Interest Rates

1
Interest Rates
2
Types of Rates

• Three really important rates that serve as the
  basis for various yield curves
• Treasury (Sovereign) Rates
• Rate of return on debt issued by government. US
  Treasury is assumed to be the least risky of all
  of these.
• LIBOR London Interbank Offer Rate
• Rate at which banks are willing to make wholesale
  deposits to each other (i.e. loans.)
• Essentially AA credit.
• Repo Rates
• Rate for very short term, secured borrowing.

3
Repo Markets

• Repurchase agreements are a form of highly
  collateralized borrowing. This is the primary
  method through which government securities
  dealers finance their inventory.
• Size of this market is about 600 Billion.
• In a repurchase agreement, Party A buys
  securities from Party B, and simultaneously
  agrees to sell them back to Party B at a later
  time.
• Reverse Repurchases are opposite note that for
  every repo there must be a reverse repo.
• Repos are used by a number of investors, not
  just securities dealers funding inventory.

4
Repo Markets

• One important issue is that repos settle on the
  trade date, not three days later like most other
  trades.
• The price at which the security will be sold back
  to the original owner will be higher than the
  price at which it was sold, with the difference
  representing the cost of financing.
• The original owner of the security, however,
  continues to receive the coupon payments on the
  instrument.

5
Repo Example

• June 10, 1986
• Securities dealer purchases a bond for a price of
  94.03 and accrued interest. The invoice price
  works out to be 94.5422 (you dont have enough
  information to calculate that here). Since the
  face amount is 10 million, this means they must
  deliver 9,454,220 to the seller.
• To finance this, the dealer turns to the repo
  market. A repo dealer agrees to finance the deal,
  they will charge a repo rate of 6.
• To protect against the dealer defaulting, the
  dealer will take a 0.5 haircut, meaning that
  they will only provide the dealer with 99.5 of
  the value of the bond (in this case 9,406,948),
  the dealer must fund the rest.

6
Repo Example

• On June 13, the Securities Dealer re-takes
  possession of the bond from the Repo Broker. The
  Securities dealer must now pay the Repo Broker
  the original amount of the deal plus the interest
  earned at the repo rate
• 9,406,948.90 0.06 (3/360) 4,703.47So the
  total amount that must be paid is
• 9,411,652.37
• Remember that the Securities Dealer continues to
  earn the coupon on the bond, so they actually
  made 5,910 during this time.
• This is called a positive cost of carry.

7
Spot and Forward Rates

• In normal, everyday usage, when we say the phrase
  interest rate we are talking about a spot
  rate.
• Somewhat formally, the n-period spot rate is the
  interest rate charged on money borrowed at time 0
  and repaid at the end of time n. We will denote
  this as rn.
• Note that when rn is a zero coupon yield.

8
Spot and Forward Rates
Graphically, this can be shown on a timeline. For
example, r4 the 4 year spot rate is the rate
extending from time 0, through the end of the
fourth year.
Spot rate covers this time.
0 1 2 3
4 5
9
Spot and Forward Rates

• A forward rate, however, is the interest rate
  associated with a loan that you contract to
  today, but which will occur at some future date.
• Note that once you sign the forward agreement,
  you are bound to it, that loan will occur at the
  specified terms.
• We have to denote the beginning and ending points
  of the forward rate. To do this we will use the
  notation fm,n where m is the beginning date for
  the loan and n is the ending date. You enter into
  the forward contract at time 0.

10
Spot and Forward Rates
Graphically, this can also be shown on a
timeline. For example, there is a forward rate
for a 3 year loan which begins in exactly 1 year,
f1,4.
f1,4 covers this time.
Sign forward loan agreement at time 0
0 1 2 3
4 5
11
Spot and Forward Rates

• Our goal is to understand the fundamental
  relationship between forward rates and spot
  rates, and how the market enforces this
  relationship.
• Recognize that we define f0,1 rm. That is, the
  initial forward and future rates are the same.
• We also have to realize that there is no
  fundamental difference between borrowing money on
  with a two period spot rate, or by contracting to
  two consecutive forward rates.

