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Derivatives Interest Rate Futures Professor André Farber Solvay Brussels School of Economics and Management Université Libre de Bruxelles

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Derivatives 2011 04 IR & GovBonds Futures.pptProfessor André
Farber Solvay Brussels School of Economics and Management
Université Libre de Bruxelles

21 January 2012 Derivatives 04 IR Derivatives |2

Interest Rate Derivatives

• Forward rate agreement (FRA): OTC contract that allows the user to "lock in" the current forward rate.

• Treasury Bill futures: a futures contract on 90 days Treasury Bills

• Interest Rate Futures (IRF): exchange traded futures contract for which the underlying interest rate (Dollar LIBOR, Euribor,..) has a maturity of 3 months

• Government bonds futures: exchange traded futures contracts for which the underlying instrument is a government bond.

• Interest Rate swaps: OTC contract used to convert exposure from fixed to floating or vice versa.

21 January 2012 Derivatives 04 IR Derivatives |3

Term deposit as a forward on a zero-coupon

0 T = 0.50 T* = 0.75

M = 100

τ = 0.25

M (1+RS × τ)

100(1+6%× 0.25) = 101.50

Profit at time T* = [M(RS – rS)τ ] = [100 (6% - rS) 0.25] Profit at time T = [M(RS – rS)τ ] / (1 + rS τ)

21 January 2012 Derivatives 04 IR Derivatives |4

FRA (Forward rate agreement)

• OTC contract • Buyer committed to pay fixed interest rate Rfra • Seller committed to pay variable interest rate rs

• on notional amount M • for a given time period (contract period) τ • at a future date (settlement date or reference date) T

• Cash settlement at time T of the difference between present values • CFfra = M[ (rS – Rfra) τ] / (1+rS τ)

• Long position on FRA equivalent to cash settlement of result on forward loan (opposite of forward deposit)

• An FRA is an elementary swap

21 January 2012 Derivatives 04 IR Derivatives |5

Hedging with a FRA

• Cy X wishes to set today 1/3/20X0 • the borrowing rate on $ 100 mio • from 1/9/20X0 (=T) to 31/8/20X1 (1 year)

• Buys a 7 x 12 FRA with R=6% • Settlement date 1/9/20X0 • Notional amount : $ 100 m • Interest calculated on 1-year period

• Cash flows for buyer of FRA • 1) On settlement date r=8% r = 4%

Settlement : 100 x (8% - 6%) / 1.08 100 x (4% - 6%) / 1.04 = + 1.852 = - 1.923 Interest on loan: - 8.00 -4.00 FV(settlement) +2.00 -2.00 TOTAL - 6.00 -6.00

21 January 2012 Derivatives 04 IR Derivatives |6

Treasury bill futures

• Underlying asset 90-days TB • Nominal value USD 1 million • Maturities March, June, September, December • TB Quotation (n days to maturity)

– Discount rate y% – Cash price calculation: St = 100 - y × (n/360 ) – Example : If TB yield 90 days = 3.50%

• St = 100 - 3.50 × (90/360) = 99.125 • TB futures quotation:

• Ft = 100 - TB yield

Example : Buying a June TB futures contract quoted 96.83

• Being long on this contract means that you buy forward the underlying TBill at an implicit TB yield yt =100% - 96.83% = 3.17% set today.

• The delivery price set initially is: K = M (100 - yt×τ)/100 = 1,000,000 [100 - 3.17 × (90/360)]/100 = 992,075

• If, at maturity, yT = 4% (⇔FT = 96)

• The spot price of the underlying asset is: ST = M (100 - yT×τ)/100 = 1,000,000 [100 - 4.00 ×(90/360)]/100 = 990,000 • Profit at maturity: fT = ST - K = - 2,075

21 January 2012 Derivatives 04 IR Derivatives |8

TB Futures: Alternative profit calculation

• As forward yield is yt = 100 - Ft yield at maturity yT = 100 - FT = 100 - ST profit fT = ST - K = M (100 - yT×τ)/100 - M (100 - yt×τ)/100

profit can be calculated as: fT = M [(FT - Ft)/100] τ

• Define : TICK ≡ M ×τ ×(0.01/100) Cash flow for the buyer of a futures for ΔF = 1 basis point (0.01%) For TB futures: TICK = 1,000,000 × (90/360) ×(0.01/100) = $25 • Profit calculation:

Profit fT = ΔF × TICK ΔF in bp In our example :ΔF = 96.00 - 96.83 = - 83 bp fT = -83 × 25 = - 2,075

fT =1,000, 000! 96.00" 96.83

100 #

3 Month Euribor (LIFFE) Euro 1,000,000

Settle Open int.

