BOND PRICING

1
BOND PRICING

• Review of Basic Principles
• Another Way of Looking at Bond Prices

2
Basic Present Value/Future Value
Rearranging gives
3
More General Relationship
i annual nominal rate p of periods per
year i/p periodic discount rate n of
years np Total of periods
4
Example

• i6
• p1
• n5
• PV100
• FV100(1.06/1)51100(1.338226)
• FV133.8226
• If we change p to 12
• FV100(1.06/12)60
• FV134.89

5
Effective Annual Yield

• The Effective Annual Yield (Rate) is the annual
  rate of interest, with no compounding, that gives
  you the same end of year wealth as the periodic
  discount rate with compounding.

6
Example Effective Annual Yield

• Consider a 6 mortgage and a 6 bond.
• Which is the better investment ?
• First, we will figure out the answer directly
  without using EAY
• Assume both are purchased at par or face value
• Answer the question by calculating future wealth
• Bond
• FV(100)(1.06/2)2106.09
• Mortgage
• FV(100)(1.06/12)12106.17
• Note that I dont have to worry about how long I
  hold the investment (as long as both have same
  maturity). If mortgage beats bond in year 1 it
  will keep winning in future years
• Only need to look at future wealth at the end of
  1 year

7
Example Effective Annual Yield

• Next, we will use EAY
• Bond From our previous calculations, it is clear
  that I would be indifferent (i.e., my future
  wealth would be the same) between owning the bond
  and owning a simple annual CD that paid 6.09
  interest annually.
• Mortgage Similarly, I would be indifferent
  between owning the mortgage and a 6.17 CD
• EAYBond 6.09
• EAYMortgage 6.17

8
Effective Annual Yield (Rate)
or
9
EAY Why do I only see EAY discussed in
textbooks???

• EAY is mathematically correct way to compare and
  rank yields- but it is rarely used in practice.
  Why??
• 1. Most of the time we are comparing investments
  within the same class
• Bonds to Bonds
• Mortgage to Mortgages
• In these cases, the annual nominal rate, i, is
  good enough to rank investments
• Only when you are comparing across classes, do
  you have to worry about controlling for
  compounding
• Matters more at higher yields

10
EAY

• EAY is mathematically correct way to compare and
  rank yields- but it is rarely used in practice.
  Why??
• 2. Most fixed income securities are compared to
  Treasuries
• The i for bonds with semiannual compounding has
  a special name BEY
• It has become customary to put other investments
  on the same basis That is to quote the BEY of a
  mortgage.
• The idea is the same as for EAY, but, we need to
  modify the basic question to ask What annual
  nominal rate on a bond (i.e., the coupon of the
  bond), assuming semiannual compounding at the
  same rate, would generate the same future wealth
  (FV) as the mortgage?

11
BEY of a Mortgage

• We already saw that FVMortgage100(1.06/12)12106
  .17
• What annual nominal rate (or coupon rate) on a
  bond would give the same FV? (Jargon we call
  that annual nominal rate a Bond Equivalent
  Yield or BEY)
• More generally

12
Basic Bond Prices/Yields

• Two Objectives
• Calculate Price, given Yield
• Calculate Yield, given Price
• A bond is simply a contractual agreement between
  2 parties whereby one party (issuer) promises to
  make certain payments at various future dates.
  In return, issuer will receive something of value
  now Usually
• Three Key Attributes that Define a Standard Bond
• Maturity
• Date at which all obligations to make payments
  end
• Coupon Rate (annual nominal rate)
• The interest rate the issuer agrees to pay each
  year on the outstanding debt amount
• Note If bond pays interest semi-annually,
  interest duePar(CR/2)
• Par or Face Value (Par, Principal, Face Value,
  Maturity Value)
• The amount of money the borrower (issuer) agrees
  to repay by the maturity date.
• Note that the paid by the buyer at issuance
  need not equal the face value of the bond.
  However, when it does, we say that the bond sold
  at par
• Discount paid at issuance lt Face Value
• Premium paid at issuance gt Face Value

13
General Principle of Valuation

• Step 1 Identify (estimate) the future cash flows
  
• paid by the issuer
• Step 2 Determine the appropriate discount rate
• Step 3 Calculate the PV of Future cash flows
• Issue Should there be a single discount rate for
  all future cash flows or should there be a
  different one for cash flows received at
  different times?