12
Spot and Forward Rates
Clearly all of the cash flows and risks will be
the same. Thus f0,1 and f1,2 must have an
equivalence with r2.
Using 1 spot rate, you lock in your borrowing
rate for both periods 1 and 2.
0 1 2 3
4 5
Using 2 forward contracts (f0,1 and f1,2), you
lock in your borrowing rates at time 0 for both
periods 1 and 2.
0 1 2 3
4 5
13
Spot and Forward Rates

• Perhaps the easiest way to see this is with an
  example. Lets say that you observe the following
  term structure of interest rates

14
Spot and Forward Rates

• To keep things simple, we will assume an annual
  compounding frequency.
• Consider if you invested 1 at the two year spot
  rate. At the end of the second year, you would
  have 1.188.

15
Spot and Forward Rates

• Similarly, if you were to lock in a series of two
  forward rates, you would invest first at 8 for 1
  year and then at 10.0009 for the second year (but
  you lock in both rates at time 0!)

16
Spot and Forward Rates

• What if this were not the case? What if instead
  you found a bank that were willing to loan to you
  one year forward at 9. How could you exploit
  this opportunity for arbitrage?
• Clearly the bank is not charging enough in the
  second year, so you want to borrow from the bank
  and lend to the market.

17
Spot and Forward Rates

• You do this in the following way. First, you
  contract with the bank at their (incorrect)
  forward rate of 9.
• You then simultaneously borrow 1 in the spot
  market for 1 year (at 8) and lend in the spot
  market for 2 years at 9.

18
Spot and Forward Rates
You can see the timing of the events here.
Lend at 9 for 2 years
Borrow 1 in the spot market at 8, and
immediately lend it in the spot market at 9.
Borrow at 8 for 1 year
0 1
2
Borrow at 9 in year 2
Lock in the forward rate of 9 from the bank for
the second year.
0 1
2
19
Spot and Forward Rates

• The cash flows are relatively easy to work out
• Borrow 1 at 8 for 1 year. At the end of the
  year you must pay back 1(1.08) 1.08.
• You must, therefore borrow 1.08 from the bank at
  9 for the second year. You will have to pay back
  a total of 1.08(1.09) 1.177.
• You invested 1 at time 1 for 2 years at 9. You
  will receive 1(1.09)2 1.188 when that loan
  matures.

20
Spot and Forward Rates

• We can thus look at the payments on all legs of
  this trade in particular note the net
  position

21
Spot and Forward Rates

• Literally there is no risk to you, and you are
  guaranteed an positive cash flow later hence
  this is an arbitrage as the old Dire Straights
  song goes money for nothing.
• Clearly arbitragers would quickly take advantage
  of this mispricing in the market, and
  discipline the bank for this.

22
Spot and Forward Rates

• To see a little more complicated example, lets
  look at a real-world case.
• This is taken from the Wall Street Journal of
  January 22, 2000.

23
Spot and Forward Rates

• On January 22, we observed the following zero
  coupon bond prices
• Zero maturing in February, 2011 58.6875
• Zero maturing in February, 2012 55.125
• The yields on these bonds are
• 58.6875 100/(1r20 /2)20 r20 5.40
• 55.125 100/(1r22/2)22 r22 5.488

24
Spot and Forward Rates

• The one year spot rate beginning in year 10, i.e.
  in period 20 (since we are now back to
  semi-annual compounding) is given by

25
Spot and Forward Rates

• Lets say that you now found a bank that was
  offering forward rates of 9 for year 10. How
  could you exploit this?
• You would clearly want to borrow at the market
  rate of 6.47 and lend to the bank at their
  incorrect rate of 9.

26
Spot and Forward Rates

• The trade you would put together would be as
  follows
• Borrow 1000 in the spot market for 11 years at
  5.488
• Lend 1000 in the spot market for 10 years at
  5.40.
• Use the forward rate to lock in to lend to the
  bank at time 10 for 1 year at a rate of 9.

27
Spot and Forward Rates

• The cashflows would be
• Note that if you lend 1000 at 5.40 for 10 years,
  at the end of the 10 years you would receive
  1000(1.054/2)20 1,703.76.
• You would then lend this amount to the bank at 9
  for the 11th year. At the end of the 11th year
  the bank would pay back to you 1703.76(1.09/2)2
  1,860.55
• You would then have to pay back the original
  1000 you borrowed at 5.488, which would be
  1000(1.05488/2)22 1,814.02.

28
Spot and Forward Rates

• We wind up at the end of year 11, then, receiving
  1860.55 and paying back to the market 1814.02
  a net gain of 46.52
• So just like before we can see that we earn
  money for nothing.