Est vol 1,216,471; open int 4,171,204

21 January 2012 Derivatives 04 IR Derivatives |10

21 January 2012 Derivatives 04 IR Derivatives |11

21 January 2012 Derivatives 04 IR Derivatives |12

Interest rate futures vs TB Futures

• 3-month Eurodollar (IMM & LIFFE) • 3-month Euribor (LIFFE)

• Similar to TB futures è Quotation Ft = 100 - yt

with yt = underlying interest rate è TICK = M ×τ ×(0.01/100) è Profit fT = ΔF × TICK • But: • TB futures Price converges to the price of a 90-day TB TB delivered if contract held to maturity • IRF Cash settlement based on final contract price:

• 100(1-rT) with rT underlying interest rate at maturity

21 January 2012 Derivatives 04 IR Derivatives |13

IRF versus FRA

• Consider someone taking a long position at time t on an interest rate future maturing at time T.

• Ignore marking to market. • Define : R : implicit interest rate in futures quotation Ft

R = (100 – Ft) / 100 • r : underlying 3-month interest rate at maturity

rT = (100 – FT) / 100

100 ×

21 January 2012 Derivatives 04 IR Derivatives |14

Hedging with an IRF

• A Belgian company decides to hedge 3-month future loan of €50 mio from June to September using the Euribor futures contract traded on Liffe.

• The company SHORTS 50 contracts. Why ? • Interest rate # Interest rate$ • Short futures F$ ΔF <0 Gain F$ ΔF>0 Loss • Loan Loss Gain

• F0 = 94.05 => R = 5.95% • Nominal value per contract = € 1 mio • Tick = €25 (for on bp)

21 January 2012 Derivatives 04 IR Derivatives |15

Checking the effectiveness of the hedge

rT 5% 6% 7%

FT 95 94 93

CF/contract -2,375 +125 +2,625

Interest -625,000 -750,000 -875,000

Short 50 IRF, F0 = 94.05, Tick = €25 (for one bp)

21 January 2012 Derivatives 04 IR Derivatives |16

A further complication: Tailing the hedge

• There is a mismatch between the timing of the interest payment (September) and of the cash flows on the short futures position (June).

• Net borrowing = $50,000 – Futures profit • Total Debt Payment = Net borrowing × (1+r × 3/12) • Effective Rate = [(Total Debt Payment/50,000,000)-1] × (12/3) • €X in June is equivalent to €X(1+rτ) in September. • So we should adjust the number of contracts to take this into account. • However, r is not known today (in March). • As an approximation use the implied yield from the futures price. • Trade 100/(1+5.95% x 3/12) = 98.53 contracts

21 January 2012 Derivatives 04 IR Derivatives |17

GOVERNMENT BOND FUTURES

• Example: Euro-Bund Futures • Underlying asset: Notional bond • Maturity: 8.5 – 10.5 years • Interest rate: 6% • Contract size: € 100,000 • Maturities: March, June, September, December • Quotation: % (as for bonds) - • Clean price (see below) • Minimum price movement: 1 BASIS POINT (0,01 %) • 100,000 x (0,01/100) = € 10 • Delivery: see below

21 January 2012 Derivatives 04 IR Derivatives |18

Example: Euro-BUND Futures (FGBL)

• Contract Standard A notional long-term debt instrument issued by the German Federal Government with a term of 8½ to 10½ years and an interest rate of 6 percent. Contract Size : EUR 100,000 Settlement A delivery obligation arising out of a short position in a Euro-BUND Futures contract may only be satisfied by the delivery of specific debt securities - namely, German Federal Bonds (Bundesanleihen) with a remaining term upon delivery of 8½ to 10½ years. The debt securities must have a minimum issue amount of DEM 4 billion or, in the case of new issues as of 1.1.1999, 2 billion euros.

• Quotation :In a percentage of the par value, carried out two decimal places.

