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Chapter 5 :BOND PRICES AND INTEREST RATE RISK Mr. Al Mannaei Third Edition What is Bond ? • Debt instrument issued by a government or a corporation in order to finance projects or activities. • A form of loan. – Issuer : lender : • At maturity borrower will pay lender face value + interest. • Negotiable instruments. 2 What is Bond ? Why issuing bonds? • Governments : to finance infrastructure projects: schools, roads, power stations..etc. • Corporations : to finance commercial project, expand the business, increase capital and reduce tax. 3 What is Bond ? Mona bought a bond for $980 issued by ALBA with 3 years maturity , the coupon rate is 6% , paid annually ? – – – – – Who are the borrower & the lender? What is the $980 ? How much mona will receive every year* ? Can Mona sell the bond before maturity ? How much mona will receive at maturity ? Year 1 Year 2 * in other word , how much alba have to pay each year. Year 3 (Maturity) 4 What is Bond ? • What is the Maturity (Time) of bonds ? Range : from to • What is the frequency of payment ? • Is payment guaranteed ? • Risk & Return ? • Why corporation pay higher interest than government ? • Listed vs. over the counter (OTC) ?! 5 How to price a bond ? • The price of a bond is the present value of the future cash flows promised, discounted at the market rate of interest. C1 C2 CN + F N PB = + + ... 1 2 N (1 + i) (1 + i) (1 + i) 6 How to price a bond ? C1 C2 CN + F N PB = + + ... 1 2 N (1 + i) (1 + i) (1 + i) Where PB = price of bond or present value of promised payments; Ct = coupon payment in period t, where t = 1, 2, 3,…, n; Fn = par value (principal amount) due at maturity; i = market interest rate (discount rate or market yield); and n = number of periods to maturity. 7 Coupon rate & Market Yield What is the difference between coupon rate & Market Yield ?! Example : • You buy 1 year government bond for $1030. • The bond pays 7% coupon annually. • What is your profit ?! In percentage (%) ?! 2 0 1 4 2 0 1 5 C1 + F N PB = 1 (1 + i) 8 Coupon rate & Market Yield Example (Continue) : if you sell the bond after 6 months for $1050. Calculate the market yield ?! 2 0 1 4 2 0 1 5 9 How to price a bond ? • Example 1 : Consider 3 Years Bond with face value of $1000 and Coupon rate 8% , Current market rate is 10% , Calculate the price of the Bond ? C1 C2 CN + F N PB = + + ... 1 2 N (1 + i) (1 + i) (1 + i) 10 How to price a bond ? • Example 1 : Consider 3 Years Bond with face value of $1000 and Coupon rate 8% , Current market rate is 10% , Calculate the price of the Bond ? 11 How to price a bond ? • Example 2 : Consider 3 Years Bond with face value of $1000 and Coupon rate 5% , Current market rate is 5% , Calculate the price of the Bond ? C1 C2 CN + F N PB = + + ... 1 2 N (1 + i) (1 + i) (1 + i) 12 How to price a bond ? • Example 2 : Consider 3 Years Bond with face value of $1000 and Coupon rate 5% , Current market rate is 5% , Calculate the price of the Bond ? 13 How to price a bond ? Example 3: Consider a 1 year bond with a face value of $1000 and a coupon rate of 8% compounded annually, current market yield is 5% , calculate the price of the bonds ? 14 How to price a bond ? Example 3: Consider a 1 year bond with a face value of $1000 and a coupon rate of 8% compounded annually, current market yield is 5% , calculate the price of the bonds ? 1. Press (CMPD) bottom press EXE 2. Press EXE on (Set) choose END press EXE 3. Press EXE on (n) entre 1 press EXE 4. Press EXE on (I) entre 5 press EXE 5. Ignore PV ( press down ) 6. Press EXE on PMT ENTRE 80 press EXE 7. Go back for PV press SOLVE. 15 How to price a bond ? Example 3: Consider a 1 year bond with a face value of $1000 and a coupon rate of 8% compounded annually, current market yield is 5% , calculate the price of the bonds ? n= 1 I=5 PV= ? PMT=80 FV=1000 C/Y=1 16 Coupon rate & Market IR How Does the Market IR & Coupon rate affect Bond price ? 17 Coupon rate & Market IR How Does the Market IR & Coupon rate affect Bond price ? If : Coupon rate Greater Market IR >>> bond price higher than par ( issued at premium ) Coupon rate Lower Market IR >>> bond price lower than par ( issued at discount) Coupon rate = Market IR >>> bond price equal par (issued at par) 18 Market IR & Bond Price Negative relationship between i & Bond Price – Increasing i ; decrease Bond Price. – Decreasing i ; increase Bond Price. Positive relationship between Coupon & Bond Price – Increasing Coupon ; increase Bond Price. – Decreasing Coupon ; decrease Bond Price. 