S

_e

_

_

_{(0,T )}

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 –

 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:

Pricing with convenience

 _{And if,} _{= c –}

 _{Then, F0,T = S0e}(r+-c)T

F



_S

_e

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 engaging}

in arbitrage

S

₀

e

(r







_{c )T}

_

_F

0,T



S

0

e

(r





)T

 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 engaging}

Interest rate parity

 Interest Rate Parity (IRP)

 _{Where; rDC = Domestic currency rate}

r_FC = Foreign currency rate

T FC DC

r

r

Spot

Forward

_

















1

1

 _{Where; rDC = Domestic currency rate}

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 outstandingCumulative $ 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

Foundation of Risk Management

Foundation of Risk Management

Expected Return and Standard Deviation of Portfolio

 _{Standard Deviation of Portfolio}





k

1

to

N

R

W

R













W

σ





W

W

σ

σ

P



k



1

to

N;

i



1

to

N;

k



i











Portfolio Variance for two asset portfolio

• Var(w_Ak_A+ w_Bk_B) = w_A2_σ

A2+ wB2σB2+ 2 wAwB σAσB ρAB

 Where ρ is correlation coefficient between A and B

 w_A,w_B 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

 For two-asset portfolio

• Var(w_Ak_A+ w_Bk_B) = w_A2_σ

A2+ wB2σB2+ 2 wAwB σAσB ρAB

 Where ρ is correlation coefficient between A and B

 w_A,w_B 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

Correlation and Portfolio Diversification

• ρ =1 & Var (w_Ak_A+ w_Bk_B)= (w_Aσ_A+ w_Bσ_B)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

 _{Perfect Positive Correlation}

• ρ =1 & Var (w_Ak_A+ w_Bk_B)= (w_Aσ_A+ w_Bσ_B)2

 _{Perfect Negative Correlation}

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

 _{Zero Correlation}

• Correlation between two assets is zero

 _{Moderate Positive Correlation}

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





Risk Free Asset Introduced

Rf

Efficient Frontier CML

Return

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





R

β

R

R

R











 

cov

_Var

R

_R

,

R

β



Relaxing Assumptions of CAPM

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





Beta

 _{Sensitivity of the return of the asset to the market return is known as Beta}  Beta is calculated as

follows:-



 

i

cov

_Var

R

_R

,

R

β



Portfolio Beta

 Beta can also be calculated for portfolio

Beta

 _{Sharpe ratio:} Sharpe ratio

• R_p= portfolio return, Rf = risk free return

• The higher the Sharpe measure, the better the portfolio





• 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





• 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





• R_p= portfolio return, R_c= return predicted by CAPM

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

c

p R

R α  

Measures of performance

(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 Deviation from MAR where R <MAR

P

E TE   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 Deviation from MAR where R <MAR









 

_





Quantitative Analysis

Quantitative Analysis

Counting Principle

 Number of ways of selecting r objects out of n objects

 n_C r

 n!/(r!)*(n-r)!

 _{Number of ways of giving r objects to n people, such that repetition is allowed}  Nr

Some definitions and properties of Probability

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

]

1

,

0

[

)

(

_A



P

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

)

(

1

)

(

A

P

A

P







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:

)

(

)

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(

)

(

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(

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(

)

(

)

(

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_B

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P

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(

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B

P

A

P

A

B

P

B

A

P



 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:

(

)

)

(

*

)

/

(

)

/

(

B

P

A

P

A

B

P

B

A

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



Mean

 _{The expected value(Mean) measures the central tendency, or the center of gravity of the}

population

 _{It is given by:}

 The graph shows the mean of normal distributions

N x X E n i i



Standard Normal Distribution

= 0,= 2 = 1,= 1 0 2 4 -4 -2 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0

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

n n

X

X

X

X

G









...

Properties of Expectation

 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]

 E(cX) = E(X) x c

 E(X+Y) = E(X) + E(Y)

 _{E(X2) ≠ [E(X)]2}

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

x

VAR







)

(



 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) = _{VAR (x)}

N

x









)

(





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 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 Y X Y X XY E Y X Cov Y X E Y X Cov          ) ( ) , ( )] )( [( ) , (

 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 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

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

)

(

)

(

_aX

_b

_a

_Var

_X

Var











X

Var

X

Var

)

(

)

(





 Covariance between same variables is also their variance

 _{For independent or uncorrelated variables,}

• covariance or correlation = 0







X

Var

X

Var

)

(

)

(

)

,

(

2

)

(

)

(

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_bY

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_X

_b

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_Y

_abCov

_X

_Y

Skewness

 Skewness describes departures from symmetry

 _{Skewness can be negative or positive.}

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

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

3 1 3

)

(











x

S

 _{Skewness can be negative or positive.}

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

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

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 hence}

create large kurtosis. Such a distribution is called leptokurtic, or fat tailed

 _{A kurtosis of 3 is considered average}

 High kurtosis indicates a higher probability of extreme movements 4 1 4 ) (







 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 hence}

create large kurtosis. Such a distribution is called leptokurtic, or fat tailed

 _{A kurtosis of 3 is considered average}

 High kurtosis indicates a higher probability of extreme movements 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 Platykurtic K<3 Leptokurtic K>3 Mesokurtic K=3

