Interest Rates and price determination

1
Interest Rates and price determination

• Fin 288
• Futures Options and Swaps

2
PV and FV in continuous time

• e 2.71828 y lnx x ey
• FV PV (1k)n for yearly compounding
• FV PV(1k/m)nm for m compounding periods per
  year
• As m increases this becomes
• FV PVern PVert let t n
• rearranging for PV PV FVe-rt

3
Compounding periods

• The PV of 100 assuming 8 per year a various
  compounding periods for 5 years.

of periods per year PV
2 44.63
4 45.29
12 45.05
52 44.96
continuous 44.93
4
Some useful conversions
Given Rccontinuous compounding
rate Rmequivalent rate with m compounding
periods
5
A General Valuation Model

• The basic components of valuing any asset are
• An estimate of the future cash flow stream from
  owning the asset
• The required rate of return for each period based
  upon the riskiness of the asset
• The value is then found by discounting each cash
  flow by its respective discount rate and then
  summing the PVs (Basically the PV of an Uneven
  Cash Flow Stream)

6
The formal model (Discrete Time)

• The value of any asset should then be equal to

7
Components of the General Model

• Cash Flow at time t (CFt) the expected future
  cash flow that the owner of the asset expects to
  receive at time t.
• The future cash flow may not be known with
  certainty.
• The Discount Rate The return that investors in
  the market are requiring for owning the asset.
• The discount rate should reflect the risks faced
  by the investor. What risks are faced???

8
The Discount Rate

• The required rate can be seen as an aggregation
  of the different forces that impact the riskiness
  of owning the asset
• rRRIPDPMPLPEP
• RR The Real Rate of Interest (reward for saving
  or investing instead of consuming)
• IP The Inflation Premium
• DP Default Risk Premium
• MP Maturity Premium
• LP Liquidity Premium
• EP Exchange Rate Risk Premium

9
The General Model

• Note
• The interest Rate and Cash Flows can change with
  each period.
• We will start with a basic bond, assuming that
  the discount rate is constant across periods.

10
Applying the general valuation formula to a bond

• What component of a bond represents the future
  cash flows?
• Coupon Payment The amount the holder of the bond
  receives in interest at the end of each specified
  period.
• The Par Value The amount that will be repaid to
  the purchaser at the end of the debt agreement.

11
Basic Bond Mathematics

• Given
• r The interest rate per period or return paid on
  assets of similar risk
• CP The coupon payment
• MV The Par Value (or Maturity Value)
• n the number of periods until maturity
• The value of the bond is represented as

12
Applying the formula

• Assume that we bought a 9 yearly coupon bond
  with 20 years left to maturity and one year later
  the required return decreased to 7.
• What is the value of the bond?
• 19 N 7 I 90 PMT 1,000 FV PV1,206.71

13
Why 1,206.71?

• New bonds of similar risk are only paying a 7
  return. This implies a coupon rate of 7 and a
  coupon payment of 70.
• The old bond has a coupon payment of 90,
  everyone will want to buy the old bond, (the
  increased demand increases the price)
• Why does it stop at 1,206.71?
• If you bought the bond for 1,206.71 and received
  90 coupon payments for the next 19 years you
  receive a 7 return.

14
Quick Facts for Review

• If the level of interest rates in the economy
  increases the bond price decreases and vice
  versa.
• If rgtCoupon rate the price of the bond is below
  the par value - it sells at a discount.
• If rltCoupon rate the price of the bond is above
  the par value - it sells at a premium.
• Keeping everything constant the value of the bond
  will move toward par value as it gets closer to
  maturity.

15
The formal model (Continuous Time)

• The value of any asset should then be equal to

16
A Package of Zero Coupon Bonds

• You can think of a bond as a package of zero
  coupon bonds. Each payment received represents a
  zero coupon bond.
• Since yields differ with maturity, this implies
  that it does not make sense to use only one rate
  (the YTM) to value the bond.
• The value of the bond should be the same
  regardless of which way it is valued.

17
Stripped Coupons

• If the value of the individual coupons is
  different from the entire bond it would be
  possible to buy the bond and sell the coupons as
  a security making a risk free profit.
• To value the stripped coupons each would be
  valued at a rate that matches its maturity, the
  rate should also represent a zero coupon bond.
• We can use the information in the market to
  create a zero coupon yield curve.

