The New Slides

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Introduction to Bond Math
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Goals of Our Introduction to Bond Math

• Our goal is to learn enough bond math so we can
• Understand the basic concept of bond pricing
• Understand the inverse relationship between bond
  prices and interest rates
• Know how the yield of a bond is calculated given
  the price, and vice versa
• know how to make a simple value comparison
  between two bonds
• know how to measure the price risk of a bond as
  interest rates move
• Note even if youre not going into the bond
  business,
• its good to know some bond math as an
  investor

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Definition of a Bond

• A bond is a security whereby the issuer of the
  bond borrows money, called the principal, and
  agrees to
• 1) pay the bond holder (the lender) periodic
  interest payments (coupons) based on the
  outstanding amount of principal
• 2) return the principal through a lump sum
  payment, periodic payments over time, or in the
  case of a perpetual bond, not at all
• Example a 3 yr. bond with 100 principal, and
  10 coupon (interest) paid annually

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The Time Value of Money Future Value Present
Value

• Example 1 I have 100 today. I can invest it at
  5 and receive 105 back a year from now.
• 105 is called the Future Value (FV), a year from
  now, of the 100 today
• 100 is called the Present Vale (PV) of the 105
  received a year from today
• Thus, the basic premise for the concept time
  value of money is
• Future Value is greater than Present Value as
  long as interest rates are positive that is,
• As long as one can invest money today and receive
  a positive interest plus the principal at a
  specified future time

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Future Value the Concept of Compounding
Frequency

• Example 2 I have 100 today. Whats the Future
  Value of this 100 in a year?
• Assume a 12 interest rate
• A) compounded annually FV Principal
  Interest
• once a year at 12
• B) compounded semi-annually
• once every 6 months at 6
• C) compounded monthly
• once a month at 1

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Future Value of 100 (Today) in 30 Years

• Example 3 I have 100 today. Whats the FV of
  this 100 in 30 year?
• Assume a 12 interest rate
• A) compounded annually
• once a year at 12
• B) compounded semi-annually
• once every 6 months at 6
• C) compounded monthly
• once a month at 1
• The impact of more frequent compounding higher
  future value, 20 from (A) to (C)

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Present Value the Concept of Compounding
Frequency

• Example 4 I will receive 100 in 1 year. Whats
  the Present Value of that 100? Assume a 12
  interest rate
• A) discounted annually
• once a year at 12
• B) discounted semi-annually
• once every 6 months at 6
• C) discounted monthly
• once a month at 1

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Present Value of 100 to be Received in 30 Years

• Example 5 I will receive 100 in 30 years.
  Whats the PV of that 100?
• Assume a 12 interest rate
• A) discounted annually
• once a year at 12
• B) discounted semi-annually
• once every 6 months at 6
• C) discounted monthly
• once a month at 1
• The impact of more frequent compounding lower
  present value, -17 from (A) to (C)

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PV vs. Interest Rate an Inverse Relationship

• lt- rates going higher
• Observations
• FV is higher when interest rates are higher
• PV is lower when interest rates are higher
• Note the inverse relationship between PV and
  interest rate
• The higher the rate, the lower the PV, and
  vice versa
• This is the basis for the inverse relationship
  between bond price and interest rates
• (rates up ? bond price down and rates down
  ? bond price up)

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Price and Yield of a Bond

• Pricing of a bond is based on the concept of
  Present Value
• the price of the bond is the sum of the PVs of
  all the future cash flows
• Example Cash flow of a 3 yr. bond w/a 10
  semi-annual coupon and a principal of 100.
• Price sum of the PVs of all the cash flows
  (discounted at a rate, e.g., 8)
• Yield-to-Maturity (YTM)
• - the one discount rate whereby the PV
  of the bonds cash flows equals the bond price

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Price vs. Yield an Inverse Relationship

• Note yields up ? bond price down yields down ?
  bond price up
• Same Example 3 yr. bond with a 10 semi-annual
  coupon and a principal of 100
• Suppose I paid 100 for a 3 year bond with 10
  coupon last month.
• At what price can I sell the bond today?
• Scenario 1 Rates are higher now, one can buy a
  new 3 year bond with 12 coupon at 100
• People wont pay 100 for my bond, as it only has
  10 coupon so I have to lower the price
• Scenario 2 Rates are lower now, one can buy a
  new 3 year bond with 8 coupon at 100
• People will pay more than 100 for my bond, as it
  has 10 coupon so I can raise the price

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Relationships of Coupon, Yield and Price
Same Example a 3 yr. semi- annual 10 coupon
bond
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Calculating the Price of a Bond with an HP12C

• Example Calculate the price of a 3yr. bond with
  10 coupon and yield-to-maturity of 8
• Price 105.2421