29
Spot and Forward Rates

• We can thus look at the payments on all legs of
  this trade in particular note the net
  position

30
Pricing Conventions

• If you pick up the Wall Street Journal or some
  other source of Treasury Bond prices, the prices
  that you find in there are the quoted prices
  (sometimes called the clean or flat price.)
• If you were to buy one of those bonds, you would
  have to pay the quoted price plus the accrued
  interest.
• Calculating the accrued interest is not
  particularly difficult, just quirky.
• First, realize that Treasury Bonds/Notes pay ½ of
  their stated coupon every six months. Thus an 8
  bond pays 4 of the principal amount (1000)
  every 6 months.

31
Pricing Conventions

• The convention in the market is that if the bond
  is sold between coupon payment dates, the accrued
  interest is calculated as equal to the percentage
  of the time between coupon dates that the seller
  held the bond.
• Thus, if you are two-thirds of the way through
  the coupon period, the accrued interest that
  would have to be paid would be equal to
  two-thirds of the coupon payment that would be
  made at the next coupon date.
• This percentage is calculated on an
  actual/actual basis, meaning that you take the
  exact number of days since the last payment and
  divide it by the exact number of days between
  coupon payments.
• One effect of this is that since the number of
  days between payments will vary (from 178 to 184)
  the daily interest accrual rate changes from
  period to period!

32
Pricing Conventions
T
ND
LD

• Let LC stand for the last coupon payment date,
  let NC stand for the next coupon payment date,
  and let T stand for today. C is the annual coupon
  on the bond.
• To calculate the accrued interest, simply do the
  following

33
Pricing Conventions

• So lets say that we had an 8 bond, with a face
  value of 1000, that pays interest on February 15
  and August 15 of every year. If we purchase this
  bond on January 22, how much accrued interest
  would we owe?
• There are 184 days between August 15 and February
  15, and 160 days between August 15 and January
  22.
• The accrued interest, therefore is

34
Pricing Conventions

• Note, however, that since normally prices are
  quoted as a percentage of face (par) value, the
  accrued interest will also be quoted that way.
• This means that the 34.78 would be quoted as
  3.478 if prices were quoted in terms of par. Thus
  if the bond were quoted as a price of 103.5, the
  accrued interest would be quoted as 3.478.

35
Pricing Conventions

• It is very common in debt markets to quote bonds
  in terms of yield instead of price. Since the two
  are (generally) monotonic transformations of each
  other, traders use whichever is convenient. Using
  yield avoids confusion in quotes because of
  differences in par amounts, etc.
• Yield for Treasury Bonds and Notes are the same
  they are bond equivalent yields, and are quoted
  under the assumption that interest is paid on a
  semi-annual basis.
• The market quotes Treasury Bills differently.
  They are quoted on a discount basis.

36
Pricing Conventions

• Lets say that it is January 22, 2003, and you
  are quoted a rate of 1.25 on a T-Bill maturing on
  April 15, 2003. This trade will settle on January
  23, 2003.
• The actual price you would pay for the bill is
  given by the following formula
• Where SD is the settlement date and MD is the
  maturity date, and D is the rate of 1.25.

37
Pricing Conventions

• Notice that unlike the Treasury Bond, Treasury
  Bills pay interest on what is called the
  actual/360 basis. Thus, if you held a Treasury
  for exactly 1 calendar year, you would earn
  slightly more than the quoted rate!
• Converting between the discount yield and the
  bond equivalent yield is cumbersome, and depends
  on how many days the bill has outstanding.
• The book covers this in great detail, and you
  will implement this as part of the first project
  set.

38
Yield Conventions

• Treasury Bonds and notes are quoted on a Yield to
  Maturity convention, and Treasury Bills are based
  on a discount rate convention.
• Bonds that have a callable feature will be quoted
  on a yield to call basis meaning assume the
  bond is called on its call date and solve for
  yield.
• Callable bonds can also be quoted on a yield to
  worst basis, meaning solve for yield to maturity
  and yield to each call date (there may be more
  than one), and assume you will get the lowest of
  all of those yields.

39
Yield Conventions

• Recall that the yield curve is just a plot of
  each bonds yield against its maturity date for a
  given set of bonds.
• The following are the yield curves for January
  13, 2003, and July 21, 2003.
• The Federal Reserve releases interest rate data
  daily on their web site at http//www.federalreser
  ve.gov/releases/h15/update/
• They also have historical data available there.
  You will need this site to collect data for some
  of the projects.