18
Zero Coupon (spot) Rates

• The n year spot interest rate is the rate that
  would be earned today on an investment lasting n
  years
• Assumes that no interest or coupon payments are
  made.
• The Forward Rate will be the rate between two
  points in time in the future implied by the zero
  coupon yield curve

19
Spot Rate Yield Curve

• The spot rate yield curve will be the value of
  the pot rate at different maturities.
• Often this is calculated for treasury securities
  based on the assumption that treasuries are
  risk free

20
Theoretical Spot Rate Curves

• Two main issues
• Given a series of Treasury securities, how do you
  construct the curve?
• Linear Extrapolation
• Bootstrapping
• Other
• What Treasuries should be used to construct it?
• On the run Treasuries
• On the run Treasuries and selected off the run
  Treasuries
• All Treasury Coupon Securities and Bills
• Treasury Coupon Strips

21
Observed Yields

• For on-the-run treasury securities you can
  observe the current yield.
• For the coupon bearing bonds the yield used
  reflects the yield that would make it trade at
  par. The resulting on the run curve is the par
  coupon curve.
• However, you may have missing maturities for the
  on the run issues. Then you will need to
  estimate the missing maturities.

22
Example

• Maturity Yield
• 1 mo 1.7
• 3 mo 1.69
• 6 mo 1.67
• 1 yr 1.74
• 5 yr 3.22
• 10 yr 4.14
• 20 yr 5.06

• The yield for each of the semiannual periods
  between 1 yr and 5 yr would be found from
  extrapolation.

23

• 5 yr yield 3.22 1 yr yield 1.74
• 8 semi annual periods

24
Bootstrapping

• To avoid the missing maturities it is possible to
  estimate the zero spot rate from the current
  yields, and prices using bootstrapping.
• Bootstrapping successively calculates the next
  zero coupon from those already calculated.

25
Treasury Bills vs. Notes and Bonds

• Treasury bills are issued for maturities of one
  year or less. They are pure discount instruments
  (there is no coupon payment).
• Everything over two years is issued as a coupon
  bond.

26
Bootstrapping example

• Assume we have the following on the run treasury
  bills and bonds
• Assume that all coupon bearing bonds (greater
  than 1 year) are selling at par (constructing a
  par value yield curve)
• Maturity YTM Maturity YTM
• 0.5 4 2.5 5.0
• 1.0 4.2 3.0 5.2
• 1.5 4.45 3.5 5.4
• 2.0 4.75 4.0 5.55

27
Bootstrapping continued

• Since the 6 month and one year bills are zero
  coupon instruments we will use them to estimate
  the zero coupon 1.5 year rate.
• The 1.5 year note would make a semiannual coupon
  payment of 100(.0445)/22.225
• Therefore the cash flows from the bond would be
• t0.52.225 t12.225 t1.5102.225

28
Bootstrapping continued

• A package of stripped securities should sell for
  the same price (100 par value) as the 1.5 year
  bond to eliminate arbitrage.
• The correct semi annual interest rates to use
  come from the annualized zero coupon bonds
• r0.5 4/2 2 r1.0 4.2/2 2.1

29
Bootstrapping continued

• r0.5 4/2 2 r1.0 4.2/2 2.1
• The price of the package of zero coupons should
  equal the price of the theoretical 1.5 year zero
  coupon

30
Bootstrapping continued
31
Bootstrapping continued

• The semi annual rate is therefore 2.2293 and the
  annual yield would be 4.4586
• Similarly the 2 year yield could be found
• the coupon is 4.75 implying coupon payments of
  2.375 and cash flows of
• t0.52.375 t1.02.375 t1.52.375
  t2.0102.375

32
Bootstrapping continued
33
Bootstrapping continued

• What is the 2.5 year par value zero coupon rate?
  The coupon is 5

34
Bootstrapping part 2 continuous time

• Now assume you know the following information
  concerning 100 par value bonds
• Time Annual Coupon Bond Price
• .25 0 97.50
• .50 0 94.90
• 1.0 0 90.00
• 1.5 8 96.00
• 2.0 12 101.6

35
The zero spot rates

• If you purchase the 3 month bond today for 97.50,
  you receive 100 in 3 months.
• This implies a 2.50/97.5 .0256 return over the
  three months.
• The yearly continuous compounding return would
  then be
• 4ln(1.0256) .1013

36
Zero Coupon Rates

• Similarly the 6 month rate would be .1047 and the
  one year rate would be .1054

37
Bootstrapping

• The 1.5 year bond makes 8 yearly coupon payments
  each year of 4 each 6 months.
• Given the current price and the zero coupon
  rates the 1.5 year rate can be found

38
Bootstrapping

• You can then solve for the 2 year rate using the
  1.5 year rate and the information from the 2 year
  bond.

