Investment Fixed Income Securities

Investment: Fixed Income Securities 14-Sep-22 Introduction • Bonds are debt or fixed income securities • When corporations require debt financing, they can either take out a loan or issue bonds • Short term: just borrow from banks → loan • Long term: issue bonds • Writer/Issuer: Seller: Borrower= • Holder: Buyer: Lender= 2 Introduction • Bond price (B), coupons (C), face/par value (F) • Buyer pays B, receives C every period and also F at maturity → coupon paying bonds → can be sold at a discount, at par, or at a premium • No C → zero coupon bonds or pure discount bonds • Treasury Notes and Bills are shorter term bonds 3 1 1/14 Investment: Fixed Income Securities 14-Sep-22  • Bond price (Dirty Price), B: present value (usually quoted as a percentage of the face value) • Par or Face Value, F: future value • Maturity • Coupon rate → coupon rate*face value = C (for annual coupon payments, adjust for other time periods) • (Floating Rate Bonds) • C → annuity → annual for many European bonds and semi annual for most US bonds • Discount rate → yield to maturity (YTM) or simply, yield → discount all future cash flows to present value, B CF • Bond price = present value = B= n (1 + r ) 4 Bond Valuation: Zero Coupon Bonds B= Market price of the Bond of bond F = Face value R= Annual percentage rate; → YTM m= compounding periods per year (annual → m = 1, semi annual → m = 2,&) r = Effective periodic interest rate; r = R/m → YTM per period T= Maturity (in years) n= Number of compounding periods; n = T*m • Two cash flows to purchaser of bond: F B= • B at time 0 (1 + r) n • F at time T • What is the price (dirty price) of a bond? Use present value formula: 5 2 2/14 Investment: Fixed Income Securities 14-Sep-22 Valuing Zero Coupon Bonds • What is the current market price of a U.S. Treasury strip that matures in exactly 5 years and has a face value of $1,000. The yield to maturity is rd=7.5%. 1000 1.075 5 = $696.56 • What is the yield to maturity on a U.S. Treasury strip that pays $1,000 in exactly 7 years and is currently selling for 1000 $591.11? 591. 11 = (1 + rd ) 7 6 Bond Yields and Prices: Zero coupon bonds • Consider three zero-coupon bonds, all with • face value of F=100 • yield to maturity of r=10%, compounded annually. Bond 1 Bond 2 Bond 3 We obtain Time / Bond value 10% $90.91 $75.13 $62.09 the following: 1 100 0 2 3 4 5 0 100 0 0 0 0 100 7 3 3/14 Investment: Fixed Income Securities 14-Sep-22 Valuing Coupon Bonds • Immediately after coupons are paid: n F 1 − (1 + r ) − ]+ B = C[ = CAn |r + F( 1+ )r r (1 + r) n −n • What is the market price of a U.S. Treasury bond that has a coupon rate of 9%, a face value of $1,000 and matures exactly 10 years from today if the required yield to maturity is 10% compounded semiannually? • Coupon payments = C = coupon rate * face value 1 − (1.05)− 20 1000 B = 45[ = $937.69 ]+ 0.05 1.0520 8 The Impact of Price Responses • Suppose the yield would drop suddenly to 9%, or increase to 11%. How would prices respond? The longer Yield 10% 9% % change 11% % change Bond 1 Bond 2 Bond 3 1 Year 3 Year 5 Year $90.91 $75.13 $62.09 $91.74 $77.22 $64.99 0.91% 2.70% 4.46% $90.09 $73.12 $59.35 -0.91% -2.75% -4.63% the term, the more sensitive to r changes • Bond prices move up if the yield drops, decrease if yield rises → inverse relationship between price and yield • Prices respond more strongly for higher maturities → the longer the term, the more sensitive to r changes 9 4 4/14 Investment: Fixed Income Securities 14-Sep-22 Bond Yields and Price: Coupon bonds • Suppose you purchase the U.S. Treasury bond described earlier (10 years maturity, 9% coupon, face value $1,000)and immediately thereafter interest rates fall so that the new yield to maturity on the bond is 8% compounded semi annually. What is the bond’s new market price? • Suppose the interest rises, so that the new yield is 12% compounded semi annually. What is the market price now? • Suppose the interest equals the coupon rate of 9%. What do you observe? 10 Valuing Coupon Bonds • New semi annual yield = 8%/2 = 4% n F 1 − (1 + rd ) − B = C[ ]+ rd (1 + rd) n • What is the price of the bond if the YTM is 8% compounded semi annually? 20 B = 45.