Interest Rates and Forward Rate Agreements

Read Hull Chapter 4

Outline

• Interest rate types
• Measuring interest rates
• Continuous compounding
• Zero rates / zero curve
• Bootstrap
• Forward rate
• Forward rate agreement (FRA)
• Term structure of interest rates

Interest Rate Types

• Common interest rates
  • Treasury rates - Rates on government instruments in its own currency
  • London Interbank Offered Rate (LIBOR) rates
    • LIBOR is the rate of interest at which a bank is prepared to deposit money with another bank
    • The second bank typically possesses an AA rating
    • The British Bankers Association compiles LIBOR once a day on all major currencies for maturities up to 12 months
  • The London Interbank Bid Rate (LIBID)
    • LIBID is the rate which a AA bank is prepared to pay on deposits from another bank
  • Repurchase agreement (Repo) rates
    • Repurchase agreement is an agreement where a financial institution that owns securities agrees to sell them today for X and buy them back in the future for a slightly higher price, Y
    • Has low risk because the securities act as collateral
    • Invented to circumvent regulations
      • U.S. banks could not pay interest on business checking accounts
      • The financial institution obtains a temporary loan and the securities become the collateral
      • The interest equals the difference between selling price of security and the higher price the bank buys the next day
      • Known as the repo rate
  • The Risk-Free Rate
    • Derivatives practitioners traditionally use the short-term risk-free rate, LIBOR
    • Many consider the Treasury rate artificially low
      • Regulations may require banks to hold highly-liquid securities
      • Treasuries may have tax advantages
      • Government has power to tax if it experiences budget problems
    • As will be explained in a later lecture
      • Financial wizards use Eurodollar futures and swaps to extend the LIBOR yield curve beyond one year
      • The overnight indexed swap rate is increasingly being used instead of LIBOR as the risk-free rate
        • Credit markets for LIBOR froze during the 2008 Global Financial Crisis

Measuring Interest Rates

• Compounding frequency
  • How often interest payments or receipts are made
  • The compounding frequency for an interest rate serves as the unit of measurement
• Impact of Compounding
  • A loan compounded m times per year at the rate R causes present value (PV) of a loan to grow to future value (FV) = PV(1+R/m) ^m in one year

Continuous Compounding

• In the limit, we obtain continuously compounded interest rates as we compound more and more frequently
  • We can think a continuous rate is one in which a loan compounds interest every fraction of a second
  • $1,000 grows to $1,000e ^RT when invested at a continuously compounded rate R for time T
  • $1,000 received at time T discounts to $1,000e ^-RT at time zero when the continuously compounded discount rate is R
• Discrete
  • Future value (FV)
  • Present value (PV)
    • FV=PV(1 + R/m) ^{m t}
    • m – how many payments per year
• Continuous compounding
  • m→∞

• Operations with e
  • Future value
    • e ^Rt similar to (1 + R) ^t
  • Present value
    • e ^-Rt = 1 / e ^RT similar to (1+R) ^-t = 1 / (1+R) ^t
  • Derivatives and Forward Rate Agreements (FRA) use continuous interest rates
• Conversion Formulas
  • Continuously compounded rate, R _c
  • Same rate with compounding m times per year, R _m
    • R _c=m ln(R _m / m+1)
    • R _m=m (e ^{Rc / m}–1)
  • Examples
    • 12% with semiannual compounding equals 2 ln( 0.12 / 2 + 1 ) = 0.1165 with continuous compounding
    • 20% with continuous compounding equals 4 ( e ^0.20/4 –1 ) = 0.2051 with quarterly compounding
    • Rates used in option pricing are nearly always expressed with continuous compounding
  • Easy to derive formula
    • Set FV = PV (1 + R _m/m) ^{t m} = PV e ^{t Rc}
  • Trick question on exams
    • Problem gives interest rates that are not continuously compounded
    • Must convert interest rates to continuous compounding

Zero Rates / Zero Curve

• A zero rate (or spot rate), for maturity T is the rate of interest earned on an investment that provides a payoff only at time T
• Discount bonds
• Zero curve – treats bonds as if they make one payment at maturity
• Must convert coupon bonds into discount bonds or zero coupon bonds
• Example of bond pricing is below

• Bond Pricing
  • To calculate the cash price of a bond we discount each cash flow at the appropriate zero rate
    • A bond has cash flows at different times
    • Use appropriate interest rate to calculate present value
    • In our example, the theoretical price of a two-year bond providing a 8% coupon semiannually and has a face value of 100 yields

PV=4e ^-0.06x0.5+4e ^-0.065x1.0+4e ^-0.070x1.5+104e ^-0.075x2.0

PV=100.74

• Bond Yield
  • The bond yield is the discount rate that makes the present value of the cash flows on the bond equal to the market price of the bond
  • Suppose that the market price of the bond in our example equals its theoretical price of 100.74
  • We solve for the bond yield by finding the value for y with continuous compounding that gives a discounted value of 100.74
  • Can solve iteratively
    • Set y to any number such as y = 0.1
    • If the PV of cashflows equal 100.74, then you are done
    • If PV is higher than 100.74, then select a lower number such as y = 0.08
    • Then repeat until PV of cash flows equals 100.74
    • Excel goal seek can solve these problems
      • Goal seek finds y = 0.07454