39
Forward Rates

• Using the theoretical spot curve it is possible
  to determine a measure of the markets expected
  future short term rate.
• Assume you are choosing between buying a 6month
  zero coupon bond and then reinvesting the money
  in another 6 month zero coupon bond OR buying a
  one year zero coupon bond.
• Today you know the rates on the 6 month and 1
  year bonds, but you are uncertain about the
  future six month rate.

40
Forward Rates

• The forward rate is the rate on the future six
  month bond that would make you indifferent
  between the two options.
• Let z1 the 6 month zero coupon rate
• z2 the 1 year zero coupon rate (semiannual)
• f the rate forward rate from 6 mos to 1 year.

41
Returns

• Return on investing twice for six months
• (1z1)(1f)
• Return on the one year bond
• (1z2)2
• If you are indifferent between the two, they must
  provide the same return
• (1z1)(1f) (1z2)2
• or
• f ((1z2)2/(1z1))-1

42
Forward Rates

• Forward rates do not generally do a good job of
  actually predicting the future rate, but they do
  allow the investor to hedge
• If their expectation of the future rate is less
  than the forward rate they are better off
  investing for the entire year and lock in the 6
  month forward rate over the last 6 months now.

43
Forward rate - continuous time

• Assume you know the following zero coupon rates
  continuous rates
• 1 year 3 2 years 4
• The one year return on 100 is
• 100e.03 103.04545
• The total value after two years is
• 100e.04(2) 108.33

44
Forward rate

• The forward rate (f) is the rate that would make
  the investment from time 1 to time 2 have the
  same return as the two year return or
• (100e.03)ef 108.33
• 103.04545ef 108.33
• ef1.05127
• ln(ef)fln(1.05127)
• f .05

45
Continuous compounding

• Notice that in the previous example the two year
  rate was the average of the one year rate and the
  forward rate.
• This occurs because we are looking at time 1 to
  time 2 and continuous compounding allows for a
  nice generalization.

46
Generalization Forward Rates

• Given
• R1 and R2 The zero rate for maturity t1 and t2
• T1 and T2 The number of periods t1 and t2
• Rf The forward rate between the periods 1 and 2

47
Treasury Yield Curve

• The most commonly investigated and used term
  structure is the treasury yield curve. (will
  want to look at zero rates)
• Treasuries are used since they are
• Considered free of default, and therefore differ
  only in maturity
• The benchmark used to set base rates
• Extremely liquid

48
Yield Curves Over the Last Year
49
US Treas Rates Jan 1990 Dec 2003
50
Three Explanations of the Yield Curve

• The Expectations Theories
• Segmented Markets Theory
• Preferred Habitat Theory

51
Pure Expectations Theory

• Long term rates are a representation of the short
  term interest rates investors expect to receive
  in the future. In other words the forward rates
  reflect the future expected rate.
• Assumes that bonds of different maturities are
  perfect substitutes
• In other words, the expected return from holding
  a one year bond today and a one year bond next
  year is the same as buying a two year bond today.
  (the same process that was used to calculate our
  forward rates)

52
Pure Expectations

• Given a two period model in continuous time we
  just showed that the 2 period rate will be equal
  to the average of the 1 period rate and the
  forward rate.

53
Expectations Hypothesis R2 (RfR1)/2

• When the yield curve is upward sloping (R2gtR1) it
  is expected that short term rates will be
  increasing (the average future short term rate
  is above the current short term rate).
• Likewise when the yield curve is downward sloping
  the average of the future short term rates is
  below the current rate. (Fact 2)
• As short term rates increase the long term rate
  will also increase and a decrease in short term
  rates will decrease long term rates. (Fact 1)
• This however does not explain Fact 3 that the
  yield curve usually slopes up.

54
Problems with Pure Expectations

• The pure expectations theory ignores the fact
  that there is reinvestment rate risk and
  different price risk for the two maturities.
• Consider an investor considering a 5 year horizon
  with three alternatives
• buying a bond with a 5 year maturity
• buying a bond with a 10 year maturity and holding
  it 5 years
• buying a bond with a 20 year maturity and holding
  it 5 years.

55
Price Risk

• The return on the bond with a 5 year maturity is
  known with certainty the other two are not.
• The longer the maturity the greater the price risk

56
Reinvestment rate risk

• Now assume the investor is considering a short
  term investment then reinvesting for the
  remainder of the five years or investing for five
  years.
• Again the 5 year return is known with certainty,
  but the others are not.

