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

MATH3090/7039: Financial mathematics Lecture 5

Interest rate swaps
Credit default swaps

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.

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?

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.

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 = 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:
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:
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:
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