40
Yield Curves
41
Measuring Risk in Fixed Income

• The most basic risk in fixed income is price
  risk. That is, that they price of the asset will
  change because of a change in interest rates.
• Normally, yields and prices are inversely
  related.
• The primary methods that finance people use to
  measure price risk is a concept known as
  duration, and its related concept of convexity.
• In the next slides we will examine these
  concepts.

42
Duration

• Duration is a measure of how much the price of a
  bond or other fixed-income asset will change when
  the discount rate changes.
• What duration measures is the instantaneous rate
  of change in price with respect to yield (i.e.
  the discount rate.)
• In other words, what we want to measure is the
  rate at which the bond price changes when yield
  changes.
• This means we want to know the slope of the price
  curve.
• Note that technically, the price of a bond is a
  mathematical function of interest rates.

43
Duration

• Recall that in general the price of a fixed
  income asset is given by the following formula
• Note that we are denoting price as a function of
  r P(r).

44
Duration

• For our purposes, it perhaps more convenient to
  write this as a product instead of as a quotient.
• A couple of rules from differential calculus are
  also useful to remember

45
Duration

• First, the derivative of a sum is equal to the
  sum of the derivatives. This means that we can
  treat each term of our summation independently.
• Second, when dealing with an equation of the
  form

46
Duration

• In this context, g(x) is (1r/m). So that means
  our derivative will be
• Notice that the 1/m term come from the fact that
  we have to take the derivative of (1r/m), which
  is simply 1/m.

47
Duration

• Reassembling this into a perhaps more
  conventional form

48
Duration

• Thus, the first derivative of price with respect
  to r is
• The first derivative tells us the instantaneous
  rate at which P is changing that is, it is the
  rate at which P is changing given a specific
  value of r.
• The derivative of a specific bond calculated at
  two different values of r will be different.
• Lets work a couple of examples to see exactly
  how this is calculated.

49
Duration

• Lets start with the simple example of a Treasury
  Bond that matures in exactly 1 year. Lets assume
  a coupon rate of 6, and that the current yield
  is 4.
• This bond will pay 30 in 6 months
• 1000 .06/2 30
• And 1030 in 1 year.
• The price of the bond, therefore is

50
Duration

• The first derivative of the bond with respect to
  price, therefore is given by
• or

51
Duration

• The first derivative of price with respect to r
  is frequently referred to in Finance as dollar
  duration.
• By convention the negative sign is usually
  omitted, so that dollar duration is quoted as a
  positive number.
• The reason that it is referred to as dollar
  duration is that you can use it to predict the
  dollar change in price for a given change in
  interest rate.
• To do this, you simply multiply the dollar
  duration by the change in rate (but you must keep
  in mind the sign of the change and dollar
  duration!).

52
Duration

• Mathematically this means
• Or, in this specific case
• So for a 10 drop in rate, you would expect the
  price of the bond to rise approximately 0.985

53
Duration

• In reality if rates fell from 4 to 3.9, the
  bonds price will rise from 1019.41 to 1020.40, a
  change of .98573.
• The reason that this is not exact, of course, is
  because duration uses a linear approximation to
  the curved price function we make a tradeoff
  between ease of calculation and accuracy.

54
Duration

• To demonstrate this, let us use another example,
  one using a longer-maturing treasury bond.
• In particular let us use a 30 year Treasury bond
  with a coupon of 8.
• If the yield on this bond is 8, then the bond is
  worth 1000, but at 10 it is worth 810.71.
• The following graph shows the price for all
  interest rates between 1 and 20.

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56
Duration

• Since there are 60 cash flows associated with a
  30 year treasury bond, it is probably easiest to
  work with the present value of an annuity formula
  to get this price.

• Which in this case would work out to be (at a 10
  yield)

You may wish to verify for yourself that if
r10, the price is 1000.
57
Duration

• We could use the same formula for the derivative
  that we did in the original equation, but, with
  60 cash flows, it is cumbersome to do so.
• Instead we can use a variation of that formula
  that is based on the present value of annuity
  formula we just used. That formula is

58
Duration

• So, at 10, the first derivative of the bond with
  respect to yield would be given by
• Or

59
Duration

• At a yield of 8, the first derivative of the
  treasury bond is 11311.7
• Recall, that the first derivative tells us the
  slope of the curve for an instantaneous change in
  rate. The next slide presents these slopes

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61
Duration

• From the graph we can see that as the interest
  rate increases, the curve becomes less steep
  indicating that as price of the bond is less
  sensitive to interest rate changes.
• By looking at the graph you can see that the rate
  at which price changes in not constant.
• What we want to do is develop a measure of the
  rate of change given a specific yield.
• Dollar duration provides this measure, but it
  does have some drawbacks.