• Minimum Price Movement :0.01 percent, representing a value of EUR 10. Delivery Day The 10th calendar day of the respective delivery month, if this day is an exchange trading day; otherwise, the immediately following exchange trading day. Delivery Months The three successive months within the cycle March, June, September and December. Notification Clearing Members with open short positions must notify Eurex which debt instruments they will deliver, with such notification being given by the end of the Post-Trading Period on the last trading day in the delivery month of the futures contract.

21 January 2012 Derivatives 04 IR Derivatives |19

Time scale

Quotation

• Spot price Cash price = Quoted price + Accrued interest Example: 8% bond with 10.5 years to maturity

(⇒ 0.5 years since last coupon) Quoted price : 105 Accrued interest : 8 × 0.5 = 4 Cash price : 105 + 4 = 109

• Forward price: Use general formula with S = cash price If no coupon payment before maturity of

forward, cash forward Fcash = FV(Scash) If coupon payment before maturity of forward,

cash forward Fcash = FV(Scash -I) where I is the PV at time t of the next

coupon Quoted forward price Fquoted : Fquoted = Fcash - Accrued interest

21 January 2012 Derivatives 04 IR Derivatives |21

Quotation: Example

• 8% Bond, Quoted price: 105 • Time since last coupon: • 6 months • Time to next coupon : • 6 months (0.5 year) • Maturity of forward: • 9 months (0.75 year) • Continuous interest rate: 6%

• Cash spot price : 105 + 8 × 0.5 = 109

• PV of next coupon : 8 × exp(6% × 0.5) = 7,76

• Cash forward price : • (109 - 7.76) e(6% × 0.75) = 105.90 • Accrued interest : • 8 × 0.25 = 2 • Quoted forward price: • 105.90 - 2 = 103.90

21 January 2012 Derivatives 04 IR Derivatives |22

21 January 2012 Derivatives 04 IR Derivatives |23

21 January 2012 Derivatives 04 IR Derivatives |24

21 January 2012 Derivatives 04 IR Derivatives |25

Delivery:

• Government bond futures based on a notional bond • In case of delivery, the short can choose the bonds to deliver from a list of

deliverable bonds ("gisement") • The amount that he will receive is adjusted by a conversion factor • INVOICE PRICE

– = Invoice Principal Amount – + Accrued interest of the delivered bond

• INVOICE PRINCIPAL AMOUNT – = Conversion factor x FT x 100,000

21 January 2012 Derivatives 04 IR Derivatives |26

Conversion factor: Definition

• price per unit of face value of a bond with annual coupon C • n coupons still to be paid • Yield = 6% • n : number of coupons still to be paid at maturity of forward T • f : time (years) since last coupon at time T

21 January 2012 Derivatives 04 IR Derivatives |27

Conversion factor: Calculation

• Step 1: calculate bond value at time T-f (date of last coupon payment before futures maturity): BT-f =PV of coupon + PV of principal : (C/y)[1-(1+y)-n] + (1+y)-n

• Step 2: Conversion factor k = bond value at time T : • k = FV(BT-f) - Accrued interest = BT-f (1+y)f - C× f

21 January 2012 Derivatives 04 IR Derivatives |28

21 January 2012 Derivatives 04 IR Derivatives |29

Cheapest-to-deliver Bond

• The party with the short position decides which bond to deliver: Receives: FT × kj + AcIntj

=(Quoted futures price) × (Conversion factor) + Accrued int. Cost = cost of bond delivered: sj + AcIntj

= Quoted price + Accrued interest • To maximize his profit, he will choose the bond j for which: Max (FT × kj - sj) or Min (sj - FT × kj) j j • Before maturity of futures contract: CTD= Max (F × kj - sj) or Min (sj - F × kj) j j

21 January 2012 Derivatives 04 IR Derivatives |30

• Duration of a bond that provides cash flow c i at time t i is

where B is its price and y is its yield (continuously compounded)

• This leads to

= − Δ

Duration

Duration Continued

• When the yield y is expressed with compounding m times per year • The expression

is referred to as the “modified duration”

1 BD yB y m Δ

Δ = − +

Convexity

2 2

1 2

BC B y B

∂ ∂

−

== =

Duration Matching

• This involves hedging against interest rate risk by matching the durations of assets and liabilities

• It provides protection against small parallel shifts in the zero curve

M Sn N F

FF FD yΔ = − Δ

Source: Dura+on hedge.xls

21 January 2012 Derivatives 04 IR Derivatives |2

Interest Rate Derivatives

• Forward rate agreement (FRA): OTC contract that allows the user to "lock in" the current forward rate.

