Cheryl Mew FINS2624 Portfolio Management Semester 1, 2011 LECTURE 1 BOND PRICING

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1 LETURE 1 BOND PRIING WHAT IS A BOND? A bond is a claim on some fixed future cash flows. A commonwealth government bond (GB) is a bond which pays semi-annual coupons, in which the maturity date/ coupon payment date is on the 15 th of every month. A zero coupon bond is a bond with no coupons. The important information of a bond: 1. Transaction date: T 2. Settlement date:t+2 3. oupon payment dates 4. Maturity date 5. YTM 6. oupon rate um-interest or Ex-interest? 1. If<=7 days to the next coupon payment-----> ex-interest 2. If> 7 days to the next coupon payment-----> cum-interest YIELD TO MATURITY The Yield to Maturity (YTM) of a bond is: Interest rate that makes the present value of the bond s payments equal to its price. Determined by the market, reflecting annual rate of return required by market. The Relationship between YTM and Bond Price: YTM = Price AND Price Sensitivity YTM = Price AND Price Sensitivity When YTM = = 10%, P = FV = $100 o = YTM, P = FV Par Bond o < YTM, P < FV Discount Bond o > YTM, P > FV Premium Bond NO ARBITRAGE PRINIPLE An arbitrage is a set of trades that generate zero cash flows in the future, but a positive and risk free cash flow today. This is done through the violation of law of one price. An arbitrage trade is done by selling the real instrument, and buying a synthetic instrument (replicating strategies or portfolios). By constructing a synthetic bond and buy the under-priced real bond and selling 1

2 overpriced synthetic bond, an arbitrage opportunity exists, where people can earn money, whilst not incurring any risk. In Fins 2624, we employ the No Arbitrage Principle i.e. same bonds will have the price. BOND PRIING The value of a bond (like any financial security) is the present value of all future cash flows. A bond produces two different cash flows: - oupon payments - Face value (paid at maturity) Thus all we have to do is find the present value of all coupon payments and the face value! P0 = ( 1 1+ ) + 1+ = oupon r = required rate of return (YTM) t = time periods Make sure that you use the correct periodic required rate of return and periods e.g. A 5 year GB bond that pays coupons on a semi-annual basis, has an annual required rate of return of 20%. t in this case will be 10 *5 x 2+ and r in this case will be 10% *20%/2+ since payments are made semiannually!!! EX-INTEREST BONDS BG Bonds are ex-interest if the settlement date (2 days after transaction date) is within 7 days of the next coupon payment. ALULATING THE PRIE OF AN EX-INTEREST BOND 1. alculate the Value of the Bond as at oupon Date = P P = ( 1 1+ ) Find fraction of period before coupon payment = f = (coupon settlement date) / total days in period 3. Discount P to find the Price as at settlement date = P P = P 1+r f 2

3 QUOTED PRIE OF EX-INTEREST BOND By convention, the market does not quote the settlement price P. This is because, if we were the buyer, and the interest was continuous, we would get some of the interest, even though it s ex-interest. Hence, the market price = P adj P adj = P + PP. f UM-INTEREST BONDS BG Bonds are cum-interest if the settlement date (2 days after transaction date) is more 7 days until the next coupon payment. ALULATING THE PRIE OF A UM-INTEREST BOND 1. alculate the Value of the Bond as at oupon Date = P P = ( 1 1+ ) Find fraction of period before coupon payment = f = (coupon settlement date) / total days in period 3. Discount P + the next coupon to find the Price as at settlement date = P P = P +PP 1+r f QUOTED PRIE OF EX-INTEREST BOND By convention, the market does not quote the settlement price P. This is because, if we were the buyer, and the interest was continuous, we would get lose of the interest, even though it s cum-interest. Hence, the market price = P adj P adj = P PP. 1 f 3

4 LETURE 2 TERM STRUTURE OF INTEREST RATES YTM AND HPR Yield to maturity (YTM) reflects the return required and set by the market on the assumption that the bond is held to maturity. In equilibrium, it is also the return that investors can expect to earn over the life of the bond. Holding Period Return (HPR) is the expected return over a future period, and is not based on the assumption that the bond is held to maturity. SIMILARITIES Both expressed as annualised returns (not effective rates) Use the settlement price as cost base Total returns accounting for both the coupon interest component and the capital gain/loss component DIFFERENES YTM is observed and set by the market, HRP is not YTM assumes that coupon interests are all invested at the same rate as quoted YTM, HPR allows for different reinvestment interest rates received at different times YTM assumes that bond is held to maturity, HPR assumes that bond is to be sold before maturity ALULATING HPR HRP is used for comparing the expected return among alternate investments over the same predetermined holding period. However, to accomplish this task, future interest rates for different lengths are required. P0(1 + HPR m )t = It + Pt P 0 current price of the bond; m the number of coupon interest payments in 1 year; t the number of holding periods; I t the total coupon interest and reinvestment income; P t the price of the bond after t periods; I t + PI t = Terminal Value of the investment TERM STRUTURE OF INTEREST RATES The Term Structure of Interest Rates shows the relationship between the yield and time the maturity of bonds that belong to the same risk class at the same point in time. It is commonly displayed graphically as the yield curve (graph that shows the relationship between yield and maturity). Nb. The shape of the yield curve can change over time as market participants continuously revise their expectations on future inflation, which is one of the key determinants of interest rates. 4

5 The Term Structure of Interest Rates offers investors an objective way of inferring future interest rates from the yield curve. These future interest rates are called forward rates, denoted f(n,t) the rate of a n year bond in t years. For example, f(2,3) is the interest rate of a 2 year bond in 3 years time. YTM, SPOT RATES AND FORWARD RATES YTM Yield to maturity of a general coupon bond Spot Rates (SR) urrent Interest Rates period annual rate Yield of maturity of zero-coupon rate From time 0 to period t Forward rates (FR) Future interest rates implied by the term structure objective rollover rate- intermediate In between 2 periods P = 1+YTM + +FV 1+YTM 2 = + +FV = 1+SP1 1+SP2 2 1+FR1 + +FV 1+FR1 1+FR2 THE RELATIONSHIP BETWEEN FORWARD RATES AND SPOT RATES (1 y ) 1) n n ( 1 fn) n1 (1 yn f n = y n = one-year forward rate for period n spot rate/yield for a security with a maturity of n Example: Look at Henry Yip s Textbook pg SPOT RATES AND TERM STRUTURE OF ZERO OUPON BONDS The term structure of coupon bonds may also be used to infer yields to maturity of zero-coupon bonds. These yields are known as spot rates in the marketplace. If investors buy and hold a zero coupon bond to maturity, they know the cost and are certain of the amount received at maturity. The price P 0 is based on the spot rate, and there are no coupons, so the terminal value received at the end of maturity is certain. This means that spot rate is a risk-free rate. 5