62
Duration

• One drawback in particular is that it is
  difficult to compare the relative risk of two
  bonds that have different face amounts.
• A bond with a 5000 face amount will have a
  derivative that is 5 times larger than one with a
  1000 face amount.
• It would be nice if we could have a somewhat more
  standardized way of measuring the risk.

63
Duration

• There are several other variants of duration
  other than dollar duration. They include
• DV01
• Modified Duration
• Macaulays duration
• Usually finance textbooks will provide you with
  either Modified or Macaulays duration (which is
  why the number so far may have seemed a little
  odd-looking to those of you that have seen
  duration in other courses.)
• Let us examine each of these in detail.

64
Duration

• DV01
• Since dollar duration numbers tend to be large in
  absolute terms, it is more convenient to scale
  them. One way of scaling them is to multiply
  them by a small yield amount. One choice is to
  use 1 basis point. This will tell you the
  approximate change in an instruments price for a
  1 basis point change in yield.

65
Duration

• DV01
• This measure is known as the Dollar Value of an
  01 or simply as DV01.
• It is used primarily to compare the magnitude of
  dollar changes across assets.
• Unfortunately, it does not take into account the
  scale of the underlying asset. That is, an asset
  with a face amount of 100,000 would have a DV01
  100 times greater than an identical asset with
  face amount of 100,000.

66
Duration

• DV01
• In the example presented earlier the DV01 for the
  bond would be
• DV01 (dp/dr) .0001 -7877.63 -.0001
  .07877
• Note that this is expressed in Dollars.

67
Duration

• Modified Duration
• Modified duration is a way of taking into account
  the scale of the asset being measured.
• Essentially it is dollar duration divided by
  price.
• When a trader or most data sources refer to
  duration they normally mean modified duration.
• This is also the variant of duration that can be
  viewed as a true time measure

68
Duration

• Modified Duration
• If you multiply modified duration by a change in
  interest rates, it gives you the approximate
  percentage change in price for the asset.
• Using modified duration to measure the interest
  rate risk in an asset lets one avoid the scaling
  difficulty of the DV01 measure.
• In our previous example, modified duration would
  be
• Mod. Duration dp/dr 1/p 7877.63
  1/810.71 9.7169.
• Remember that, in general, the larger the
  duration number, the greater the interest rate
  risk.

69
Duration

• Macualays Duration
• The first derivation of duration was made in the
  1930s by an economist named Macaulay. He was
  not thinking of it as a risk measure per se, but
  rather as the price elasticity of a bond with
  respect to interest rates. As such his measure
  is given by
• One interesting fact is that for any bond with
  only one cash flow the Macaulays duration of
  that bond will exactly equal its maturity!

70
Duration

• So there are actually at least four measures of
  duration
• Dollar Duration (dp/dr)
• DV01 (dp/dr .0001)
• Modified Duration (dp/dr 1/p)
• Macaulays Duration (dp/dr (1r/m)/p).
• Note that many books refer to duration as a time
  measure. It is possible to construe it that way,
  but I think it is much more useful to think of it
  as a rate of change.
• Also, recall that it is really a negative number
  (in most cases), it is just the convention in
  finance that we omit the negative sign.

71
Duration

• Complications with Duration
• The example we have worked with so far considers
  a case where the cash flows from the bond are
  certain. What if they are not?
• If the cash flows do not vary with interest
  rates, then you would calculate duration as
  normal just realize there may be risks which
  duration is not capturing.
• For example, some companies issue bonds that have
  contract rates which depend upon the price of
  some factor of production some ski resorts have
  issued bonds where the interest rate is a
  function of the amount of snow they get.
• You can still calculate duration as normal just
  realize that interest rate risk is not the only
  risk in the bond.