• Treasury Bill futures: a futures contract on 90 days Treasury Bills

• Interest Rate Futures (IRF): exchange traded futures contract for which the underlying interest rate (Dollar LIBOR, Euribor,..) has a maturity of 3 months

• Government bonds futures: exchange traded futures contracts for which the underlying instrument is a government bond.

• Interest Rate swaps: OTC contract used to convert exposure from fixed to floating or vice versa.

21 January 2012 Derivatives 04 IR Derivatives |3

Term deposit as a forward on a zero-coupon

0 T = 0.50 T* = 0.75

M = 100

τ = 0.25

M (1+RS × τ)

100(1+6%× 0.25) = 101.50

Profit at time T* = [M(RS – rS)τ ] = [100 (6% - rS) 0.25] Profit at time T = [M(RS – rS)τ ] / (1 + rS τ)

21 January 2012 Derivatives 04 IR Derivatives |4

FRA (Forward rate agreement)

• OTC contract • Buyer committed to pay fixed interest rate Rfra • Seller committed to pay variable interest rate rs

• on notional amount M • for a given time period (contract period) τ • at a future date (settlement date or reference date) T

• Cash settlement at time T of the difference between present values • CFfra = M[ (rS – Rfra) τ] / (1+rS τ)

• Long position on FRA equivalent to cash settlement of result on forward loan (opposite of forward deposit)

• An FRA is an elementary swap

21 January 2012 Derivatives 04 IR Derivatives |5

Hedging with a FRA

• Cy X wishes to set today 1/3/20X0 • the borrowing rate on $ 100 mio • from 1/9/20X0 (=T) to 31/8/20X1 (1 year)

• Buys a 7 x 12 FRA with R=6% • Settlement date 1/9/20X0 • Notional amount : $ 100 m • Interest calculated on 1-year period

• Cash flows for buyer of FRA • 1) On settlement date r=8% r = 4%

Settlement : 100 x (8% - 6%) / 1.08 100 x (4% - 6%) / 1.04 = + 1.852 = - 1.923 Interest on loan: - 8.00 -4.00 FV(settlement) +2.00 -2.00 TOTAL - 6.00 -6.00

21 January 2012 Derivatives 04 IR Derivatives |6

Treasury bill futures

• Underlying asset 90-days TB • Nominal value USD 1 million • Maturities March, June, September, December • TB Quotation (n days to maturity)

– Discount rate y% – Cash price calculation: St = 100 - y × (n/360 ) – Example : If TB yield 90 days = 3.50%

• St = 100 - 3.50 × (90/360) = 99.125 • TB futures quotation:

• Ft = 100 - TB yield

Example : Buying a June TB futures contract quoted 96.83

• Being long on this contract means that you buy forward the underlying TBill at an implicit TB yield yt =100% - 96.83% = 3.17% set today.

• The delivery price set initially is: K = M (100 - yt×τ)/100 = 1,000,000 [100 - 3.17 × (90/360)]/100 = 992,075

• If, at maturity, yT = 4% (⇔FT = 96)

• The spot price of the underlying asset is: ST = M (100 - yT×τ)/100 = 1,000,000 [100 - 4.00 ×(90/360)]/100 = 990,000 • Profit at maturity: fT = ST - K = - 2,075

21 January 2012 Derivatives 04 IR Derivatives |8

TB Futures: Alternative profit calculation

• As forward yield is yt = 100 - Ft yield at maturity yT = 100 - FT = 100 - ST profit fT = ST - K = M (100 - yT×τ)/100 - M (100 - yt×τ)/100

profit can be calculated as: fT = M [(FT - Ft)/100] τ

• Define : TICK ≡ M ×τ ×(0.01/100) Cash flow for the buyer of a futures for ΔF = 1 basis point (0.01%) For TB futures: TICK = 1,000,000 × (90/360) ×(0.01/100) = $25 • Profit calculation:

Profit fT = ΔF × TICK ΔF in bp In our example :ΔF = 96.00 - 96.83 = - 83 bp fT = -83 × 25 = - 2,075

fT =1,000, 000! 96.00" 96.83

100 #

3 Month Euribor (LIFFE) Euro 1,000,000

Settle Open int.

