Commerce 4FJ3

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Commerce 4FJ3

• Fixed Income Analysis
• Week 9
• Bonds with Options

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Traditional Yield Spread

• Traditional yield spread is 109 basis points
• Ignores term structure of interest rates
• The bond, if called in 10 years, should be
  compared to a 10 year treasury

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Static Spread

• Find the treasury spot rate term structure using
  the bootstrapping method
• Find the present value of the cash flows for the
  bond using the spot rate plus a spread
• Solve for the spread that gives the current price
• Called the static or zero volatility spread

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Static Spread Example
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Value of Static Spread

• Gives the return in excess of treasury over the
  entire term structure
• Takes into account that there are expected
  different reinvestment rates at different points
  in the future
• Assumes that the required spread over treasury is
  constant over time

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Static vs. Traditional

• If yield curve is flat, no difference
• If yield curve is rising, the static spread will
  be higher, with bigger differences for long
  maturities and steeper term structures
• Static may be smaller if inverted yield curve
• Bigger differences in spread for amortizing
  securities (MBS, asset backed securities)

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Callable Bonds

• Two main disadvantage for buyers
• Extra reinvestment risk if the bond is called
  before the investors time horizon they will face
  extra reinvestment risk, probably at a lower rate
• Price compression if yields fall, the bond price
  will not rise as much as it should because the
  bond can be bought back at a fixed price

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Traditional Valuation

• As mentioned earlier, calculate yield to each
  call, yield to worst, and then decide on the
  price that is reasonable
• Assumes bond will be called at that date
• Ignores reinvestment rates
• can be mitigated by comparing to treasuries of
  the maturity of the called bond

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Price vs. Yield for Callable
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Negative Convexity

• The normal price/yield curve is convex
• With price compression the level of convexity can
  become negative (technically it is now concave)
• Price change from increasing interest rates
  becomes larger than the change from falling
  interest rates

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Price vs. Call Price

• Note that the price of the bond can still be
  higher than the face value plus call premium if
  the bond has time before the call
• 13 bond, callable at 5 premium in one year,
  market rate is 5

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Bonds as Bundles

• Bonds with embedded options can be seen as a
  package of bonds and options
• Callable bond package of long an option free
  bond and short a call option on bond
• Putable (retractable) bond a package of long an
  option free bond and long a put option on the
  same bond

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Value of Options

• The option value is difficult to calculate since
  most pricing models assume that the price
  volatility of the underlying asset does not
  change over time
• The price volatility (modified duration) of the
  bond changes with time and also with the level of
  interest rates

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Another Problem

• Call options on bonds are American options while
  pricing models are based on European options
• Argument for using Euro for models is that it is
  usually not worth exercizing early due to loss of
  time value does not hold here since there are
  intermediate cash flows

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Interest Rate Volatility

• The major influence on the price of a bond is
  interest rates
• Changes in interest rates can be measured over
  time and the volatility can be estimated
• Can be used to create an interest rate model
• Textbook model is single factor, lognormal random
  walk, binomial interest ladder or lattice,
  estimating potential forward rates

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Interest Rate Lattice

• A bond can be valued by taking the present value
  of each cash flow, discounted by the product of
  all applicable forward rates
• The model assumes that the forward rate will take
  one of two equally likely values
• The higher rate lower rate x e2s
• Rates are found for each node using trial and
  error

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Option-Free Value

• Once the interest rate lattice has been
  constructed, other bonds can be analysed
• Starting with the final cash flows (since the
  intermediate prices can not be determined in
  advance), fill in the nodes on the lattice
• The price found should be identical to the one
  found using the static spread analysis

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Valuation with Options

• As with the option-free bond, add the value of
  the bond plus coupon to each node, but if the
  bond is likely to be called (greater than call
  price refunding cost), replace that value with
  the call price
• As above, but replace market values below the put
  price with the put price

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Modelling Risk

• If the assumptions that the model is based on is
  incorrect, the values derived from the model will
  not be useful
• The volatility assumption is critical
• The higher the volatility, the higher the value
  of an option, the lower the price of a callable
  bond
• It is important to stress test the model