57
Local expectations

• Local expectations theory says that returns of
  different maturities will be the same over a very
  short term horizon, for example three months.

58
Return to maturity expectations hypothesis

• This theory claims that the return achieved by
  buying short term and rolling over to a longer
  horizon will match the zero coupon return on the
  longer horizon bond. This eliminates the
  reinvestment risk.

59
Liquidity Theory

• This explanation claims that the since there is a
  price risk associated with the long term bonds,
  investor must be offered a premium. Therefore
  the long term rate reflects both an expectations
  component and a liquidity premium.
• This tends to imply that the yield curve will be
  upward sloping as long as the premium is large
  enough to outweigh an possible expected decrease.

60
Segmented Markets Theory

• Interest Rates for each maturity are determined
  by the supply and demand for bonds at each
  maturity.
• Different maturity bonds are not perfect
  substitutes for each other.
• Implies that investors are not willing to accept
  a premium to switch from their market to a
  different maturity.
• Therefore the shape of the yield curve depends
  upon the asset liability constraints and goals of
  the market participants.

61
Preferred Habitat Theory

• Like the liquidity theory this idea assumes that
  there is an expectations component and a risk
  premium.
• In other words the bonds are substitutes, but
  savers might have a preference for one maturity
  over another (they are not perfect substitutes).
• If there are demand and supply imbalances then
  investors might be willing to switch to a
  different maturity.

62
Preferred Habitat Theory

• The long term rate should include a premium
  associated with them. To attract savers who
  prefer a shorter maturity, the long term bond
  will need to pay an additional amount or term
  (liquidity) premium.
• Thus according to the theory a rise in short term
  rates still causes a rise in the average of the
  future short term rates. Therefore the long and
  short rates move together (Fact 1).

63
Preferred Habitat Theory

• The explanation of Fact 2 from the expectations
  hypothesis still works. In the case of a
  downward sloping yield curve, the term premium
  (interest rate risk) must not be large enough to
  compensate for the currently high short term
  rates (Current high inflation with an expectation
  of a decrease in inflation). Since the demand
  for the short term bonds will increase, the yield
  on them should fall in the future.

64
Preferred Habitat Theory

• Fact three is explained since it will be unusual
  for the term premium to be so small that the
  yield curve slopes down.

65
Price Determination in Forward and Futures Markets

• Fin 288
• Futures Options and Swaps

66
Determining the delivery price

• The delivery price will be determined by the
  participants expectations about the future price
  and their willingness to enter into the contract.
  (Todays spot price most likely does not equal
  the delivery price).
• What else should be considered?
• They should both also consider the time value of
  money

67
Theoretical Pricing of Futures Contracts

• The theoretical price Is based upon the
  elimination of arbitrage opportunities.
• Start with a simple example
• Assume transaction costs are zero
• Assume that storage costs are zero
• You have a choice today of purchasing or selling
  a given asset or entering into a contract to buy
  or sell it in the future.

68
Theoretical Price

• Assume you want to own the asset at a given point
  in time in the future, You can enter into a long
  futures position or buy the asset today and hold
  on to it.
• If you enter into the futures contract you can
  invest your cash today and earn interest ( r)

69
Basic Relationship

• The Forward Price (F) should equal the spot price
  (S) plus any interest that could be received on
  an amount of cash equal to the spot price or
• Note The book uses continuous compounding to
  illustrate the same result

70
Eliminating Arbitrage

• If the forward price is greater than the spot
  plus interest an arbitrage opportunity exists.
• Borrow to buy the underlying asset in the spot
  market and take a short position in the futures
  contract. (for now we will use forward and
  futures price as if they are the same thing)

71
Numerical Example

• Consider an asset that is currently selling at
  30 The asset has a two year futures price of
  35. The risk free rate is 5

At Time 0 Borrow 30 (will need to repay
30(1.05)233.075 Buy asset for 30 Take Short
Futures Position
At Time 2 Deliver Asset in Futures Receive
35 Payoff loan with 33.075 Profit 35-33.075
1.925
72
Example cont

• Increased demand for short contracts, the of
  participants willing to sell in two years will be
  greater than the number willing to buy.
• Those willing to sell will compete by lowering
  their price therefore the futures price
  declines...

