Financial Derivatives

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 • 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

Example

Discount yield • Given a discount yield of d, the dollar price of a discount security having a face value F is

Discount yield

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

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

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Financial Derivatives