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Is it reasonable to believe that the right
discount rate for cash flows received thirty
years from now is the same as that for
discounting cash flows received in 6 months? The
right answer is different ones but traditional
bond pricing uses a single discount rate so we
will too for a while.
15
Example Calculate Price, Given Required Yield

• Consider an annual pay bond that promises to
  pay interest of 10/yr for 4 years plus 100 at
  the end of the 4th year
• Maturity 4
• Principal 100
• Coupon 10
• Step 1 Layout the Cash Flows

Year Cash Flow
1 10
2 10
3 10
4 10100110
16
Example Calculate Price, Given Required Yield

• Step 2 Assume investor required yield is 8
• Obtain from market observation of trading of
  similar quality/maturity bonds
• Step 3 Calculate Present Value
• Using a calculator to make life easier

N 4
PMT 10
FV 100
I 8
PV -106.6243
17
Calculating Yield Given Price

• Consider the same bond
• 10, 4 yr, annual pay
• Assume an investor pays 115 for that bond
• What yield to maturity will she earn?
• Same basic formula applies we know CF and we
  know PV question is what is r?
• Solve for r using calculator.

N 4
PMT 10
FV 100
PV -115.00
I 5.70
Note that almost all calculators Return the
periodic discount rate, r, expressed in not
decimal. However, EXCEL works in decimal Rates
of return, not .
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Basic Principles of Bond Pricing

• Because bond prices are Present Values, prices
• vary with the Discount Rate in a predictable way

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Basic Principles of Bond Pricing

• Prices Decline with Discount Rate
• Inverse relationship between prices and yields
• On Friday 9/7/07, Treasury yields fell sharply
  driving the prices of Treasuries up
• Prices decline at a declining rate
• Price curve is convex function of the yield
• Whenever Coupon rate Discount rate,
  price 100 or
  pricePar
• Coupon lt Discount Rate PVlt Face or
  Discount Bond
• Coupon gt Discount Rate PVgt Face or
  Premium Bond

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Basic Principles of Bond Pricing

• As a bond moves closer to maturity, its value
  moves toward par
• Assumes the investor yield at original purchase
  remains constant over time

21
Basic Principles of Bond Pricing

• In the previous graph, the discount rate was held
  constant as time changed
• In practice, the discount rate (investor required
  yield) changes with time
• Bond prices reflect the combination of both
  effects
• Time
• Discount rate
• Consider another example 4-yr, 10 annual pay
  bond.
• At t0, required yield is 8
• Bond price is 106.624254

N4 i8 PMT10 FV100 Solve for PV106.624254
22
Basic Principles of Bond Pricing

• Assume now that we are pricing the same bond at
  the end of the first year. However, instead of
  assuming that the investor required yield stays
  the same, assume that it increases to 9
• Change in Price 102.53-106.62-4.09
• Change due to maturity -1.47
• Change due to discount rate -2.62
• Total Change -4.09

N3 i9 PMT10 FV100 Solve for PV102.5313
23
Discounting at a Constant Rate?

• Treasury yield curve is not flat
• Two different Treasury curves
• Par or Coupon Curve
• Strips Curve
• Suppose we observe the following Treasury Strip
  Curve

Maturity Price Yield ()
1 93.6330 6.8
2 87.0183 7.2
3 80.2718 7.6
4 73.5030 8.0
Notice that the investor required yield for cash
flows received four years From now is 8.
However, the required yields for earlier cash
flows are Less than that.
24
Pricing Off the Strip Curve

• Consider again the simple 10, 4-yr, annual pay
  bond
• Unbundle the bonds cash flows into the four
  distinct cash flows received at the end of each
  year.

The price calculated this way (106.95) is greater
than the price calculated using a constant 8
)106.62) because the early cash flows Are
discounted at lower required yields
25
Arbitrage Opportunity

• Assume that the price was set at the single yield
  price of 106.62
• Calculated as the present value of future cash
  flows discounted at 8
• Also assume an active Treasury strips market with
  prices and yields as described in the earlier
  table
• Yields ranging from 6.8 for 1 year to 8 for 4
  year cash flows
• How can traders profit from this?
• Buy the 4 year bond at 106.62
• Create a trust and strip off the individual
  cash flows
• Sell the four separate cash flows at the prices
  indicated earlier.