Est vol 1,216,471; open int 4,171,204

21 January 2012 Derivatives 04 IR Derivatives |10

21 January 2012 Derivatives 04 IR Derivatives |11

21 January 2012 Derivatives 04 IR Derivatives |12

Interest rate futures vs TB Futures

• 3-month Eurodollar (IMM & LIFFE) • 3-month Euribor (LIFFE)

• Similar to TB futures è Quotation Ft = 100 - yt

with yt = underlying interest rate è TICK = M ×τ ×(0.01/100) è Profit fT = ΔF × TICK • But: • TB futures Price converges to the price of a 90-day TB TB delivered if contract held to maturity • IRF Cash settlement based on final contract price:

• 100(1-rT) with rT underlying interest rate at maturity

21 January 2012 Derivatives 04 IR Derivatives |13

IRF versus FRA

• Consider someone taking a long position at time t on an interest rate future maturing at time T.

• Ignore marking to market. • Define : R : implicit interest rate in futures quotation Ft

R = (100 – Ft) / 100 • r : underlying 3-month interest rate at maturity

rT = (100 – FT) / 100

100 ×

21 January 2012 Derivatives 04 IR Derivatives |14

Hedging with an IRF

• A Belgian company decides to hedge 3-month future loan of €50 mio from June to September using the Euribor futures contract traded on Liffe.

• The company SHORTS 50 contracts. Why ? • Interest rate # Interest rate$ • Short futures F$ ΔF <0 Gain F$ ΔF>0 Loss • Loan Loss Gain

• F0 = 94.05 => R = 5.95% • Nominal value per contract = € 1 mio • Tick = €25 (for on bp)

21 January 2012 Derivatives 04 IR Derivatives |15

Checking the effectiveness of the hedge

rT 5% 6% 7%

FT 95 94 93

CF/contract -2,375 +125 +2,625

Interest -625,000 -750,000 -875,000

Short 50 IRF, F0 = 94.05, Tick = €25 (for one bp)

21 January 2012 Derivatives 04 IR Derivatives |16

A further complication: Tailing the hedge

• There is a mismatch between the timing of the interest payment (September) and of the cash flows on the short futures position (June).

• Net borrowing = $50,000 – Futures profit • Total Debt Payment = Net borrowing × (1+r × 3/12) • Effective Rate = [(Total Debt Payment/50,000,000)-1] × (12/3) • €X in June is equivalent to €X(1+rτ) in September. • So we should adjust the number of contracts to take this into account. • However, r is not known today (in March). • As an approximation use the implied yield from the futures price. • Trade 100/(1+5.95% x 3/12) = 98.53 contracts

21 January 2012 Derivatives 04 IR Derivatives |17

GOVERNMENT BOND FUTURES

• Example: Euro-Bund Futures • Underlying asset: Notional bond • Maturity: 8.5 – 10.5 years • Interest rate: 6% • Contract size: € 100,000 • Maturities: March, June, September, December • Quotation: % (as for bonds) - • Clean price (see below) • Minimum price movement: 1 BASIS POINT (0,01 %) • 100,000 x (0,01/100) = € 10 • Delivery: see below

21 January 2012 Derivatives 04 IR Derivatives |18

Example: Euro-BUND Futures (FGBL)

• Contract Standard A notional long-term debt instrument issued by the German Federal Government with a term of 8½ to 10½ years and an interest rate of 6 percent. Contract Size : EUR 100,000 Settlement A delivery obligation arising out of a short position in a Euro-BUND Futures contract may only be satisfied by the delivery of specific debt securities - namely, German Federal Bonds (Bundesanleihen) with a remaining term upon delivery of 8½ to 10½ years. The debt securities must have a minimum issue amount of DEM 4 billion or, in the case of new issues as of 1.1.1999, 2 billion euros.

• Quotation :In a percentage of the par value, carried out two decimal places.

• Minimum Price Movement :0.01 percent, representing a value of EUR 10. Delivery Day The 10th calendar day of the respective delivery month, if this day is an exchange trading day; otherwise, the immediately following exchange trading day. Delivery Months The three successive months within the cycle March, June, September and December. Notification Clearing Members with open short positions must notify Eurex which debt instruments they will deliver, with such notification being given by the end of the Post-Trading Period on the last trading day in the delivery month of the futures contract.