72
Duration

• Complications with Duration
• Of course some assets, like mortgages, have cash
  flows that do vary with interest rates.
• This means that the simple derivative formula
  does not work cash flow itself must be treated
  as a function of r, and so one must, at a
  minimum, use the chain rule to extend the
  derivative.
• Frankly, this is not commonly done. The reason
  is that most good prepayment models are so
  complex that they do not have easily computed
  derivatives.
• Analysts can do one of two things, therefore.
  They can either
• Ignore that cash flows are a function of r
• Approximate duration

73
Duration

• Complications with Duration
• Both ways are fairly common, although if you
  ignore the fact that cash flow is a function of
  interest rates, you will misstate duration. If
  you feel the misstatement is small enough, you
  may choose to do this.
• You approximate duration by approximating the
  derivative. To do this you calculate the price
  of the asset at two points on either side of the
  current rate
• For example, if the discount rate were at 10,
  you would determine the price at 9.9 and 10.10,
  and then divided the difference in prices by the
  20 basis point difference in yield. This
  approximates the slope and hence the derivative.
• Example in our previous example, the price of
  the bond at 9.9 is 818.65 and at 10.10 is
  802.90.

74
Duration

• Complications with Duration
• We can approximate duration as follows
• Clearly this yields an approximate duration which
  is very close to the true duration.
• This numerical approximation for duration is
  commonly used in financial modeling and financial
  modeling software packages.

75
Duration and Taylors Theorem

• Fundamentally duration is an application of
  Taylors Theorem from mathematics. Taylors
  theorem says simply that if you know the value of
  a function and all of its derivatives at a given
  point (x), then you can calculate its value at
  any other point (xh). The exact formula is

76
Duration and Taylors Theorem

• What we do when we use duration is we simply use
  the first two terms of this formula and drop the
  rest (although we frequently will add the second
  term it is called convexity).
• That is, for a bond price (r), we use

77
Duration and Taylors Theorem

• What this says is that if we have an asset with a
  known price at a given interest rate, (p(r)),
  then if we change r by dr, the price at that new
  rate p(rdr), will be approximately equal to the
  old price plus the change in rate times the first
  derivative of the pricing function (which we call
  dollar duration!).
• The reason our value is not an exact match is
  because we drop those higher order terms.
• This can be extremely useful if are told the
  price of the bond and want to determine its
  yield.

78
Duration and Taylors Theorem

• To see this, consider the first bond that we used
  in this section.
• Recall that that bond had a coupon rate of 6. It
  had a yield of 4, and thus had a price of
  1019.41.
• Now, instead lets say that you were simply told
  that the price of the bond were 1005.00, and were
  asked to find its yield, which we will denote as
  Y. How could you do this?
• One approach would be to use Taylors Theorem.
• We begin by simply guessing a yield, say 5, and
  then determining the price of the bond at a yield
  of 5

79
Duration and Taylors Theorem

• The price if the is 5 is
• and the first derivative of price at a yield of
  5 is

80
Duration and Taylors Theorem

• Recall that Taylors theorem says that the value
  of a function at a point x (i.e. f(xh)) is given
  byor, ignoring the higher order terms by
• Realize that we know the price of the bond at
  5 and we know its derivative at 5, we also know
  that when we find the current yield of the bond,
  its price will be 1005 (we were given that!).

81
Duration and Taylors Theorem

• We can think of the price when the yield is 5 as
  being f(x), the derivative of the price when the
  yield is 5 as being f(x), and the price of the
  bond at the (still unknown) correct yield as
  being f(xh) (h is the difference between the
  correct yield Y and our guessed yield of 5.)
  Thus,
• f(x) 1009.63
• f(x) -970.73
• f(xh) 1005.00
• If we insert these into Taylors equation, we get
  the following

82
Duration and Taylors Theorem

• Recall the formulaSo we get
• We can then solve for h, the difference between
  5 and the yield which will set the price of the
  bond to 1005

83
Duration and Taylors Theorem

• So we add .0047710 to our initial guess of 5 for
  an updated guess of 5.4771
• Which is obviously pretty close to the correct
  yield, since when we use 5.4771, we get a price
  of 1005.0217.
• We can repeat this process a second time to get
  an even closer price, first, we the first
  derivative at 5.4771.

84
Duration and Taylors Theorem

• Plugging back into Taylors theoremSo we
  get
• We can then solve for h, the difference between
  5 and the yield which will set the price of the
  bond to 1005

85
Duration and Taylors Theorem

• So we add .00002251 to our latest guess of
  5.4771 for an updated guess of 5.4779351
• Which is close enough for our purposes. If you
  needed a more accurate answer, you can repeat the
  process to any level of accuracy required.
• This is the exact process that your calculator
  and Excel use to solve for yields (or IRRs,
  which are the same thing.)

