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
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
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
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
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 )(
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
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)
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 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
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
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*
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
…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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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
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)=
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
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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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
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
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
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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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
27
)(2
)(')( 210 dNrKe
TdNSPut rT
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
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
29
2)1('
2/)2^1( dedN
1 4 7 10131619222528313437404346490
468
10121416
2
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)
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
32
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
33
F0,T S0e(r )T
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
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 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
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 )
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 –
37
Pricing with convenience
So, if:
And if, = c –
Then, F0,T = S0e(r+ -c)T
F0 ,T S0e( r )T
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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
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
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
Foundation of Risk Management
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Foundation of Risk Management
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
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
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
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
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
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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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
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
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
Quantitative Analysis
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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
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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
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
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
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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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
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)(
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
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
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
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
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
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
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
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
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
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
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
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
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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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
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
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
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
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
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
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
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)
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
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
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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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
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
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
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
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
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
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
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?
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
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’
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)94.0*94.0(*06.0*94.0*06.0 21
21
221 tttt
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.
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2221 ttLt V
Long-term average Volatility
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
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
Valuations and Risk Models
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Valuations and Risk Models
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 **%
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:
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***w2w w abbaba2b
2b
2a
2aportfolio w
*)(%VAR*)(%VAR*w2w)(%VAR w)(%VAR%)(inVaR abbaba2
b2b
2a
2aportfolio w
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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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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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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