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# Bonds and Swaps PowerPoint PPT Presentation

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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#### Presentation Transcript

FIN285a: Lecture 5.1a

Fall 2008

### Outline

• Coupon bonds

• Currency swap

• Fixed/floating swaps

• 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(\$)

+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