[ 1 − (1.04) − 1000 = $1067.95 ]+ 0.04 1.0420 • Similarly, • If r = 12%, B = $827.95 • If r = 9%, B = $1,000 11 5 5/14 Investment: Fixed Income Securities 14-Sep-22 Bond Prices and Yields A bond sells at par if its coupon rate equals the YTM (discount rate) A bond sells at a premium if its coupon is above the YTM • A bond sells at a discount if its coupon is below the YTM But both premium and discount bond prices converge at maturity → The (Clean) prices of both bonds approach the par value as maturity date approaches (no more C) Bond prices are inversely related to interest rates (or yields). Which of premium bonds, discount bonds or bonds selling at par offer better returns? It doesn’t matter! * It is the YTM that determines the return over the term/life of the bonds. So an investor gets the same returns from premium bonds, discount bonds, or bonds at par as long as they have the same YTM. • • 12 Current Yield and Yield To Maturity • Yield is the return on your investment in bonds • Current yield is the annual coupon payments divided by the bond price • At maturity, have to consider also par value to calculate total return/overall yield over the entire life of the bond • The total return on an investment in bonds over its entire life depends on both the interest income and any capital gains/losses → Yield To Maturity (YTM) • YTM is the interest rate for which the present value of the bond’s payments equals the bond price 13 6 6/14 Investment: Fixed Income Securities 14-Sep-22 Current Yield and Yield To Maturity • Premium bonds: Bond Price > Par Value • Premium bonds will suffer capital losses • Receive more now from coupon payments rather than par value at maturity • Hence, current yield tends to over-estimate total return (YTM current yield) (coupon rate > current yield > YTM) • Discount bonds: Bond Price Par Value • Discount bonds will suffer capital gains • Receive more later from capital gain • Hence, current yield tends to under-estimate total return (YTM > current yield) (coupon rate < current yield < YTM) 14 Interest Rate Risks • The risk in bond prices due to fluctuations in interest rates • But not all bonds are equally affected by interest rate changes. • Longer term bonds are more sensitive to interest rate changes • (inverse relationship between bond price and interest rates) 15 7 7/14 Investment: Fixed Income Securities 14-Sep-22 The Term Structure of Interest Rates: Bond Prices and Yields (Inverse relationship) • Bond Price • Prices respond more strongly for longer maturities → the longer the term, the more sensitive to r changes • Longer term bonds are more sensitive to changes in interest rates than shorter term bonds. Shorter maturity → Yield when interest rate increases, the decrease in bond price is less for shorter maturity bonds Interest rate 16 YTM • Yields are often annualized. Bond dealers typically annualize semi annual rate by doubling it → like APR • Compare with effective annual yield • YTM is calculated by trial and error → try different rates until you equate the PV of bond’s payments to bond price • Excel → =PRICE to find bond price (Clean or Flat Price) • → answer: % of face value (set redemption as 100) • Excel → =YIELD to find YTM • But note that settlement (beginning) and maturity dates are typed as Excel date formula → =DATE(Year,Month,Day) 17 8 8/14 Investment: Fixed Income Securities 14-Sep-22 Using EXCEL for Bond valuation • PRICE → (Clean Price) Price of a $100 par value bond or (% of the face value) • SETTLEMENT → beginning date • MATURITY → maturity date • RATE → coupon rate • YLD → yield to maturity → < discount rate= • REDEMPTION → par value → type 100 • FREQUENCY → number of coupons a year • Par Value is entered as a percentage, just type in 100 for par value → and the Bond price is returned as a percentage of the Par Value • Example: If Par value is $1000, still type in 100 for par value, and when bond price is given as 90, bond price is 90% of the par value = $900 18 Flat Flat/C /C /Clea lea lean n Price and Dirty Price • Invoiced Price = Dirty Price • Flat Price = Clean Price • Invoice or Dirty Price at any time t, is the true PV of the cashflows from the bonds at time t, or the PV of the coupon payment and bond price on the next coupon paying date • Dirty Price = Flat Price + Accrued Interest • Between coupon payment dates, interest is accrued • On coupon payment dates, no accrued interests, hence, clean price = dirty price • Excel : =PRICE → gives the clean price (flat price) • * Convergence of bond price (< clean price=) towards par value at maturity 19 9 9/14 Investment: Fixed Income Securities 14-Sep-22 Bond Valuation: Coupon Bonds • Prices of bonds between coupon paying dates: • Example: A semi annual coupon paying bond (coupon rate 8%, YTM 10%, par value $1,000) is maturing on 31st December 2030. • Find the clean and dirty prices of the bond on the • 31st December 2020 immediately after the coupon is paid, • 30th June 2021 immediately after the coupon is paid, • 31st January 2021, and • 30th April 2021. 