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Interest Rates and price determination

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If r Coupon rate the price of the bond is below the par value - it sells at a discount. ... They are pure discount instruments (there is no coupon payment) ... – PowerPoint PPT presentation

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Title: 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
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