86
Duration and Taylors Theorem

• This type of search algorithm is known as a
  Newton-Raphson method, although frequently it is
  called simply Newtons method. It is one of a
  general category of search algorithms known as
  Gradient Descent algorithms.
• These search algorithms work well for most
  financial problems.
• The general rule of the algorithm, therefore is
  as follows

87
Duration and Taylors Theorem

• If you know the price of the bond and want to
  solve for the yield
• First select an initial guess for the yield.
• With that guessed value determine the price and
  first derivative of the bond.
• Use Taylors Theorem to update your guessed
  value
• Update your initial guess by h, and see if you
  are close enough.
• If you are close enough, then stop, otherwise
  repeat steps 2-4.

88
Convexity

• As mentioned earlier, Taylors Theorem uses
  higher order derivatives. It is common in finance
  to use only the first, although occasionally we
  will use the second derivatives as well.
• The second derivative is generally known as
  Convexity, and it measures the rate at which the
  first derivative (duration) changes when the
  underlying interest rate changes.

89
Convexity

• The book defines convexity to be
• There are a couple of items to note with this
  definition.
• First, its a modified convexity, that is, it
  is quoted in terms of percentage. A dollar
  convexity would omit the 1/P term.
• Second, it already takes part of the Taylor
  series (the ½) into account.

90
Convexity

• Recall that earlier we noted that the first
  derivative with respect to price was
• The second derivative, therefore must be given
  by

91
Convexity

• Recall that Taylors theorem states
• Or, putting it into books terms
• Or in percentage terms

92
Convexity

• Now, you have to be a little bit careful of one
  other issue, and that has to do with compounding
  frequency.
• Writing out Taylors theorem as we did in the
  last slide, one has to recognize that we are
  working in periodic interest rates. Recall from
  equation (2) of two slides ago that buried within
  it is a 1/m term. For semi-annual paying bonds,
  this will be ½, so there are two ½ terms in the
  convexity portion of the Taylor expansion for the
  semi-annual paying bond.

93
Convexity

• To see the effect this has, lets consider a
  simple example of a 100 bond with a 10 coupon
  that pays interest semi-annually and that has one
  year to maturity. If current discount rates are
  12, the price function is
• And the first derivative with respect to price is

94
Convexity

• And the second derivative with respect to price
  is
• Lets say the annual discount rate changes from
  12 to 10. Based on Taylors theorem we can
  approximate the change using just duration as

95
Convexity

• If we incorporate the convexity term, we will
  wind up with the following estimated price

96
Convexity

• The actual price would be exactly par
• So clearly the estimate with convexity is closer
  that the estimate without convexity.
• Although we have spent a lot of time working with
  the exact cash flows, its worth noting that his
  way of estimating convexity is usually quite
  convenient and is widely used on Wall Street

97
Theories of the Term Structure

• Over the years various researchers have attempted
  to relate spot rates to forward rates that is
  one period future spot rates to currently
  observed forward rates.
• Four primary hypotheses have emerged
• The Expectations Hypothesis
• Liquidity-Premium Hypothesis
• Market Segmentation Hypothesis
• Local-Expectations Hypothesis
• Lets briefly examine each one.

98
Expectations Hypothesis

• There are multiple variants of this hypothesis.
  They ultimately all try to develop the notion
  that forward rates are some form of the markets
  estimate of future spot rates.
• The unbiased expectations hypothesis formally
  states just that, i.e. fk,k1 EtRk.
• The difficulty is, that empirically forward rates
  are terrible predictors of futures spot rates. If
  these are the markets expectations, the market
  consistently sets its expectations of future
  forward rates too high.

99
Liquidity-Preference Theory

• This says that lenders would, all things equal,
  prefer to lend for shorter periods of time. To
  induce them to lend long term, therefore,
  borrowers must pay an additional premium, which
  shows up in the form of an upward sloping yield
  curve, which generates forward rates that are
  greater than the markets actual expected spot
  rate, i.e. fk,k1 EtRk pk,k1.
• The difficulty is that sometimes the yield curve
  is inverted, which would imply that investors
  preferences changed.
• This is, in some ways, the most convincing of
  these arguments.

100
Market-Segmentation

• This is not so much a theory as an anti-theory
  basically it says that the long-term and
  short-term markets are different, and that only
  if they are grossly out of line with each other
  will participants in one market cross over to
  participate in the other.
• Basically it says that the fundamental premise of
  a yield and/or forward rate coupon curve is
  flawed that the curve tends to give the illusion
  that there is a relationship that is not there.
• Generally this theory is not particularly useful,
  especially in the context of fixed income pricing.

