fin285a lecture 5 1a fall 2008
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Bonds and Swaps

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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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Presentation Transcript
outline
Outline
  • Coupon bonds
  • Currency swap
  • Fixed/floating swaps
software
Software
  • bondvar.m
  • bpswaphist.m
  • bpswapbs.m
  • fixfloat.m
bond pricing assumptions
Bond Pricing: Assumptions
  • Flat term structure
  • Yields
    • Geometric random walk
    • Rate = Tbond + 5% (risk spread)
    • Volatility = 1.75*tbond volatility
bond structure
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
Matlab Program
  • bondvar.m
  • Features
    • Government bond data file
    • Aggregate 12 months to get 1 year changes
outline7
Outline
  • Coupon bonds
  • Currency swaps
  • Fixed/floating swaps
currency swap
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
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
Coupon Payments
  • BP coupon : 6 months and 1 year
    • c(BP)
  • $ coupon : 6 months and 1 year
    • c($)
cash flow
Cash Flow
  • Today
    • 20BP, -($/BP)20BP
  • 6 Month (coupons)
    • -c(BP) = Libor(BP) + 1%
    • +c($)=Libor($) + 2%
  • 12 Month
    • -20BP-c(BP), ($/BP)20BP+c($)
slide13
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
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
Risk Factors
  • Exchange rate ($/BP)
  • r(BP): British interest rate
    • Flat term structure
  • r($): US interest rate
    • Flat term structure
data set
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
Matlab Code
  • Historical VaR
    • bpswaphist.m
    • Note: impact of FX
  • Bootstrap values
    • bpswapbs.m
multiple risk factors
Multiple Risk Factors
  • X = % Change [FX r(BP) r($)]
  • Historical
    • Use matrix of changes
    • Keep changes in each component of X together in time
multiple risk factors21
Multiple Risk Factors
  • X = % Change [FX r(BP) r($)]
  • 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 factors22
Multiple Risk Factors
  • X = % Change [FX r(BP) r($)]
  • Bootstrap 2
    • Assume independence
    • Sample separately
      • xbs = sample(x(:,1),n)
multiple risk factors23
Multiple Risk Factors
  • X = % Change [FX r(BP) r($)]
  • Monte-carlo
    • Assume normality
    • Estimate mean vector
    • Estimate variance/covariance matrix
    • Simulate multivariate normals
    • Find valuations V(x(t+1))
multiple risk factors24
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
Multi Factor Challenges
  • Which factors are important?
  • How do they move together?
    • Covariances??
outline26
Outline
  • Coupon bonds
  • Currency swaps
  • Fixed/floating swaps
interest rate 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
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
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
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
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 float37
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 code40
Matlab Code
  • fixfloatswap.m
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