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Bonds and Swaps PowerPoint PPT Presentation


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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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Bonds and Swaps

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Fin285a lecture 5 1a fall 2008 l.jpg

FIN285a: Lecture 5.1a

Fall 2008

Bonds and Swaps


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Outline

  • Coupon bonds

  • Currency swap

  • Fixed/floating swaps


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Software

  • bondvar.m

  • bpswaphist.m

  • bpswapbs.m

  • fixfloat.m


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Bond Pricing: Assumptions

  • Flat term structure

  • Yields

    • Geometric random walk

    • Rate = Tbond + 5% (risk spread)

    • Volatility = 1.75*tbond volatility


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Bond Structure

  • Principal = 1000

  • Coupon = 8% = 80 (starting in 1 year)

  • Maturity = 3 years

  • Problem:

    • Find VaR and ETL over 1 year period


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Matlab Program

  • bondvar.m

  • Features

    • Government bond data file

    • Aggregate 12 months to get 1 year changes


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Outline

  • Coupon bonds

  • Currency swaps

  • Fixed/floating swaps


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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


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Valuation of Swap


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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)


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Coupon Payments

  • BP coupon : 6 months and 1 year

    • c(BP)

  • $ coupon : 6 months and 1 year

    • c($)


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Cash Flow

  • Today

    • 20BP, -($/BP)20BP

  • 6 Month (coupons)

    • -c(BP) = Libor(BP) + 1%

    • +c($)=Libor($) + 2%

  • 12 Month

    • -20BP-c(BP), ($/BP)20BP+c($)


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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


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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)


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Risk Factors

  • Exchange rate ($/BP)

  • r(BP): British interest rate

    • Flat term structure

  • r($): US interest rate

    • Flat term structure


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Valuation in US $ (today)


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Valuation in US $ (1 month future)


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Data Set

bp.dat

  • Date (matlab format)

  • $/BP exchange rate

  • R(BP) = 1 Month interbank (London)

  • R($) = 1 Month eurorate (London)

  • Source: Datastream


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Matlab Code

  • Historical VaR

    • bpswaphist.m

    • Note: impact of FX

  • Bootstrap values

    • bpswapbs.m


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Multiple Risk Factors

  • X = % Change [FX r(BP) r($)]

  • Historical

    • Use matrix of changes

    • Keep changes in each component of X together in time


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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)


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Multiple Risk Factors

  • X = % Change [FX r(BP) r($)]

  • Bootstrap 2

    • Assume independence

    • Sample separately

      • xbs = sample(x(:,1),n)


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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))


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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)


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Multi Factor Challenges

  • Which factors are important?

  • How do they move together?

    • Covariances??


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Outline

  • Coupon bonds

  • Currency swaps

  • Fixed/floating swaps


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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


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Swap Example

  • Structure

    • Receiving fixed payments

    • Paying floats

    • Semiannual payments

    • Units: semiannual compounding

    • 1 year to maturity

      • Payments in 6 and 12 months


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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)


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Semiannual Compounding


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Bond Specifics

  • Fixed:

    • Principal = 1000

    • Coupon = 5% (semi-annual)

    • Maturity = 1 year

  • Float:

    • Principal = 1000

    • Coupon = Libor (initial = 5%) semi-annual

    • Maturity = 1 year


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Valuing Fixed at Issue


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Valuing Float at Issue


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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)


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Valuing Float 1 Month in the Future


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Valuing Swap (1 Month Future)


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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.


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Valuing Float 7 Months in the Future


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Valuing Swap (7 Month Future)


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Matlab Code

  • fixfloatswap.m


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