# CS代写 MATH3090/7039: Financial mathematics Lecture 5 – cscodehelp代写

MATH3090/7039: Financial mathematics Lecture 5

Interest rate swaps

Credit default swaps

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Interest rate swaps

Credit default swaps

A swap is a financial contract to exchange cashflow obligations or finanical exposure from one basis to another:

• Swap between fixed and floating interest rates.

• Swap between payments denominated in different currencies. • Swap or alter credit risk exposures.

Almost any set of cashflows or exposures can be swapped between market participants. We restrict ourselves to:

• Fixed-for-floating (vanilla) interest rate swaps. • Credit default swaps.

Interest rate swaps

A fixed-for-floating or plain-vanilla interest rate swap (IRS) involves: • One party borrowing at a fixed rate, but desires floating.

• Another party borrowing at a floating rate, but desires fixed.

• They are matched by a swap dealer to swap their yield exposures.

IRS: comparative advantage argument

Consider the following swap table:

Fixed Floating

Firm A borrowing cost 14.50%

Yield curve + 0.50%

Firm B borrowing cost 16.50%

Yield curve + 1.70% Net difference

Difference 2.00% 1.20% 0.80%

Firm A borrows at a lower absolute cost. But

• Firm A has a comparative advantage in borrowing fixed.

• Firm B has a comparative advantage in borrowing floating.

What if firm A desires floating exposure, and firm B fixed exposure?

IRS: comparative advantage argument

Suppose the parties enter into a fixed-for-floating swap:

• Firm A borrows fixed at 14.50% in the market.

• Firm B borrows yield curve + 1.70% in the market.

• They enter into a swap with each other, agreeing that:

◦ Firm A pays firm B yield curve + 1.70% in the swap. ◦ Firm B pays firm A fixed 16.25% in the swap.

IRS: comparative advantage argument

Both firms win:

• Firm A net gain is 0.55% per annum:

◦ Pays 14.50% and receives 16.25% fixed: 1.75% gain.

◦ Pays yield curve + 1.70% instead of + 0.50%: 1.20% loss.

• Firm B net gain is 0.25%:

◦ Pays 16.25% fixed instead of 16.50%.

IRS: mechanics

• Interest payments are calculated based on a notional principal. ◦ Fixed PMTs = fixed rate 16.25% × notional.

◦ Floating PMTs = (1-period forward rates + 1.7%) ×

• Net interest is paid in arrears as the fixed payment minus the floating payment based on the spot rate at the start of each period.

• Floating rate is based on a reference yield curve: BBSW or LIBOR.

• The fixed rate is called the swap rate.

IRS: pricing

• We calculate what’s called the swap value.

• Pricing: Setting the swap rate so the swap value equals zero. • Application of both DCF and arbitrage/replication principals:

◦ Price off an arbitrage-free yield curve.

◦ Swap value is present value of future net cashflows.

An example will best illustrate.

IRS: example

• Construct the 1-period forward yield curve from the zero one. • Suppose semiannual compounding over 4 years.

1 y0,1 = 0.1264

2 y0,2 = 0.1289

3 y0,3 = 0.1316

4 y0,4 = 0.1349

5 y0,5 = 0.1372

6 y0,6 = 0.1410

7 y0,7 = 0.1448

8 y0,8 = 0.1480

1-period forward yields y0,1 = 0.1264

y1,2 = 0.1314

y2,3 = 0.1370

y3,4 = 0.1448

y4,5 = 0.1464

y5,6 = 0.1601

y6,7 = 0.1677

y7,8 = 0.1705

Spot yields

IRS: example (cont.)

• Firm A pays 1-period forward + 1.70%, Firm B pays fixed 16.25%.

• Notional principal is $1, 000, 000, 4 years, semiannual payments.

