380 likes | 676 Views
Financial Derivatives. Interest Rates. Interest Rates. Why bother with interest rates? Role in pricing derivatives Derivatives on fixed income instruments Key markets Treasuries Bills Notes and Bonds TIPS Fed Funds and Repo Eurodollars (LIBOR and LIBID). Measures of yield.
E N D
Interest Rates • Why bother with interest rates? • Role in pricing derivatives • Derivatives on fixed income instruments • Key markets • Treasuries • Bills • Notes and Bonds • TIPS • Fed Funds and Repo • Eurodollars (LIBOR and LIBID)
Measures of yield • Discount basis • The discount yield on a single payment security is defined as • The difference between the amount payable at maturity and the price of the security now • Divided by the amount payable at maturity • Annualized
Discount yield • Given a discount yield of d, the dollar price of a discount security having a face value F is
Discount yield • The value of an 01 measures the price change caused by a .01% (one basis point) change in the discount yield • Index of interest rate risk • For $1,000,000 face value of a 90-day security, the value of an 01 is $25
Simple interest yield • The simple interest yield is defined as • The difference between the amount payable at maturity and the price of the security now • Divided by the price of the security now • Annualized using either 360-day or 365-day year When the simple yield is annualized using a 360-day year, it is called a “money market yield”
Simple interest yield • The simple interest yield on a security having a discount yield of d is
Simple interest yield • Suppose you invest P dollars at the simple interest yield y for a period of 1/n years. • At the end of the period, you will receive
Compound interest • If you invest P for 1/n years at the simple rate y, • And then reinvest the result for another 1/n years, • At the end of 2/n years the investment is worth
Effective annual interest • If you continue this process n times, • At the end of n/n = 1 year the investment is worth Where y* is the effective annual rate of interest:
Effective annual interest • The more frequent the compounding, the higher the effective annual yield:
Continuous compounding • As the compounding frequency n tends to infinity (or the fraction of a year 1/n over which the simple interest yield y is earned tends to zero), the effective annual yield becomes where r = ln(y*+1) is the continuously compounded instantaneous rate of interest
Continuous compounding • The continuously compounded rate r corresponding to a simple interest yield y compounded n times per year is Conversely, y given r is
Examples • A loan is quoted at 5% simple interest compounded monthly. • What is the annual effective yield? • What is the equivalent continuously compounded yield? • A loan is quoted at 6% compounded continuously. Interest on the loan is payable semi-annually. • What is the equivalent simple interest compounded semi-annually? • How much interest is due each period? • What is the annual effective yield?
Bond equivalent yield • In the US, bonds pay interest semi-annually, but are quoted on a simple interest basis. • A bond paying 6%, for example, actually will pay 3% (=6%/2) every six months. • The effective annual yield on such a bond would be
Bond equivalent yield • The bond equivalent yield for any security is its yield stated as a simple interest yield compounded semi-annually. • Any simple interest yield compounded n times per year can be expressed as a bond equivalent yield using:
Example • A mortgage backed bond pays 8% interest compounded monthly. • What is the bond equivalent yield for this security?
Bond coupons • Bonds pay periodic interest (coupons) to their owners. • The amount of each payment is determined by the bond’s stated coupon rate, c, its par or face value, F, and the periodicity of the payments. • In the US, for example, where bonds pay interest semi-annually, a bond holder receives a coupon payment of (c/2)F dollars every six months. • The coupon yield on a bond is the bond’s annual coupon payment divided by the bond’s current price:
Bond yield to maturity • The price of a bond should be equal to the present discounted value of the bond’s cash flows. • The bond’s yield to maturity, y, is the single discount rate (usually stated on a bond equivalent basis) that makes the present value of the bond’s cash flows equal to its (full) market price. Here, T is measured in years, so t indexes semi-annual periods.
Between coupon payments • This calculation can be adjusted for partial period discounting when the settlement date is between coupon payment dates. • S = the number of coupon payments left on the bond, and • q = days until the next coupon payment expressed as a fraction of the total days in the current coupon period. The maturity of the bond, T, in years is (S-1+q)/2.
Between coupon payments • For computation in Excel, this formula can be rewritten The Excel formula for this is: =-PV(y/2, S, c/2, 100, 0)/(1+y/2)^-(1-q)
Continuously compounded ytm • ytm is normally expressed as a bey, but it can also be expressed as a continuously compounded rate, r:
Zero-coupon bonds • A zero-coupon unit bond (a “zero”) is a pure discount security with a face value of $1. • Let B(0,T) be the price today (time t = 0) of a zero maturing (paying $1) at time T. • For now, let’s assume that such bonds are actively traded for all T>0. • The ytm on this bond is the T-year spot rate, r(0,T):
The discount function • The set of zero prices regarded as a function of time to maturity is called the discount function:
The law of one price • The law of one price states that all portfolios having the same payoff (set of future cash flows) have the same present price. • The payoff on a coupon bond is the set of coupon payments {c/2} and return of principal {F}, which arrive at future times {t1, t2, t3, … tn}. • This is the same as the payoff on a portfolio of zeros consisting of • c/2 units of each zero maturing at times {t1, t2, t3, … tn}, and • F units of the zero maturing at time tn. • By the law of one price, these two portfolios should have the same present price.
The law of one price • The law of one price implies that we can use the discount function to price a coupon bond. In the top expression, payments are discounted using the sequence of spot rates {r(0,(t-1+q)/2), t=1,…,S}, and in the bottom, all payments are discounted at the single ytm, r.
The spot rate curve • The spot rate curve (or term structure) is a plot of spot rates r(0, t) as a function of t.
Forward rates • Given spot rates, the law of one price implies sets of forward rates that connect any two points on the spot curve. • Example: • The value of $1 invested today for 2 years is FV2 = e2r(0,2) • The value of $1 invested today for 1 year is FV1 = er(0,1). If this amount is then reinvested for one more year at the one-year rate of r(1,2), the final amount will be FV2* = FV1er(1.2). • The law of one price requires
Forward rates • The forward rate r(1,2) is called the “one-year rate one year forward.” • r(1,2) is not a future rate (it is a feature of the present term structure, which could change in the future), but it can be “locked in” at present. • According to the pure expectations theory of the term structure, r(1,2) is the one-year spot rate expected to obtain one year from now. • Generally, the forward rate r(t1,t2) is
The forward rate curve • One-year forward rates plotted along with the spot curve:
Par yield • Given the discount function, the par yield associated with a particular maturity is the coupon rate c that would make the value of a coupon bond with that maturity equal to par. • That is, cpar is the par yield for maturity T=(S-1+q)/2 iff You can solve for cpar in Excel using the Goal Seek tool or Solver.
Interest Rates • Lkfjals;k
Next: Futures