Unformatted text preview: Ch. 8: Interest Rates and Bond Valuation 3/04/2015 Bonds and Bond Valuation • Bonds – debt securities frequently issued or sold by corporations and governments as a way of borrowing money o An interest-‐only loan where the interest is paid regularly every period, but none of the principal is repaid until the end of the loan. o Coupons / level coupon bond – regular interest payments § Coupon rate – annual coupon divided by the face value o Face value / par value – the amount repaid at the end of the loan § $1,000 for corporate bonds; when sold at this amount they are referred to as “par value bonds” o (Bonds’ time to) Maturity – number of years until the face value is paid • Value of bond fluctuates due to CF of the bond staying the same over the course of its maturity. o ↑ Interest rate, ↓ PV of bond’s remaining CF (value ↓) § ↓ Interest rate, ↑ PV of bond’s remaining CF (value ↑) o Determine the value of a bond § No. periods until maturity, face value, coupon, market interest rate • Yield to Maturity (YTM) – interest rate required to in the market on a bond o r = unknown discount rate / YTM § Must try different discount rates through Trial & Error until calculated bond value equals the given value o Calculate PV of CF to estimate current market value o CF have an annuity components (coupons) and lump sum (face value paid at maturity) § = + −
+ = +
+ • C = coupon paid each period, r = discount rate per period, T = number of periods, F = bond’s face value o Discount bond – bond rate < going rate; bond sells for < face value § Investors willing to lend only something less than $1,000 promised repayment o Premium bond – bond rate > going rate; bond sells for > face value o Semiannual coupons – quoted rate equals actual rate per period multiplied by the number of periods Interest rate risk – when the fluctuating interest rates of a bond arise risk for bond owners; level of risk depends on how sensitive the price of the bond is to interest rate changes o The longer the time to maturity, the greater the interest rate risk o The lower the coupon rate, the greater the interest rate risk o Slope of the line connecting price § Steepness indicates a relatively small change in interest rate leads to a substantial change in bond’s value • PV isn’t greatly affected by a small change in interest rate if the amount received in 1 year, but after a long period of compounding, there’s a great effect on PV Ch. 8: Interest Rates and Bond Valuation 3/04/2015 Interest rate risk increases at a decreasing rate Bonds with lower coupons have greater interest rate risk § Depends on PV of coupons and face amount § Value of lower-‐coupon bond proportionately more dependent on the face amount to be received at maturity • Bond with higher coupon has larger CF early in life, so value is less sensitive to changes in discount rate • Current yield – discount bond ignores built-‐in gain, making it too low; premium bond ignores built-‐in loss, making it higher = • Zero coupon bonds – bond that pays no coupon and must be offered at a price much lower than face value o YTM (y) on a Zero under Annual Compounding $ = $, ( + )
o Expresses yield as an effective annual yield – use semiannual periods to be consistent for coupon bonds • Settlement date – date you pay the bond o Maturity date – date the bond actually matures o
o Government and Corporate Bonds • Government Bonds and treasury notes are sold to the public when the government wants to borrow money for more than one year. o US Treasury issues have no default risk, because the they can always come up with money to make payments § Treasury issues are exempt from state income taxes (but not federal income taxes) • Coupons received are only taxed from a federal level § Need to compare aftertax yields on two bonds to show treasury bond has slightly better yield o Municipal notes / bonds (munis) – Sold by the local governments when they want to borrow money § Varying degree of default risk § Break-‐even rate is the tax rate at which an investor would be indifferent between a taxable and nontaxable issue % = % ∗ ( − ∗ ) • Corporate Bonds – face he possibility of default, which generates a wedge between the promised yield and the expected return on a bond o Expected payoff from the bond at maturity = [ ∗ ] + [ ∗ ] o Promised yield calculation assumes bondholder will receive the full amount, thus ignoring the probability of default – need the coupon rate, par value, and maturity § Promised yield doesn’t indicate what the bondholder expects to receive o Expected return takes probability of default into account § Both are equal in a risk-‐free envi • Government bonds vs. corporate bonds Ch. 8: Interest Rates and Bond Valuation 3/04/2015 Yield on gvmnt bond is below corp bond because corporate bond has default risk, while gvmnt bond doesn’t § Coupons will not be paid in full if there is a default § Promised yield equals expected return of gvmnt bond; promised yield is greater than expected return on corp bond. Interest rate risk – the risk of a change in value of a bond resulting from a change in interest rate o The rating of a bond determines its degree of risk § On the S&P and Moody bond rating board, those with higher ratings (AAA) have less risk than those who have lower ratings (D) o • Bond Markets • Trading in bonds over the counter (OTC) – no particular place; generally dealers