4e ^-0.5y+4e ^-1.0y+4e ^-1.5y+104e ^-2.0y= 100.74

• Par Yield
  • The par yield for a certain maturity is the coupon rate that causes the bond price to equal its face value
  • In our example we solve for c, which equals c = 7.594, semi-annual compounding

• Par Yield
  • If m is the number of coupon payments per year
  • d is the present value of $1 received at maturity
  • A is the present value of an annuity of $1 on each coupon date

• Example
  • m = 2
  • d = 0.86071
  • A = 3.66855
• Refer to derivation below. Notice where the terms m, d, and A come from

The Bootstrap Method

• Data to Determine Zero Curve
• Half the stated coupon is paid each year*

• Bootstrap – method to solve for zero rates from the interest rates
• The bond holder earns $ 2.00 on the 98.0 during 3 months
  • First Method
    • The 3-month interest rate is 2.00/98.0 = 0.0204
    • Convert to annual with quarterly compounding 2.04% x 4 = 8.163%
    • This is 8.081% with continuous compounding
  • Second Method
    • We can also use PV = FV e ^-RxT
    • 98 = 100 e ^{-0.25 R}
    • R = 0.08081
• Similarly the 6 month and 1 year rates are 8.164% and 8.338% with continuous compounding
• To calculate the 1.5 year rate we solve

2e ^-0.08164x0.5+2e ^-0.08338x1.0+102e ^-1.5R = 96
1.92+1.84+102e ^-1.5R = 96
102e ^-1.5R=92.24
102e ^-1.5R) / 102 = 92.24 / 102
ln( e ^-1.5R) = ln (0.9043)
-1.5R = -0.10058
R = 0.06705

• R = 0.06705 or 6.7053%
• Similarly the two-year rate is 6.942%
• Zero Curve Calculated from the Data

Forward Rates

• The forward rate is the future zero rate implied by today's term structure of interest rates
  • Term structure of interest rates is what we calculated in last slide
• Formula
  • Suppose that the zero rates for time periods T ₁ and T ₂ are R ₁ and R ₂ with both rates continuously compounded.
  • Note – forward rate is a contract for an interest rate for a future time period between T ₁ and T ₂ .
  • The formula becomes approximate when rates are not expressed as continuous compounding
  • The forward rate for the period between times T ₁ and T ₂ is

• Calculate the forward rates for the interest rates in the table

• For example
  • A 4 year loan pays 7% interest for each year
  • The forward rate between Years 3 and 4 equals 10%
  • The forward rate only covers the payment or receipt between 3 and 4 year and not the whle four years
• For an upward sloping yield curve:
  • Forward Rate > Zero Rate > Par Yield
• For a downward sloping yield curve
  • Par Yield > Zero Rate > Forward Rate

Forward Rate Agreement

• An FRA is an agreement where an investor pays a fixed rate, R _K , and receives a market interest rate
  • The counter party receives fixed and pays the market interest rate
  • Zero sum game
  • Since interest rates do not have a physical existence, parties settle differences in cash
  • An FRA can be valued by assuming that the forward LIBOR interest rate, R _F , will be realized
  • We are using the forward rate as a prediction
  • We do not know the future interest rate and we must place a value on a future interest
  • Thus, the value of an FRA is the present value of the difference between one party receiving a variable interest rate R _F and paying the fixed rate R _K
    • The counterparty does the opposite
    • The parties settle the FRA at the beginning of the FRA
• Example
  • A company has agreed to pay 5% on $10 million for 6 months starting in 2 years
    • Rate is fixed, R _K
  • The forward rate for the period between 2 and 2.5 years is 6% in semi-annual compounding
    • Forward rate forecasts the market interest rate
    • Person expects to receive R _F
    • FV _FRA=notional principal x (R _F – R _K)(compounding)(discount)
    • FV _FRA=10 million x (0.06 – 0.05)(0.5) = $50,000
    • The value of the contract to the company is +$50,000 at 2.5 years
    • Discount today to get present value
  • The actual market rate is 4.5% (with semi-annual compounding)
    • We can calculate the true value of the FRA
    • FV _FRA= 10 million x (0.045 – 0.05)(0.5) = -25,000
    • The payoff is –25,000 at the 2.5 year point
    • Parties settle the FRA at the beginning period
      • -25,000 e ^-0.04450x0.5 = -24,449.88
      • Continuous interest rate = 2 ln (0.045 / 2 + 1) = 0.04450
      • We are at year 2.5 and discount back by six months
      • Could also use old school discounting, -25,000 / (1 + 0.045 / 2)=-24,449.88

Term Structure of Interest Rates

• Theories try to explain why the yield curve is usually upward sloping
  • Expectations Theory:
    • Long-term interest rates should reflect expected short-term interest rates
    • Forward rates equal expected future zero rates
  • Market Segmentation:
    • Short, medium and long rates are separate markets with their own independent supply and demand
  • Liquidity Preference Theory:
    • Investors prefer to invest with liquid securities but will invest in longer maturities with a premium
    • Forward rates higher than expected future zero rates
    • Suppose that the outlook for rates is flat and you have been offered the following choices
• Which rate would you choose as a depositor? Which maturity for your mortgage?

• Liquidity Preference Theory continued
  • To match the maturities of borrowers and lenders a bank has to increase long rates above expected future short rates
  • In our example the bank might offer