Drake DRAKE UNIVERSITY Fin 288 Interest Rates and price determination Fin 288 Futures Options and Swaps

Drake Drake University Fin 288 PV and FV in continuous time e = y = lnx x = e y FV = PV (1+k) n for yearly compounding FV = PV(1+k/m) nm for m compounding periods per year As m increases this becomes FV = PVe rn =PVe rt let t =n rearranging for PV PV = FVe -rt

Drake Drake University Fin 288 Compounding periods The PV of $100 assuming 8% per year a various compounding periods for 5 years. # of periods per year PV continuous44.93

Drake Drake University Fin 288 Some useful conversions Given R c =continuous compounding rate R m =equivalent rate with m compounding periods

Drake Drake University Fin 288 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 PV’s (Basically the PV of an Uneven Cash Flow Stream)

Drake Drake University Fin 288 The formal model (Discrete Time) The value of any asset should then be equal to:

Drake Drake University Fin 288 Components of the General Model Cash Flow at time t (CF t ): 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???

Drake Drake University Fin 288 The Discount Rate The required rate can be seen as an aggregation of the different forces that impact the riskiness of owning the asset r=RR+IP+DP+MP+LP+EP 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

Drake Drake University Fin 288 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.

Drake Drake University Fin 288 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.

Drake Drake University Fin 288 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:

Drake Drake University Fin 288 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 PV=1,206.71

Drake Drake University Fin 288 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, and received $90 coupon payments for the next 19 years you receive a 7% return.

Drake Drake University Fin 288 Quick Facts for Review  If the level of interest rates in the economy increases the bond price decreases and vice versa.  If r>Coupon rate the price of the bond is below the par value - it sells at a discount.  If r<Coupon 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.

Drake Drake University Fin 288 The formal model (Continuous Time) The value of any asset should then be equal to:

Drake Drake University Fin 288 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.

Drake Drake University Fin 288 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.

Drake Drake University Fin 288 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

Drake Drake University Fin 288 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”

Drake Drake University Fin 288 Theoretical Spot Rate Curves Two main issues 1.Given a series of Treasury securities, how do you construct the curve? a.Linear Extrapolation b.Bootstrapping c.Other 2.What Treasuries should be used to construct it? a.On the run Treasuries b.On the run Treasuries and selected off the run Treasuries c.All Treasury Coupon Securities and Bills d.Treasury Coupon Strips

Drake Drake University Fin 288 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.

Drake Drake University Fin 288 Example MaturityYield 1 mo1.7 3 mo mo yr yr yr yr5.06 The yield for each of the semiannual periods between 1 yr and 5 yr would be found from extrapolation.

Drake Drake University Fin yr yield = 3.22% 1 yr yield = 1.74% 8 semi annual periods

Drake Drake University Fin 288 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.

Drake Drake University Fin 288 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.

Drake Drake University Fin 288 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) MaturityYTMMaturityYTM 0.54%2.55.0% %3.05.2% %3.55.4% % %

Drake Drake University Fin 288 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)/2=2.225 Therefore the cash flows from the bond would be t 0.5 =2.225 t 1 =2.225 t 1.5 =

Drake Drake University Fin 288 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 r 0.5 = 4%/2 = 2% r 1.0 = 4.2%/2 = 2.1%

Drake Drake University Fin 288 Bootstrapping continued r 0.5 = 4%/2 = 2% r 1.0 = 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

Drake Drake University Fin 288 Bootstrapping continued

Drake Drake University Fin 288 Bootstrapping continued The semi annual rate is therefore % and the annual yield would be % Similarly the 2 year yield could be found: the coupon is 4.75% implying coupon payments of $2.375 and cash flows of: t 0.5 =2.375 t 1.0 =2.375 t 1.5 =2.375 t 2.0 =

Drake Drake University Fin 288 Bootstrapping continued

Drake Drake University Fin 288 Bootstrapping continued What is the 2.5 year par value zero coupon rate? The coupon is 5%

Drake Drake University Fin 288 Bootstrapping part 2 – continuous time Now assume you know the following information concerning 100 par value bonds TimeAnnual CouponBond Price

Drake Drake University Fin 288 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

Drake Drake University Fin 288 Zero Coupon Rates Similarly the 6 month rate would be.1047 and the one year rate would be.1054

Drake Drake University Fin 288 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:

Drake Drake University Fin 288 Bootstrapping You can then solve for the 2 year rate using the 1.5 year rate and the information from the 2 year bond.

Drake Drake University Fin 288 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.

Drake Drake University Fin 288 Forward Rates The forward rate is the rate on the future six month bond that would make you indifferent between the two options. Let z 1 = the 6 month zero coupon rate 2 z 2 = the 1 year zero coupon rate (semiannual) f = the rate forward rate from 6 mos to 1 year.

