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FIN285a: Lecture 5.1a Fall 2008. Bonds and Swaps. Outline. Coupon bonds Currency swap Fixed/floating swaps. Software. bondvar.m bpswaphist.m bpswapbs.m fixfloat.m. Bond Pricing: Assumptions. Flat term structure Yields Geometric random walk Rate = Tbond + 5% (risk spread)

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Outline

- Coupon bonds
- Currency swap
- Fixed/floating swaps

Software

- bondvar.m
- bpswaphist.m
- bpswapbs.m
- fixfloat.m

Bond Pricing: Assumptions

- Flat term structure
- Yields
- Geometric random walk
- Rate = Tbond + 5% (risk spread)
- Volatility = 1.75*tbond volatility

Bond Structure

- Principal = 1000
- Coupon = 8% = 80 (starting in 1 year)
- Maturity = 3 years
- Problem:
- Find VaR and ETL over 1 year period

Matlab Program

- bondvar.m
- Features
- Government bond data file
- Aggregate 12 months to get 1 year changes

Outline

- Coupon bonds
- Currency swaps
- Fixed/floating swaps

Currency Swap

- Foreign currency swap
- Trade principal and interest in one currency for another
- Borrow British pounds, lend US dollars
- Structure
- Long $ bond
- Short BP bond

Simple Swap Example

- 1 Year contract
- Interest payments at 6 months and 1 year
- BP principal = 20 million BP
- Payback in 1 year

- $ principal = 20 million BP ($/BP)

Coupon Payments

- BP coupon : 6 months and 1 year
- c(BP)

- $ coupon : 6 months and 1 year
- c($)

Cash Flow

- Today
- 20BP, -($/BP)20BP

- 6 Month (coupons)
- -c(BP) = Libor(BP) + 1%
- +c($)=Libor($) + 2%

- 12 Month
- -20BP-c(BP), ($/BP)20BP+c($)

Cash Flow PictureLet X = $ notional = E($/BP)20Fixed today at current FX rateNote: now transactions neutralize

+20 BP

+c($)

+c($)+X $

-c(BP)

-X $

-c(BP)-20 BP

12 Months

Now

+6 Months

Find 1 Month VaR

- Mark to market today (current FX and interest rates)V(t)
- Find FX and interest rates 1 month in the future (t+1)
- Use historical data and arithmetic returns

- Mark to market in one month V(t+1)
- Find VaR using P/L = V(t+1)-V(t)

Risk Factors

- Exchange rate ($/BP)
- r(BP): British interest rate
- Flat term structure

- r($): US interest rate
- Flat term structure

Data Set

bp.dat

- Date (matlab format)
- $/BP exchange rate
- R(BP) = 1 Month interbank (London)
- R($) = 1 Month eurorate (London)
- Source: Datastream

Matlab Code

- Historical VaR
- bpswaphist.m
- Note: impact of FX

- Bootstrap values
- bpswapbs.m

Multiple Risk Factors

- X = % Change [FX r(BP) r($)]
- Historical
- Use matrix of changes
- Keep changes in each component of X together in time

Multiple Risk Factors

- X = % Change [FX r(BP) r($)]
- Bootstrap
- Use matrix of changes
- Keep changes in each component of X together in time
- Sample X together
- Sample command does this (row by row)

Multiple Risk Factors

- X = % Change [FX r(BP) r($)]
- Bootstrap 2
- Assume independence
- Sample separately
- xbs = sample(x(:,1),n)

Multiple Risk Factors

- X = % Change [FX r(BP) r($)]
- Monte-carlo
- Assume normality
- Estimate mean vector
- Estimate variance/covariance matrix
- Simulate multivariate normals
- Find valuations V(x(t+1))

Multiple Risk Factors

- X = % Change [FX r(BP) r($)]
- Delta normal
- Assume normality
- Estimate mean vector
- Estimate variance/covariance matrix
- Linearly approximate distribution of V(X)

Multi Factor Challenges

- Which factors are important?
- How do they move together?
- Covariances??

Outline

- Coupon bonds
- Currency swaps
- Fixed/floating swaps

Interest Rate Swaps

- Pay fixed coupon payments
- Receive floating coupon payment (Libor * notional amt.)
- or the reverse
- Floating rate locked 6 months before payment
- Also, dealers arrange and take a spread

Swap Example

- Structure
- Receiving fixed payments
- Paying floats
- Semiannual payments
- Units: semiannual compounding
- 1 year to maturity
- Payments in 6 and 12 months

Swap Valuation

- Long a fixed rate bond
- Valuation: easy

- Short a floating rate bond
- Valuation: a little tricky, but not bad

- Swap value = PV(fixed) - PV(float)

Bond Specifics

- Fixed:
- Principal = 1000
- Coupon = 5% (semi-annual)
- Maturity = 1 year

- Float:
- Principal = 1000
- Coupon = Libor (initial = 5%) semi-annual
- Maturity = 1 year

Picture of the Float

6 Months

1000(1+r(6)/2)

C=1000*r(0)/2

r(0)

r(6)

Right after coupon is paid

PV at r(6)/2 = $1000

Valuation 1 month in

the future is easy

PV( 1000*r(0)/2 + 1000)

Picture of the Float

6 Months

1000(1+r(6)/2)

C=1000*r(0)/2

r(0)

r(6)

Right after coupon is paid

PV at r(1)/2 = $1000

Valuation 7 months in the

future trickier. Need r(6) for coupon

and r(7) for discount.

Matlab Code

- fixfloatswap.m

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