20 Duration • The duration of a financial asset is the weighted average of the times until the fixed cashflows that are generated from the asset is received. • Example: Bonds: The duration is the weighted average of the times until the bonds and par value of the bond is received. • Duration is the number of years it takes for an investor to recover the present value of the cashflows of the bond. • Macaulay Duration is the weighted average time until cash flows are received and measured in years. 21 10 10/14 Investment: Fixed Income Securities 14-Sep-22 Duration • When the price of the asset is considered as a function of its yield, then duration also measures the price sensitivity to yield, the rate of change of price with respect to yield (or the percentage change in the price). • It can also be expressed as a percentage of how sensitive of the value of the bond is to changes in interest rates. • The higher the duration, the more sensitive to interest rate changes. (The longer term to maturity, the longer the duration) • Modified Duration is the price sensitivity and is the percentage change in price for a unit change in yield 22 Macaulay Duration • Where B is the price of the bond, and ti is the time till cashflows are received, then Macaulay Duration (in years) is the weighted average of the times until the cashflows are received: �㔶�㔹�㕖 (1+�㕟)�㕖 �㔶�㔹�㕖 σ (1+�㕟)�㕖 σ ý�㕖 • �㕀�㕎ý�㕎þ�㕙�㕎�㕦�㔷þ�㕟�㕎ý�㕖Āÿ = •= •= �㔶�㔹1 ý2 1+�㕟1 �㔶�㔹�㕖 σ σ 1+�㕟�㕖 �㔶�㔹1 1+�㕟1 �㔶�㔹�㕖 1 σ 1+�㕟�㕖 ý1 + (ý )+ �㔶�㔹2 1+�㕟2 �㔶�㔹�㕖 1+�㕟�㕖 �㔶�㔹2 1+�㕟2 �㔶�㔹�㕖 2 σ 1+�㕟�㕖 + ⋯⋯+ ýĀ σ �㔶�㔹Ā 1+�㕟Ā �㔶�㔹�㕖 1+�㕟�㕖 �㔶�㔹Ā 1+�㕟Ā �㔶�㔹�㕖 ÿ σ 1+�㕟�㕖 (ý ) + ⋯⋯+ (ý ) 23 11 11/14 Investment: Fixed Income Securities 14-Sep-22 �㔶�㔹1 1+�㕟1 �㔶�㔹2 1+�㕟2 • �㕀�㕎ý�㕎þ�㕙�㕎�㕦�㔷þ�㕟�㕎ý�㕖Āÿ (ý�㕖 (ý�㕖 �㔶�㔹 �㔶�㔹 1 )+ 2 ) + ⋯⋯+ = σ σ 1+�㕟�㕖 1+�㕟�㕖 �㔶�㔹Ā 1+�㕟Ā �㔶�㔹�㕖 ÿ σ 1+�㕟�㕖 (ý ) • → PV of each cashflow as a percentage of total PV of all cashflows: weights • Macaulay Duration → weighted sum of PV of each CF by the time to the occurrence of the cashflow (1+ �㕌�㕇�㕀 1+ ) �㕐 �㕌�㕇�㕀 �㕌�㕇�㕀 +ÿ( 2 ) 1 ÿ ÿ ÿ 2 �㕐 ÿ�㕌�㕇�㕀 =] [ �㕌�㕇�㕀 • �㕀�㕎ý�㕎þ�㕙�㕎�㕦�㔷þ�㕟�㕎ý�㕖Āÿ �㕌�㕇�㕀 Ā þ (1+ ÿ ÿ ) 21+ ÿ ÿ • For zero coupon bonds, intuitively, Macaulay duration is equal to the time to maturity (in years) 24 Modified Duration • Modified Duration: = • �㕀Āþ�㕖Ā�㕖ÿþ�㔷þ�㕟�㕎ý�㕖Āÿ �㕌�㕇�㕀 (1+ ÿ �㕀�㕎�㕐 ) • Using EXCEL: • Macaulay Duration: =DURATION • Modified Duration: =MDURATION 25 12 12/14 Investment: Fixed Income Securities 14-Sep-22 Bond Issuers Federal Government and its Agencies Local Government (Municipalities) Corporations Federal Government issued bonds are considered to be default free. Corporations may have financial difficulties and may default on interest payments or even go bankrupt Default (or credit) risk is the risk that a bond issuer may default on its bonds Default premium is the additional yield on a bond investors require for bearing credit risk • • • • • • • 26 U.S. Government Bonds • Treasury Bills • • • • • No coupons (zero coupon security) Face value paid at maturity Maturities up to one year Short term Treasury Notes • Coupons paid semiannually • Face value paid at maturity • Maturities from 2-10 years • Treasury Bonds • • • • Coupons paid semiannually Face value paid at maturity Maturities over 10 years The 30-year bond is called the long bond. 27 13 13/14 Investment: Fixed Income Securities 14-Sep-22 Corporate Bonds • Secured Bonds (Asset-Backed) • Secured by real property • Ownership of the property reverts to the bondholders upon default. • Debentures • General creditors • Have priority over stockholders, but are subordinate to secured debt. • • Senior versus subordinated bonds Convertible bonds → can choose later to exchange for shares of common stock 28 Bond Ratings Moody’s S&P Quality of Issue Aaa AAA Highest quality. Very small risk of default. Aa AA High quality. Small risk of default. A A Baa BBB Ba BB High-Medium quality. Strong attributes, but potentially vulnerable. Medium quality. Currently adequate, but potentially unreliable. Some speculative element. Long-run prospects questionable. Able to pay currently, but at risk of default in the future. B B Caa CCC Ca CC High specullative quality. May be in default. C C Lowest rated. Poor prospects of repayment. D - In default. Poor quality. Clear danger of default . Investment grade Speculative grade, highyield or junk bonds 14 14/14