19 Question What is the price of a $1000 face value with a 10% coupon if the market rate of return is 10%? 20 How to price a bond ? • Example 4: Consider a 2 year bond with a face value of $1000 and a coupon rate of 5% compounded Semi-annually, current market yield is 6% , calculate the price of the bonds ? 21 How to price a bond ? • Example 4: Consider a 2 year bond with a face value of $1000 and a coupon rate of 5% compounded Semi-annually, current market yield is 6% , calculate the price of the bonds ? C1 / 2 C2 / 2 C4 / 2 + F N PB = + + ... 1 2 4 (1 + i / 2 ) (1 + i / 2 ) (1 + i) 22 Risk related with Bonds • Credit or default risk: chance that issuer may be unable or unwilling to pay as agreed. 23 Risk related with Bonds • Reinvestment risk: potential effect of variability of market interest rates on return at which payments can be reinvested when received. • Price risk: Inverse relationship between bond prices and interest rates. C1 C2 CN + F N PB = + + ... 1 2 N (1 + i) (1 + i) (1 + i) 24 Zero Coupon Bonds • No periodic coupon payments. • Issued at discount from par. • Single payment of par value at maturity. FV PB = n (1 + i) 25 Zero Coupon Bond • How to price Zero Coupon Bonds ? Annual FV PB = n (1 + i) Semi-annual, Quarterly , monthly, daily FV PB = mn i (1 + ) m PB is simply PV of FV represented by par value, discounted at market rate. 26 Example • if you want to purchase a Company XYZ zerocoupon bond that has a $1,000 face value and matures in 3 years compounded annually, and you would like to earn 10% per year on the investment, what is the price of the bond ? FV PB = n (1 + i) 27 Example • if you want to purchase a Company XYZ zerocoupon bond that has a $1,000 face value and matures in 3 years compounded semi-annually, and you would like to earn 10% per year on the investment, what is the price of the bond ? FV PB = mn i (1 + ) m 28 Example • In the previous example if the compounding frequency increase , Bond price will : – Decrease – Increase – No Change Compounding Frequency increase Yearly , semi-annually, monthly, daily. Compounding Frequency decrease 29 Problem Carol purchases a one-year discount bond with a face value of $1,000 for $862.07. What is the yield of the bond? FV PB = n (1 + i) 30 Zero bond • Mariam bought a bond mature after 3 years for 1000 BD. The coupon rate and market yield = 8%. • Marwa bought zero-bond mature after 3 years for 1000 BD. The market yield is 8%. • What if both bond defaulted after year 2 ?! Year 0 1 2 3 31 Bond Yields • Market Yield (Interest Rate) – Yield to Maturity – Expected Yield – Realized Yield C1 C2 CN + F N PB = + + ... 1 2 N (1 + i) (1 + i) (1 + i) 32 Yield to Maturity (YTM) • YTM : Investor's expected yield if bond is held to maturity and all payments are reinvested at same yield. C1 C2 CN + F N PB = + + ... 1 2 N (1 + i) (1 + i) (1 + i) • The longer until maturity, the less valid the reinvestment assumption • Example: – Bond A mature after 30 years pay 8%. – Bond B mature after 5 years pay 8%. – Which bond most likely will change the coupon rate ?! 33 Yields calculation • The Bond price , Coupon rate & maturity will be given and the Yield (i) has to be calculated . • The Yields can be calculated through – Trail & error. – Financial calculator. C1 C2 CN + F N PB = + + ... 1 2 N (1 + i) (1 + i) (1 + i) 34 YTM Example • Investor buys 5% percent coupon (semiannual payments) bond for $951.90; bond matures in 3 years. Solve the bond pricing equation for the interest rate (i) such that price paid for the bond equals PV of remaining payments due under the bond. C1 C2 CN + F N PB = 1 (1 + i) + (1 + i) 2 + ... (1 + i) N 35 YTM Example • Investor buys 5% percent coupon (semiannual payments) bond for $951.90; bond matures in 3 years. Solve the bond pricing equation for the interest rate (i) such that price paid for the bond equals PV of remaining payments due under the bond. 25 25 1,025 951.90 = + + ... 1 2 6 (1 + (i / 2) ) (1 + (i / 2) ) (1 + (i / 2) ) 36 YTM Example • Investor buys 5% percent coupon (semiannual payments) bond for $951.90; bond matures in 3 years. Solve the bond pricing equation for the interest rate (i) such that price paid for the bond equals PV of remaining payments due under the bond. I: n: FV: PV: PMT: Answer : 37 Expected Yield • Predicted yield for a given holding period • Almost same as YTM but the expected holding period is shorter . C1 C2 CN + F N PB = + + ... 