Spot Forward

1 0.1264 0.1264

2 0.1289 0.1314

3 0.1316 0.1370

4 0.1349 0.1448

5 0.1372 0.1464

6 0.1410 0.1607

7 0.1448 0.1677

8 0.1480 0.1705

Fix PMT 81,250 81,250 81,250 81,250 81,250 81,250 81,250 81,250

Float PMT 71,700 74,201 77,005 80,915 81,712 88,551 92,371 93,767 Swap

Fix-Float 9,550 7,049 4,245 335 -462 -7,301 -11,121 -12,517 value

PV @ Spot 8,982 6,221 3,506 258 -332 -4,851 -6,818 -7,071 -104.59

IRS: example (cont.)

In the calculated example,

Swap value: PV Fixed PMTs – PV float PMTs = -$104.59.

• Firm A pays floating / receives fixed: Swap has negative value.

• Firm B pays fixed / receives floating: Swap has positive value. Pricing a swap: involves setting the swap and floating rates to ensure:

• The swap value is initially equal to zero.

• It is mutually beneficial for both parties to enter into a swap. Given the floating rate, the correct fixed rate is ≈ 16.2535%.

Interest rate swaps

Credit default swaps

Credit default swaps (CDS)

A credit default swap is essentially an insurance contract in which: • The buyer pays regular premiums to the seller which are

calculated from the credit default rate.

• The buyer receives a (insurance) payout from the seller upon the

occurance of a credit event.

Credit default swaps are often called default insurance contracts.

• We first develop some terminology and notation. • We then turn to valuing or pricing CDSs.

CDS: terminology

• Reference entity: The entity over which the CDS is written.

• Reference asset: The specific asset over which the CDS is written. Example

• Eg ANZ Bank (buyer) gets a prespecified payout from a party (seller) in the event that BHP (reference entity) defaults on an interest payment on a large loan (reference asset) it has with ANZ.

Hence, in this case ANZ is insuring against the possibility of BHP defaulting on an interest payment on a loan that BHP has with ANZ.

CDS: terminology (cont.)

• Credit event: Events upon whose occurance a payout is made: ◦ Hard: Default on interest or loan payments, principal, etc. ◦ Soft: Corporate restructuring, credit rating downgrade,

corporate takeover or merger, asset writeoff, credit deterioration, etc.

• Protection buyer: The buyer in the CDS, who pays a regular premium in exchange for receiving a payout from a credit event.

• Protection seller: The seller in the CDS, who agrees to make the payout for a credit event in return for receiving the premium.

CDS: terminology (cont.)

• Notional principal or amount: Underlying ‘value’ of the CDS.

• Premium payments: The regular payment made by the buyer.

• Premium payments = credit default rate × notional principal.

• Probability of default: The probability that a credit event occurs. • Recovery rate: The percent amount recovered upon default.

• Loss given default: Dollar amount lost upon default.

• Payout: Payout made by the seller in the event of a credit event.

CDS: notation

• T is the time in years to maturity of the CDS.

• 0 = t0,t1,…,tN−1,tN = T is a set of dates.

• y0,1, y0,2, . . . , y0,N−1, y0,N is the zero coupon bond yield curve. • F is the notional principal.

• r is the credit default rate.

• Cn is the cashflow to be paid on the reference asset at time tn. • pn is the probability of default on the cashflow Cn.

• Rn is the recovery rate on cashflow Cn in the case of default.

CDS: assumptions

Assume only hard credit events: Default on the cashflows. In the case of default on cashflow Cn we set:

• The amount recovered at time tn is

recovery rate × Cn = RnCn.

• The dollar payout made at time tn is

payout = Cn − amount recovered = (1 − recovery rate)Cn

= (1 − Rn)Cn.

• The dollar payout is actually the loss given default.

We also assume the swaps no longer exists after a default event.

CDS: pricing

Similar to IR swaps:

• We calculate what’s called the credit default swap value.

• Pricing: Set the credit default rate r so the swap value equals zero.

• Application of both DCF and arbitrage/replication principals:

◦ We price off an arbitrage-free yield curve and the swap value

is the present value of the swap’s future cashflows. ◦ Priced from the perspective of the buyer.

A simple example will best illustrate.

CDS: example

• The buyer wants to insure against default on a coupon-paying

bond with principal F and annual coupon rate c.