Maturity Cash Flow Price Net Proceeds
1 10 93.6330 9.36
2 10 87.0183 8.70
3 10 80.2718 8.03
4 110 73.5030 80.85
Total 140 -- 106.94
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Another Reason for No -Arbitrage Pricing

• The Final Maturity of a security is a poor
  descriptor of the timing of cash flows.
• Consider the following three different 10-year
  Treasury bonds.

Cash Flows Cash Flows Cash Flows
Coupon Rate Coupon Rate Coupon Rate
Period 12 8 0
1-19 6 4 0
20 106 104 100
These three securities represent very different
bundles of cash flows It seems unlikely that the
market required yield for all three would be the
same With no-arbitrage pricing, the yield on
these securities will vary reflecting the Timing
of the cash flows and the current yield curve.
27
No-Arbitrage Pricing

• Observe from the Strip market the price that
  investors are willing to pay for a 1 promise to
  pay at dates in the future.
• Assume these are given by the handout (or the
  earlier table)
• Provides prices and BEY for strips securities
  maturing every six months from now until ten
  years from now.
• View these prices just like the unit prices at
  the supermarket
• Determine how much of a particular cash flow the
  security has
• Multiply that quantity of dollars by the unit
  price of those dollars
• Repeat for all cash flows in the bundle
• Sum to find the total price of your shopping
  cart

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No-Arbitrage Pricing

• The back side of the handout does this for an 8
  10-year bond
• Cash flows are 4 at time 1 (6 months from now)
• 4 at time 2
• 104 at time 20 (10 years from now)
• The price of a 1 promise to pay by the Treasury
  six months from now is .9852
• This is equivalent to a BEY of 3 or a
  semi-annual periodic discount rate of 1.5-- but
  the price is all we need
• Multiplying 4 times .9852 gives us the market
  value of a 4 promise to pay as 3.94
• Repeat this for each cash flow multiplying the
  cash flow times the unit price to find the number
  in the far right hand side column
• Finally, add all the individual values up and we
  get 115.26 as the no arbitrage price of the 8
  10 year bond.

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No-Arbitrage Pricing

• As before this price is the no-arbitrage price
  because at this price, traders cannot profit
  from breaking out the cash flows and reselling
  them
• Example Say someone priced the cash flows at a
  single yield of 6
• N20
• PMT4
• FV100
• i6/2
• PV114.8775
• An astute investor could buy the entire bundle of
  cash flows and then sell each piece based on the
  strips curve. We just saw that this astute
  investor would then generate 115.2619 for a
  riskless certain profit of .38 per 100 of face
  value.
• While it may not be worth her while to do this
  for a 100 investment, for a 10 million
  investment the profit would be 38,000. Not bad
  for a few minutes of trading
• Similar profits can be made in reverse. Assume
  you people buying and selling the 8 bond at a
  price of 115.64 higher than its no arbitrage
  price.
• Strategy now is to sell the bond short (promise
  to deliver it in the future), buy the needed cash
  flows in the strip market, reconstitute the true
  underlying bond and deliver it to the buyer.
• As is frequently the case, there are generally
  more problems with the short strategy
• Problems going short
• Harder to reconstitute a bond than it is to
  strip a bond

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No-Arbitrage Pricing

• Now, lets repeat the same pricing exercise but
  use a 10-year, 6 coupon Treasury instead of
  an 8
• Periodic cash flows are now 3 instead of 4 and
  final cash flow is 103
• Strip prices are still the same
• The mix of cash flows is different with a
  smaller of cash flows received early and a
  larger received at the end.
• Repeating the same exercise of multiplying unit
  prices by cash flows and adding up, we find a
  price of 100
• This means the internal rate of return or
  yield, as we traditionally calculate bond
  yields, is 6
• What this means is that the weighted average
  discount rates in the table, where the weights
  reflect the present values of the cash flows of
  the 6 bond, is 6
• The yield on the 8 bond (price115.2619) is
  5.95
• All that this tells us is that the weighted
  average of the discount rates in the handout
  differ when the weights change
• For the 8 bond, there is more weight on the
  early cash flows leading to the lower weighted
  average or yield
• Bottom line The price of a bond is what really
  matters. The yield we calculate given the
  price is a derived number that is sometimes
  useful, but cannot simply be used to price
  securities where the timing of cash flows differs

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No-Arbitrage Pricing 1 Last example

• Now we will price the three different 10-year
  Treasuries we discussed before.
• For each, we will first price them using the
  strip curve
• We will compare that price to the one you would
  get if you priced each bond using the 6 YTM
  calculated for the 6 bond selling at par.