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 H0is True H0is False H0is True Correct Decision Confidence Level = 1-α Type-II Error P(Type-II Error) = β H0is False Type-I Error Significance Level = α Power=1-β  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

Hypothesis tests for variances

Hypothesis Test of Variances Hypothesis Test of Variances

Test for

Single Population VarianceTest for

Single Population Variance Two Population VariancesTwo Population VariancesTest forTest for

Example Hypothesis Example Hypothesis H₀: σ2 _{= σ} 02 H_A: σ2 _{≠ σ} 02 Chi-Square Test Statistic Chi-Square Test Statistic 2 0 2 2 ) 1 (, ( _1) _{ n}  n s Example Hypothesis Example Hypothesis H₀: σ₁2 _{– σ} 22 = 0 H_A: σ₁2 _{– σ} 22≠ 0 F-test Statistic F-test Statistic 2 2 2 1 , ,

_s

s

F



Test for single population variance

different forms:

• Two Tailed Test:

• Lower Tail test:

• Upper Tail Test

H₀: σ2 _{= σ} 02 H_A: σ2 _{≠ σ} 02 H₀: σ2 _{ σ} 02 H_A: σ2 _{< σ} 02  /2 /2

 Single population variance test has 3 different forms:

• Two Tailed Test:

• Lower Tail test:

• Upper Tail Test

H₀: σ2 _{ σ} 02 H_A: σ2 _{< σ} 02 H₀: σ2 _{≤ σ} 02 H_A: σ2 _{> σ} 02 

Chi-square test statistic

 The chi-squared test statistic for a Single Population Variance is:

Where



n = sample size s2 _{= sample variance} σ2 _{= hypothesized variance} 2 2 2

σ

1)s

(n







 The chi-squared test statistic for a Single Population Variance is:

Where



n = sample size

s2 _{= sample variance}

Finding the critical value

, is found from the chi-square table:



H₀: σ2 _{≤ σ} 02

H_A: σ2 _{> σ} 02

Upper tail test:

2

Do not reject H₀ Reject H₀



2 

Lower tail or two tailed Chi-square tests

Lower tail test: Two tail test:

Do not reject H₀ Reject 2 1- 2 2 /2 Do not reject H₀ Reject 2 1-/2 2 Reject

F-test for difference in two population variances

different forms:

• Two Tailed Test:

• Lower Tail test:

• Upper Tail Test

H₀: σ₁2 _{– σ} 22 = 0 H_A: σ₁2 _{– σ} 22≠ 0 H₀: σ₁2 _{– σ} 22  0 H_A: σ₁2 _{– σ} 22< 0  /2 /2

 Two population variance test has 3 different forms:

• Two Tailed Test:

• Lower Tail test:

• Upper Tail Test

H₀: σ₁2 _{– σ} 22  0 H_A: σ₁2 _{– σ} 22< 0 H₀: σ₁2 _{– σ} 22 ≤ 0 H_A: σ₁2 _{– σ} 22> 0 

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

= Variance of Sample 1

(n1 – 1) = numerator degrees of freedom = Variance of Sample 2

(n2 – 1) = denominator degrees of freedom

2 1

s

s

s

F



 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

2 1

s

s

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.

Population linear regression

Random Error for this x value Y Observed Value of Y for X_i Predicted Value of Y for X_i Slope = β₁

u

X

Y











Mean marks for hours of study Individual person’s marks

Random Error for this x value Predicted Value of Y for X_i x_i Intercept = β₀ ui x

Population regression function

Variable IndependentVariable

u

X

β

β

Y



₀



₁



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

Linear component Random Error

component

u

X

β

β

Y



₀



₁



Sample regression function

Random Error for this x value Y Observed Value of Y for X_i Predicted Value of Y for X_i Slope = β₁

e

x

b

b

y







Random Error for this x value Predicted Value of Y for X_i x_i Intercept = β₀ x

Sample regression function

Estimate of the regression

intercept Error term

Estimated (or predicted) y value Estimate of the regression slope Independent variable

e

x

b

b

y

_i



₀



₁



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

e

x

b

b

One method to find b

and b

 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 1 0 2 2

x))

b

(b

(y

)

yˆ

(y

e









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

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x))

b

(b

(y

)

yˆ

(y

e

















OLS regression properties

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

 The sum of the squared residuals is a minimum Minimize ( )

 _{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

0

)

ˆ

(







y

y



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

 The sum of the squared residuals is a minimum Minimize ( )

 _{The simple regression line always passes through the mean of the y variable and the mean}

of the x variable

The least squares equation





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(

x

x

y

y

x

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b



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n

x

x

n

y

x

xy

b

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x

b

y

b

₀

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

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n

x

x

n

y

x

xy

b

₍

₎

x

b

y

b

₀





₁

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

X

b

b

Y





slope of the line(b₁)

x

b₀

Explained and unexplained variation

•RSS = Residual sum of squares

_ _ TSS = Total sum of squares RSS = (yi-yi)2 TSS = (y_i- y)2 Xi x y _ y _ y  ESS = (yi- y)2  _