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Option Adjusted Spread

• The spread that would explain the current price
  of a bond with an embedded option
• Can be constructed over the treasury term
  structure or the issuers term structure
• Since there is disagreement between market
  participants, knowing which assumption they are
  using is critical

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Option Value in Spread Terms

• If we have the OAS in terms of the treasury
  forward rate structure, we can calculate the
  amount of the spread that is due to the embedded
  option
• option value static spread - OAS
• Main reason for spreads is because some market
  participants prefer to talk about all investments
  in terms of rate of return

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Effective Duration and Convexity

• Found using the approximation formulas
• Similar to modified if the option is deeply out
  of the money

P- price if yield down P price if yield up P0
original price
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Finding P- and P

• Five step process for binomial model
• Calculate OAS for the bond
• Shift the treasury yield curve down/up a few
  basis points
• Construct the interest rate tree
• Add the OAS to each nodes interest rate
• Determine the value of the security

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Convertible Bonds

• Another type of embedded option
• A call option on a number of the issuers common
  share where the exercise price is the bond,
  regardless of current market value
• Number of shares is conversion ratio
• Can be physical or cash settle
• Exchangeable bonds are similar options, but on
  other companys shares

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Conversion Price

• The conversion price is simply the implied
  exercise price of the option on a per share basis
• If the bond is issued at par the conversion price
  is

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Other Features

• Conversion ratio may change over time, on a
  schedule given in the issue
• Conversion ratio is adjusted for stock splits and
  stock dividends
• Most convertibles are also callable, which may
  trigger early conversion
• Some are putable (hard or soft put)

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Sample Convertible
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Minimum Price

• The bond will trade at a minimum of the greater
  of the conversion value or straight (debt) value
• conversion value how much the stock that the
  bond can be converted to is worth
• straight value the value of the convertible if
  it did not have the conversion option

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Sample Minimum Price

• For the sample bond, conversion value
• 17 x 50 850
• Given a 14 yield on non-convertible otherwise
  similar bonds, straight value
• PVcoupons PVface 788
• This bond should trade for a minimum of 850
  since that is the higher value

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Market Conversion Prices

• Since the exercise price is the bond, the
  effective price of the common stock changes over
  time

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Sample Conversion

• Market conversion price 950/50 19
• Market conversion premium per share 19 - 17
  2
• Market conversion premium ratio 2/17 11.8

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Current Income

• One reason for not converting a convertible bond
  before maturity, are the coupon payments
• FIDPS Coupon/(conversion ratio) - dividend
• Premium payback period (break-even time)
  Market premium per share Favourable income
  differential per share

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Sample Income

• Coupon interest from bond 100
• Dividend per share 1
• Conversion ratio 50
• Favourable income differential per share
  100/50 - 1 1
• Premium Payback Period 2/1 2 years

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Downside Risk

• Often measured as the premium over straight value
• (Market value/Straight value) - 1
• Sample bond 950/788 - 1 21
• Note the investor has more than 21 downside
  risk since the YTM could increase, decreasing the
  straight value

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Jargon

• A convertible where the option is well out of the
  money is called a bond equivalent or busted
  convertible
• A convertible with a conversion value much higher
  than its straight value is called an equity
  equivalent
• Between those it is a hybrid security

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Payoff

• Share price goes up to 34Shareholder return
  100Convertible holder return 79
• Share price goes down to 7Shareholder return
  -59Convertible holder return -17
• The convertible is less risky

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Call Risk

• One reason for issuing convertible bonds is that
  the company would prefer to issue equity, but
  considers the current price to be too low to be
  worth issuing common shares
• Conversion ratio is set to reflect reasonable
  pricing
• Call options can be used to force conversion

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Takeover Risk

• If the issuer gets taken over before the price of
  the shares make conversion reasonable, the bond
  holders may be left with a bond that pays a lower
  coupon than similar corporate bonds

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Options Approach

• Similar to callable bonds, convertibles can be
  viewed as a bond and an option
• An additional problem here is that the exercise
  price on the share changes over time as the
  bonds market price is affected by changes in
  interest rates
• To make matters worse, most convertible bonds are
  also callable