Coupon Rate Coupon Rate Coupon Rate
Prices 12 8 0
Strip Curve 145.79 115.26 54.21
Par Curve 144.63 114.88 55.37
Dif 1.16 .38 (1.16)
Notes 1) Zero coupon bond is undervalued using
the six percent while the 12 coupon is
overvalued. This is because all the cash flows
from the zero come at t20 where YTM was
6.27. 2) Price difference is smaller when cash
flows are similar to those of the 6 bond 3) A 6
bond would be perfectly priced (at par).
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No-Arbitrage Pricing 1 Last Comment

• There could be a large number of different strip
  (or spot) yield curves that produce the same
  price for a given bond.
• Simply think of them as different ways to get the
  same average score.
• Consequently, knowing the weighted average (i.e.,
  the yield) tells you very little about the
  specific components.
• Therefore, knowing the YTM for coupon bonds that
  have different bundles of cash flows tells you
  relatively little about how to price bonds with
  different cash flows.

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How good are yields as measures of return?

• Sources of Return from investing in a bond
• Coupon payments made by the issuer over time
• Capital gain/loss based on the difference between
  what an investor pays at purchase and what he/she
  receives at maturity or upon sale.
• Income from reinvestment of interim cash flows

34
Yield to Maturity

• YTM is the single interest rate (discount rate)
  that makes the present value of cash flows from a
  bond equal to its market or purchase) price.
• If we know Price (P or PV) and we know CF1, CF2,
  CFN,
• Then it is a simple matter to solve for r

35
Yield to Maturity
Bond Coupon (All 10-year Maturity) Bond Coupon (All 10-year Maturity) Bond Coupon (All 10-year Maturity) Bond Coupon (All 10-year Maturity)
0 6 8 12
No-Arbitrage Price 54.214 100.000 115.262 145.786

N 20 20 20 20
PMT 0 3 4 6
FV 100 100 100 100
PV -54.214 -100.000 -115.262 -145.786
Solve for i 6.217 6.000 5.953 5.879
Note that every 10-year bond has a different YTM
because each bond represents a different
combination of elements from the same yield curve.
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Limitations of YTM

• Does not accurately price of other securities
  with different cash flow timing
• Does not consider reinvestment of interim cash
  flows
• You will only earn 6 on the 6 bond purchased at
  par if you can reinvest all coupons at 6
• Zero coupon bond has no reinvestment risk
• Assumes bond is held to maturity
• Consider the 12 bond. If we buy it today and
  sell it six months from now (right after
  collecting the 6 coupon payment), what is our
  return?

37
Holding Period Return

• Assume rates do not change (and here I mean that
  the exact same spot yield curve applies)
• What should the market price of the bond be at
  that time?
• 144.895
• How did I get this? As before CF1.9852CF2.9678
  
• What yield will I earn?
• N1
• PV-145.786
• FV144.8956150.895
• PMT0
• Solve for i3.5 (annualized7)
• Reconciliation
• Coupon 6.00
• Capital Loss (.89)
• Net 5.11 and 5.11/145.7863.5

38
Holding Period Return

• Only if I estimated the price at time of sale at
  t1 using a 5.879 YTM, would I get a holding
  period return equal to 5.879
• N19
• FV100
• PMT6
• i5.879/2
• Solve for PVSale Price144.07
• Now we have
• Coupon 6.00
• Cap Loss (1.716)
• Net 4.28 and 4.28/145.7862.9425.88
• But, for the YTM on 9.5 year bonds to be 5.879,
  the spot yield curve would have to shift (down)

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Bottom Lines

• Fixed Income Securities Trade on Price --- Not
  YTM
• YTM is a summary measure of the market return one
  can earn from a security
• No chicken and egg issue here, however. Clearly,
  the price comes first and the YTM is calculated
  based on the price.
• YTM is a weighted geometric average of all the
  different returns along the yield curve.
• Weights are the present values of the cash flows
• Using the YTM of one 10-yar bond to price other
  10-year bonds can generate very misleading values.