21 January 2012 Derivatives 04 IR Derivatives |19

Time scale

Quotation

• Spot price Cash price = Quoted price + Accrued interest Example: 8% bond with 10.5 years to maturity

(⇒ 0.5 years since last coupon) Quoted price : 105 Accrued interest : 8 × 0.5 = 4 Cash price : 105 + 4 = 109

• Forward price: Use general formula with S = cash price If no coupon payment before maturity of

forward, cash forward Fcash = FV(Scash) If coupon payment before maturity of forward,

cash forward Fcash = FV(Scash -I) where I is the PV at time t of the next

coupon Quoted forward price Fquoted : Fquoted = Fcash - Accrued interest

21 January 2012 Derivatives 04 IR Derivatives |21

Quotation: Example

• 8% Bond, Quoted price: 105 • Time since last coupon: • 6 months • Time to next coupon : • 6 months (0.5 year) • Maturity of forward: • 9 months (0.75 year) • Continuous interest rate: 6%

• Cash spot price : 105 + 8 × 0.5 = 109

• PV of next coupon : 8 × exp(6% × 0.5) = 7,76

• Cash forward price : • (109 - 7.76) e(6% × 0.75) = 105.90 • Accrued interest : • 8 × 0.25 = 2 • Quoted forward price: • 105.90 - 2 = 103.90

21 January 2012 Derivatives 04 IR Derivatives |22

21 January 2012 Derivatives 04 IR Derivatives |23

21 January 2012 Derivatives 04 IR Derivatives |24

21 January 2012 Derivatives 04 IR Derivatives |25

Delivery:

• Government bond futures based on a notional bond • In case of delivery, the short can choose the bonds to deliver from a list of

deliverable bonds ("gisement") • The amount that he will receive is adjusted by a conversion factor • INVOICE PRICE

– = Invoice Principal Amount – + Accrued interest of the delivered bond

• INVOICE PRINCIPAL AMOUNT – = Conversion factor x FT x 100,000

21 January 2012 Derivatives 04 IR Derivatives |26

Conversion factor: Definition

• price per unit of face value of a bond with annual coupon C • n coupons still to be paid • Yield = 6% • n : number of coupons still to be paid at maturity of forward T • f : time (years) since last coupon at time T

21 January 2012 Derivatives 04 IR Derivatives |27

Conversion factor: Calculation

• Step 1: calculate bond value at time T-f (date of last coupon payment before futures maturity): BT-f =PV of coupon + PV of principal : (C/y)[1-(1+y)-n] + (1+y)-n

• Step 2: Conversion factor k = bond value at time T : • k = FV(BT-f) - Accrued interest = BT-f (1+y)f - C× f

21 January 2012 Derivatives 04 IR Derivatives |28

21 January 2012 Derivatives 04 IR Derivatives |29

Cheapest-to-deliver Bond

• The party with the short position decides which bond to deliver: Receives: FT × kj + AcIntj

=(Quoted futures price) × (Conversion factor) + Accrued int. Cost = cost of bond delivered: sj + AcIntj

= Quoted price + Accrued interest • To maximize his profit, he will choose the bond j for which: Max (FT × kj - sj) or Min (sj - FT × kj) j j • Before maturity of futures contract: CTD= Max (F × kj - sj) or Min (sj - F × kj) j j

21 January 2012 Derivatives 04 IR Derivatives |30

• Duration of a bond that provides cash flow c i at time t i is

where B is its price and y is its yield (continuously compounded)

• This leads to

= − Δ

Duration

Duration Continued

• When the yield y is expressed with compounding m times per year • The expression

is referred to as the “modified duration”

1 BD yB y m Δ

Δ = − +

Convexity

2 2

1 2

BC B y B

∂ ∂

−

== =

Duration Matching

• This involves hedging against interest rate risk by matching the durations of assets and liabilities

• It provides protection against small parallel shifts in the zero curve

M Sn N F

FF FD yΔ = − Δ

Source: Dura+on hedge.xls

of 34/34

Derivatives Interest Rate Futures Professor André Farber Solvay Brussels School of Economics and Management Université Libre de Bruxelles

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