• T = 3 years and we are given a zero coupon bond yield curve

y0,1 , y0,2 , and y0,3 .

• Both probability of default p and recovery rate R are constant.

CDS: example (cont.)

There is only four possible outcomes over the life of the swap: (i) A default occurs at the end of year 1 and the swap expires. (ii) A default occurs at the end of year 2 and the swap expires.

(iii) A default occurs at the end of year 3 (at maturity). (iv) No default occurs.

The swap value equals the sum of the present value of each of these outcomes multiplied by their probabilities of occuring.

CDS: example (cont.)

Time t = 1 year:

• Cashflows upon default with probability p:

◦ Payout (1 − R)C is received, no premium paid. • Cashflows upon no default with probability 1 − p:

◦ Premium rF is paid.

Default: Swap expires and cashflow is p(1 − R)C in year 1.

No default: Swap survives so continue to work out its future cashflows.

CDS: example (cont.)

Time t = 2 years: Probability of (1 − p) of getting to t2. • Cashflows upon default with probability (1 − p)p:

◦ Payout (1 − R)C is received, no premium paid. • Cashflows upon no default with probability (1 − p)2:

◦ Premium rF is paid.

Default: Swap expires and cashflows are

−rF in year 1, p(1−R)C in year 2.

No default: Swap survives so continue to work out its future cashflows.

CDS: example (cont.)

Time t = 3 years = maturity: Probability of (1 − p)2 of getting to t3. • Cashflows upon default with probability (1 − p)2p:

◦ Payout (1 − R)(C + F ) is received, no premium paid. • Cashflows upon no default with probability (1 − p)3:

◦ Premium rF is paid.

Default: Swap matures and cashflows are

−rF in year 1, −rF in year 2, (1−R)(C +F) in year 3. No default: Swap matures and cashflows are −rF each year.

CDS: example (cont.)

rF 1+y0,1 − rF

(1+y0,2 )2

+ (1−R)(C+F) (1+y0,3 )3

1+y0,1 − rF

(1+y0,2 )2

Event Default t1 Default t2 Default t3 No Default

Swap Cashflows (1 − R)C −rF, (1−R)C

−rF, −rF, (1−R)(C +F) − −rF, −rF, −rF

PV Swap Cashflows

(1−R)C 1+y0,1

− rF + (1−R)C

Probability

p (1−p)p (1−p)2p (1−p)3

rF (1+y0,2 )2

rF (1+y0,3 )3

(1−R)C 1 + y0,1

2 +(1−p) p −

swap value = p

(1−R)C (1 + y0,2)2

(1 + y0,2)2

(1−R)(C+F) (1 + y0,3)3

1 + y0,1 (1 + y0,2)2 (1 + y0,3)3

+(1−p)− − − .

CDS: pricing

The swap value equals the sum of the present value of each of possible outcome multiplied by their probabilities of occuring.

CDS: example with numbers

• Buyer wants to insure against default on a coupon-paying bond

with principal F = 100 and annual coupon rate of c = 5%.

• T = 3 years and we are given a zero coupon bond yield curve

y0,1 = 3%, y0,2 = 4%, and y0,3 = 5%.

• The probability of default is constant at p = 25% and the recovery rate is constant at R = 60%. The swap rate is r = 3.68%

CDS: example with numbers (cont.)

Swap Cashflows

−3.68, 2 −3.68, −3.68, 42 −3.68, −3.68, −3.68 2

PV Swap Cashflows

Probability 0.25 (0.75)0.25 (0.75)2 0.25 (0.75)3

Event Default t1 Default t2 Default t3 No Default

swap value = 0.25

+ (0.75)0.25

−3.68 + 1.03

− 3.68 − 3.68 + 42 1.03 1.042 1.053 − 3.68 − 3.68 − 3.68

1.042 1.053

2 − 1.03 + 1.042

2 3.68 3.68 42

+ (0.75) 0.25 − 1.03 − 1.042 + 1.053

3 3.68 3.68 3.68

+ (0.75) − 1.03 − 1.042 − 1.053 = −0.0004 ≈ 0.

Interest rate swaps

Credit default swaps

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