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Finding the YTM of a Bond with an HP12C

• Example Find the YMT of a 3yr. bond with a
  coupon of 10 and a price of 105.2421
• YTM 8.00

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Mark-to-Market

• Suppose a bond trader paid 100 for a bond,
• if he sells the bond the same day at 98, he
  would lose 2 points on the bond
• if he still owns the bond at the end of the day,
  and the bond price is at 98,
• the 2 point loss would still be in his
  profit/loss for the day
• as long as he owns this bond, he will have an
  unrealized daily P/L on this bond
• Mark-to-Market
• evaluate a bond at its market price, and
• record the profit/loss accordingly
• Dealers are required to mark-to-market their
  trading portfolios daily
• Many financial institutions are not required to
  mark to market (insurance companies)

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Bond Price Yield Conventions

• Price
• Price represents a percent of the principal (or
  Face Value), for example
• A price of 102 means for 50mm principal you pay
  (102)(50mm) 51mm
• Prices are generally expressed in 32nds 102.50
  would be written 102-16
• A means a 64th 102-16 means 102 and 33/64
• The smallest unit is generally 1/256, 102-16
  means 102 and
• Yield
• 100 Basis Points 1.0, 1 Basis Point (B.P.)
  .01

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Some Bond Facts

• The maximum bond price is when the yield is 0
  just add up all the cash flows
• Max. Price
• The minimum bond price is zero (at an infinitely
  high yield)
• Zero coupon bonds have the formula
• Example 30 yr. zero coupon bond using a 10
  yield

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Yield as a Measure of Value
Which of these two 20 year bonds is cheaper
(more attractive), assuming same credit quality?

• Bond A
• Maturity 20 years
• Coupon 6
• Price 90

• Bond B
• Maturity 20 years
• Coupon 9
• Price 100

YTM of Bond A 6.93
YTM of Bond B 9.00
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Accrued Interest, Flat Price, and Full Price

• Accrued Interest
• If the bond is sold between coupon dates
• the seller is entitled to a portion of the next
  coupon, thus
• the buyer needs to pay for that portion of the
  coupon, called accrued interest
• Flat Price vs. Full Price
• The price of a bond is quoted without accrued
  interest.
• This is called the flat price, the clean price,
  or the market price of the bond.
• The bonds price with accrued interest is called
  the full price or the dirty price of the bond

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Absolute Price Risk of a Bond DV01

• The dollar value of an 01 (DV01)
• is change in a bonds price for a 1 bp. increase
  in its yield.
• Example - the DV01 of a 10 yr., 8 bond yielding
  7.50
• Price at 7.50 103.4741
• Price at 7.51 103.4030
• DV01 0.0710

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Absolute Price Risk of a Bond DV01

• Examples of applications using DV01
• (1) If the yield of the bond above moves up 100
  bps (1),
• the price will move down by about 7.1 (0.0710
  100)
• (2) A trader wants to hedge the interest rate
  risk of 100mm Bond A with Bond B.
• Question how many Bond B should he short to
  hedge Bond A?
• Assume DVO1s of Bond A B are 0.040 0.050
  respectively.
• Hedge Ratio (DV01 of Bond A) / (DV01 of Bond
  B) 0.04 / 0.05 0.8
• Thus, he should short 80mm of bond B to hedge
• (B is more volatile, hence fewer bonds needed)
  

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DV01 vs. Maturity
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Modified Duration

• The modified duration of a bond approximates the
  percentage price change of the bond for 1 change
  in yield - a standard measure of risk for a bond
  portfolio.
• Example I want to invest 100,000 in either
• Bond Fund X (with a Duration of 4), or
• Bond Fund Y (with Duration 7)
• Question How will interest rate changes affect
  each fund?
• Answer Fund Y is more sensitive to interest rate
  (as 7 gt 4).
• If interest rates rise by 1,
• Fund X will lose approximately 4000,
• while Fund Y will lose approximately 7000.

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Modified Duration vs. Maturity
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Positive and Negative Convexity

• A bonds price-yield curve is positively convex
  at a given yield
• if the bonds price rises more than it falls (as
  its yield moves by the same of bps)
• positive convexity is an attractive
  characteristic
• A bonds price-yield curve is negatively convex
  at a given yield
• if the bonds price falls more than rises (as
  its yield moves by the same of bps)
• negative convexity is an unattractive
  characteristic
• All non-callable bonds are positively convex
• Callable bonds, including most mortgage backed
  securities, are negatively convex
• The convexity of bond, at a given yield level,
  measures how much more
• a bonds price rises than it falls (as the yield
  changes by the same of bps)

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Bond Price vs. Yield
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