101
Local Expectations Hypothesis

• This is a special case of the expectations
  hypothesis. It basically says that all bonds (of
  the same credit quality) have the same rate of
  return for a very short period of time. This is
  the only arbitrage-free model!
• Short is normally defined to be whatever is the
  smallest time period in the model being used,
  i.e. its instantaneous for continuous time
  models, but maybe one month for discrete-time
  models.
• Some of the discrete time models really push this
  to the absolute limit.
• Formally it is EZT-(t1)/ZT-t (1rt)

102
Coupons, Zeros and Strips (oh my!)

• In the market we primarily observe coupon bearing
  bonds, although many times we wish to work with
  zero coupon bonds this is especially true in
  the context of building arbitrage-free term
  structure models.
• We really have two ways of getting zero
  information extracting them from coupon bearing
  bonds in a process known as bootstrapping, and
  directly observing them in the strips market.
• Why ever use the bootstrapping method? Primarily
  because the coupon market is still larger and
  more liquid than the strips market, and as a
  result traders have more confidence in the quotes
  from that market.

103
Bootstrapping

• Bootstrapping is simply a procedure for
  extracting zero coupon bond rates from coupon
  bearing bonds. It is most commonly used in the
  Treasuries market.
• The basic idea is that you start with an initial
  short term bond typically a 1 month or 3 month
  bond, which is truly a zero coupon bond, and you
  calculate its yield.
• You continue calculating the zero coupon bonds
  until you run into your first coupon-bearing
  bond. For an examples sake, lets say that you
  observe a six month and a twelve month zero
  (which is typical), and that your first coupon
  bearing bond matures at month 18. Denote the zero
  coupon yields as Z6 and Z12.

104
Bootstrapping

• Typically you use a par-value coupon bearing
  bond, that means one for which you know the price
  is 100. Given that you know the values of Z6 and
  Z12, what you do is find the value of Z18 that
  makes this statement true
• For example, let us say that Z610, Z12 12,
  and that the coupon on a par-priced coupon bond
  is 13. The 18 month zero would be

105
Bootstrapping

• One major problem with bootstrapping is that you
  dont really get bonds spaced evenly every six
  months, and if you are working with a model where
  you need monthly zeros, you really have trouble
  finding it.
• Constant Maturity Treasuries (CMT) help this a
  lot. They provide you on a daily basis with rates
  for bonds that have maturities of 1 year, 2 year,
  3 years, 5 years, etc.
• You still have to make assumptions about what
  happens in-between those bonds. One approach is
  to get more data i.e. find interest rate
  derivatives that do mature between the bonds.

106
Bootstrapping

• The second approach is to make some assumption
  about what is going on in-between the observed
  rates. In essence you simply assume how the rates
  will behave.
• A common (but not particularly good) assumption
  is to assume that the par bonds are linear in
  coupon between observed points. That is, if Cn
  and Cn2 are the coupons of par bonds maturity at
  times n and n2, and you need a coupon for a bond
  which matures at time n1 but there is no such
  bond trading, simply use linear interpolation to
  estimate Cn1, i.e. Cn1 (CnCn2)/2
• The problem with this approach is that it can
  lead to some rather bizarre-looking forward rate
  structures.

107
Bootstrapping

• Other commonly used methods include various curve
  fitting algorithms such as piecewise cubic
  splines, cubic discount rate generating
  functions, and statistical methods such as
  Diaments method outlined in Sundaresans book.
• While cubic-spline methods are pretty common in
  commercial software, there is still a lot of
  variation in how various firms elect to fit their
  curves. All methods involve tradeoffs, such as
  being willing to accept a more jagged curve in
  some regions in exchange for a smoother curve in
  other regions.
• There is not one universally accepted method.

108
Duration Appendix

• Recall the present value of an annuity formula
• It is convenient to rewrite this a little and
  then re-express it as a product

109
Duration Appendix
110
Duration Appendix

• Recall the product rule from differential
  calculus
• In our case we note that

111
Duration Appendix

• We can easily calculate the derivative for v(x),

112
Duration Appendix

• The derivative for u(x) is slightly more
  difficult,

113
Duration Appendix

• We now have each component we need to calculate
  the derivative for the entire pva formula

114
Duration Appendix

• So plugging in the various values yields

115
Duration Appendix

• So the formula to calculate the derivative of an
  annuity with respect to interest rates is
• If there is a final principal cash flow you must
  modify this to include that cash flow