Drake Drake University Fin 288 Returns Return on investing twice for six months =(1+z 1 )(1+f) Return on the one year bond =(1+z 2 ) 2 If you are indifferent between the two, they must provide the same return (1+z 1 )(1+f) =(1+z 2 ) 2 or f = ((1+z 2 ) 2 /(1+z 1 ))-1

Drake Drake University Fin 288 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.

Drake Drake University Fin 288 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 = The total value after two years is 100e.04(2) =

Drake Drake University Fin 288 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 )e f = e f = e f = ln(e f )=f=ln( ) f =.05

Drake Drake University Fin 288 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.

Drake Drake University Fin 288 Generalization – Forward Rates Given: R 1 and R 2 The zero rate for maturity t=1 and t=2 T 1 and T 2 The number of periods t=1 and t=2 R f The forward rate between the periods 1 and 2

Drake Drake University Fin 288 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

Yield Curves Over the Last Year

Drake Drake University Fin 288 US Treas Rates Jan 1990 Dec 2003

Drake Drake University Fin 288 Three Explanations of the Yield Curve The Expectations Theories Segmented Markets Theory Preferred Habitat Theory

Drake Drake University Fin 288 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)

Drake Drake University Fin 288 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.

Drake Drake University Fin 288 Expectations Hypothesis R 2 = (R f +R 1 )/2 When the yield curve is upward sloping (R 2 >R 1 ) 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.

Drake Drake University Fin 288 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.

Drake Drake University Fin 288 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

Drake Drake University Fin 288 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.

Drake Drake University Fin 288 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.

Drake Drake University Fin 288 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.

Drake Drake University Fin 288 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.

Drake Drake University Fin 288 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.

Drake Drake University Fin 288 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.

Drake Drake University Fin 288 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).

Drake Drake University Fin 288 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.

Drake Drake University Fin 288 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.

Drake DRAKE UNIVERSITY Fin 288 Price Determination in Forward and Futures Markets Fin 288 Futures Options and Swaps

Drake Drake University Fin 288 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. (Today’s 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

Drake Drake University Fin 288 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.

Drake Drake University Fin 288 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)

Drake Drake University Fin 288 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

Drake Drake University Fin 288 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)

Drake Drake University Fin 288 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 $30 (will need to repay 30(1.05) 2 =$ Buy asset for $30 Take Short Futures Position At Time 2 Deliver Asset in Futures Receive $35 Payoff loan with Profit = =$1.925

Drake Drake University Fin 288 Example con’t 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...

Drake Drake University Fin 288 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

Drake Drake University Fin 288 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 $30(1.05)=$ Take out a long position in the Forward Market At time 1 Receive Buy the asset in futures market for 31 Profit = =2.075

Drake Drake University Fin 288 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.

Drake Drake University Fin 288 Eliminating Arbitrage In both cases the futures price moves toward a point where arbitrage does not exist When the futures price is neither strategy is possible and arbitrage is eliminated

Drake Drake University Fin 288 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.

Drake Drake University Fin 288 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.

Drake Drake University Fin 288 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%.

Drake Drake University Fin 288 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) > S(1+r) T 985 = > 930(1.05) = What arbitrage opportunity exists?

Drake Drake University Fin 288 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 = Profit = 3.50

Drake Drake University Fin 288 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

Drake Drake University Fin 288 No Arbitrage Again the futures price is moving toward a point where there will not be an arbitrage opportunity. (F + Coupon Payment) = S(1+r) T Rearranging F = S(1+r) T - Coupon Payment F = S (1+r) T - CP(1+r) T /(1+r) T F=(S – CP/(1+r) T )(1+r) T where CP/(1+r)T is the PV of the coupon payment

Drake Drake University Fin 288 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 = (S+I)(1+r) T

Drake Drake University Fin 288 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

Drake Drake University Fin 288 Assume futures price is $930 F=$930 > ( )(1.1)= At time 0 Borrow $900 today Borrow for 6 mos Borrow 10% for 1yr Enter into short Futures position At time 6 mos Receive the $40 coupon payment Repay 6 mo loan At time 1 year Sell Bond for $930 Receive coup pay = $40 Total = $970 Repay loan (1.1) = Profit = $22.14

Drake Drake University Fin 288 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.

Drake Drake University Fin 288 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(1+r) T /(1+q) T

Drake Drake University Fin 288 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 = (S+U)(1+r) T Where U represents the present value of all costs.

Drake Drake University Fin 288 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 = (S+C)(1+r c ) T C is any cash income / costs and r c is net interest expense