73
Eliminating Arbitrage Part 2

• What if the futures price is less than the spot
  price plus interest?
• Short Sell the underlying asset and take a long
  position in the futures market

74
Numerical example

• What if the futures price is 31 instead of 35?
  Leave the spot price at 30 and r at 5

At time 0 Short sell the asset and receive
30 Place the 30 in the bank receive
30(1.05)33.075 Take out a long position in the
Forward Market
At time 1 Receive 33.075 Buy the asset in futures
market for 31 Profit 33.075-31 2.075
75
Eliminating Arbitrage

• Now there is an excess of participants willing to
  take a long position but few willing to take a
  short position.
• To facilitate trading the futures price will
  increase. As the price increases it is more
  attractive to participants willing to take a
  short position.

76
Eliminating Arbitrage

• In both cases the futures price moves toward a
  point where arbitrage does not exist
• When the futures price is 33.075 neither strategy
  is possible and arbitrage is eliminated

77
Short Sales

• What if it is not possible to short sell the
  asset?
• That is not a problem as long as there are enough
  people that hold the asset that are willing to
  sell in the futures market.

78
Paying a known cash income

• The above analysis can be extended to the case
  where the underlying asset pays a known cash
  income (a treasury bond for example)
• We are going to assume that the cash payment is
  due at the same time as the expiration of the
  forward contract.

79
Cash Income Example

• Suppose that you can purchase a treasury bond
  that makes its coupon payments yearly. If you
  purchase the bond it will pay a coupon payment of
  35 in one year. The bond has a forward price of
  950. The risk free rate is 5.

80
Know cash income

• We want to consider the coupon as a cash flow
  just like the forward price.
• Let the spot price be 930
• (F Coupon Payment) gt S(1r)T
• 985 95035 gt 930(1.05) 976.50
• What arbitrage opportunity exists?

81
Similar to before

• Borrow to buy the underlying asset in the spot
  market and take a short position in the futures
  contract.

At time 0 Borrow 930 Buy bond for 930 Enter
into short position
At time 1 Receive coupon payment 35 Sell bond
in Fut Market 950 Receive total 985 Repay loan
976.50 Profit 3.50
82
Opposite Case

• What if current price is 940?

At time 0 Short sell bond receive 940 Invest
940 at 5 Enter into Long Position in Fut
At Time 1 Receive 940(1.05) 987 Buy bond in
Fut Market 950 Close short sale pay coupon
35 Profit 2
83
No Arbitrage

• Again the futures price is moving toward a point
  where there will not be an arbitrage opportunity.
• (F Coupon Payment) S(1r)T
• Rearranging
• F S(1r)T - Coupon Payment
• F S (1r)T - CP(1r)T/(1r)T
• F(S CP/(1r)T)(1r)T
• where CP/(1r)T is the PV of the coupon payment

84
Extension

• If cash payments come at other points in time,
  all you need is a generalization of the
  relationship above.
• Let I represent the PV of all coupon payments to
  be received during the forward contract.
• F (SI)(1r)T

85
Accounting for payments

• Consider the 1 year forward contract on a bond
  that matures in 5 years. Assume that the bond
  makes semiannual coupon payments of 40 and has a
  spot price of 900.
• The 6 month rate is 9 and the 1 year rate is 10
• PV of coupon 1 40/(1.09)0.5 38.31
• PV of coupon 2 40/1.10 36.36

86
Assume futures price is 930

• F930 gt (900-39.31-36.36)(1.1)907.86

At time 0 Borrow 900 today Borrow 38.31 _at_9 for
6 mos Borrow 861.69 _at_ 10 for 1yr Enter into
short Futures position
At time 1 year Sell Bond for 930 Receive coup
pay 40 Total 970 Repay loan 861.69(1.1)
947.859 Profit 22.14
At time 6 mos Receive the 40 coupon
payment Repay 6 mo loan
87
Extensions

• If the futures price was less than the spot minus
  the PV of the coupons carried forward an argument
  similar to the earlier ones could have also been
  made
• A final case is if the income stream pays a known
  dividend income.

88
Dividend income

• Assume that the asset pays a return of q in the
  future based on the current price of the asset.
• The equilibrium is then
• F S(1r)T/(1q)T

89
Storage Costs?

• If the asset has a storage cost (more important
  for commodities than financial assets), it can be
  viewed as a negative cash income, the no
  arbitrage condition would be
• F (SU)(1r)T
• Where U represents the present value of all costs.

90
Generalization

• Thank of the net amount of any of the possible
  costs, income received, and interest as the cost
  of carrying the spot position to the future. It
  is the cost of holding the spot position instead
  of the future position.
• The equilibrium condition is then simply
• F (SC)(1rc)T
• C is any cash income / costs and
• rc is net interest expense