1 2 N (1 + i) (1 + i) (1 + i) 38 Realized Yield • Realized Yield: actual rate of return, given the cash flows actually received and their timing. C1 C2 CN + F N PB = + + ... 1 2 N (1 + i) (1 + i) (1 + i) • Differ from YTM & Expected yield , due to : – Change in the amount of promised payments. – Change in market interest rates. 39 Realized Yield Investor pays $1,000 for 10-year 8% coupon bond; sells bond 3 years later for $902.63. Solve for i such that $1,000 (the original investment) equals PV of 2 annual payments of $80 followed by a 3rd annual payment of $982.63 C1 C2 CN + F N PB = + + ... 1 2 N (1 + i) (1 + i) (1 + i) 40 Realized Yield Investor pays $1,000 for 10-year 8% coupon bond; sells bond 3 years later for $902.63. Solve for i such that $1,000 (the original investment) equals PV of 2 annual payments of $80 followed by a 3rd annual payment of $982.63 80 80 982.63 1000 = + + ... 1 2 3 (1 + i) (1 + i) (1 + i) 41 Realized Yield Investor pays $1,000 for 10-year 8% coupon bond; sells bond 3 years later for $902.63. Solve for i such that $1,000 (the original investment) equals PV of 2 annual payments of $80 followed by a 3rd annual payment of $982.63 I: n: FV: PV: PMT: Answer : 42 Bond price volatility (price risk) • Percentage change in price for given change in interest rates where %∆PB = percentage change in price Pt = new price in period t P t – 1 = bond’s price one period earlier 43 Bond price volatility (price risk) 44 Bond price volatility (price risk) 45 Bond price volatility (price risk) 46 Bond theorems • Bond prices are inversely related to bond yields. • The price volatility of a long-term bond is greater than that of a short-term bond, holding the coupon rate constant. • The price volatility of a low-coupon bond is greater than that of a high-coupon bond, holding maturity constant 47 • Price Risk • Reinvestment Risk • Price risk and reinvestment risk work against each other. You bought one year bond at par, where interest rate and coupon rate= 5%. After three months the interest rate fall to 3% ?! 48 • Price risk and reinvestment risk work against each other. – if interest rates fall : • Bond prices rise but >> Coupons are reinvested at lower return. – if interest rates rise : • Bond prices fall but >> Coupons are reinvested at higher return. 49 Duration • A measure of the volatility of bond price to a change in interest rates. • Expressed as a number of years. • It is NOT the length of time it takes to get back the original investment ( Payback Period ). 50 Bond : Duration will always be less than its time to maturity. Zero-Bond : Duration is equal to its time to maturity. Zero Bond Regular Bond 51 Duration • Duration CFt * t t t 1 (1 i) D n CFt t t 1 (1 i) – – – – n CF : Coupons payment t: time of payment i: interest n: number of years 52 Duration Example • Suppose we have a bond with a 3-year term to maturity, an 8% coupon paid annually, and a market yield of 10%. Duration is: 53 Duration • If the yield increases to 15%: What is the relationship between yield & Duration ? 54 Duration What is the relation between coupon rates and duration ?! Duration equals term to maturity for zero coupon securities. 55 Duration concepts • Higher coupon rates mean shorter duration (less price volatility). Bond A has 5 coupon rate , IR 10% , 2 years Bond B has 8 coupon rate , IR 10% , 2 years Which of the above bonds has lower duration ? • Duration equals term to maturity for zero coupon securities. What is the duration for 3 years zero bond ? 56 Duration concepts • Longer maturities mean longer durations (greater price volatility). Bond A has 8 coupon rate , IR 10% , 5 years Bond B has 8 coupon rate , IR 10% , 2 years Which of the above bonds has lower duration ? • The higher the market rate of interest, the shorter the duration. Bond A has 8 coupon rate , IR 20% , 2 years Bond B has 8 coupon rate , IR 10% , 2 years Which of the above bonds has lower duration ? 57 Duration where: wi = proportion of bond i in portfolio and Di = duration of bond i. 58 Duration 59 Duration Using the 3-year, 4% coupon bond in Exhibit 5.6 (previous slide ) , If yield increases to 12%: i %PB D 100 (1 i ) 0.02 2.88 100 5 . 24 % 1 . 10 60 Thank You Kingsoft Office published by www.Kingsoftstore.com @Kingsoft_Office kingsoftstore 61