are connected electronically • Bond markets so big because the number of bonds issued > number of stocks issued o Financial market is transparent if its prices and trading volume are easily observed • Financial Industry Regulatory Authority (FINRA) – provides daily snapshot of the data from TRACE by reporting the most active issues o Bid price – what a dealer is willing to pay for a security o Ask price – what a dealer is willing to take for it o Bid-‐ask spread – difference between the two, representative of the dealer’s profit • Buying a bond between coupon payment dates is usually more than you are quotes o Clean price – quote prices net of accrued interest, where accrued interest deducted to arrive at the quoted price § Dirty price – full / invoice price Inflation and Interest Rates • Nominal rate on an investment is the % change in the number of dollars you have. o Real rate on an investment is the % change in how much you can buy with your dollars § Percentage change in buying power + = + ∗ ( + ) = + + ( ∗ ) ≅ + R = nominal rate r = real rate (on the investment) h = inflation rate (compensation for the decrease in the value of money originally invested) r * h = compensation for the fact that dollars earned on the investment are worth less because of inflation • Inflation risk – faces by investors due to uncertainty in inflation rate o Knows how much will be received, but not how much can be bought with it o Inflation can erode real value of payments, implying that inflation risk is a serious concerns, esp. in time of high and variable inflation Ch. 8: Interest Rates and Bond Valuation • 3/04/2015 Fisher Effect – Inflation reduces purchasing power, so investor demand an increase in nominal rate before lending money o A rise in the rate of inflation causes the nominal rate to rise enough so that the real rate of interest in unaffected § Real rate is invariant to the rate of inflation § Important in determining nominal rate Determinants of Bond Yields • Term structure of interest rates – relation betw short and long-‐term interest rates o Nominal interest rateson default-‐free, pure discount bonds contain to risk of default and involve just a single lump-‐sum future payment § Term structure indicate pure time value of money for diff lengths of time § Upward sloping, with varied degree of steepness given 3 components • Real rate of interest – compensation investors demand for forgoing use of money o Factor of high expected growth (raises it), but differ across maturities o Low for short-‐term bonds and high for long-‐term • Rate of inflation – future inflation erodes value of dollars that will be returned o Inflation premium – investors demand compensation for loss in the form of high nominal rates o Interest rate risk premium – demanding extra compensation in the form of higher rates § Longer the term to maturity, greater the interest rate risk (increases with maturity) • Increases at decreasing rate, as does premium o Expected to fall in the future, and expected decline is enough to offset interest rate risk premium abd produce downward-‐sloping term structure § Still get upward sloping because of rate risk premium is rate of inflation expected to decline by only a small amount • Treasury yield curve – treasury yield relative to maturity o Reflection of the term structure of interest rates o Term structure is based on pure discount bonds, whereas the yield curve is based on coupon bond yields § Treasury yields depend on 3 components • Real rate, expected future inflation, and interest rate risk premium § Treasury notes and bonds have 3 features • Default-‐free, taxable, and highly liquid o Credit risk – possibility of default § Demand higher yield to compensate called default risk premium • Lower-‐rate bonds have higher yields (high-‐yield bonds) Ch. 8: Interest Rates and Bond Valuation 3/04/2015 Taxability premium – extra yield on a taxable bond as compensation for the unfavorable tax treatment § Liquidity premium – investors prefer liquid > illiquid, so demand too Real rate are 5 premiums representing compensation for o Expected future inflation interest rate risk, default risk, taxability, and lack of liquidity. § • Appendix: The Term Structure of Interest Rates, Spot Rates, and Yield to Maturity • Assume constant interest rate over all future period o Fluctuating inflation rates cause interest rate to vary over future periods o E.g. (1) Two zero-‐coupon bonds with spot rates r1 and r2 (where r2 > r1 due to expected increase in inflation rate) Bond A – 1 yr, face value $1,000, r1 =8% Bond B – 2 yr, face value $1,000, r2 = 10% ! = 1,000 1.08 = $925.93 ! = 1,000
= $826.45 1.10!
o E.g. (2) To determine the yield to maturity when given the spot rates of a bond, you must solve for the PV of both, then (leaving all else constant in the equation) solve for new rate y, yield to maturity. This is the single rate for the bond over the course of its life Bond A – 1 yr, face value $1,000, r1 =8% Bond B – 2 yr, face value $1,000, r2 = 10% Cash Flows – $50 @ yr 1, $1,050 @ yr 2 Step 1 !,! = 50 1.08 + 1,050 1.10! = 914.06489 Step 2 914.06489 = 50 1 + + 1,050 (1 + )! = 0.0995 9.95% What this does (1) gives us price of the bond given the market spot rates, whereas (2) provides the yield to maturity, an average of the two market rates, given the price of the bond. Two bonds with the same maturity will have different yields to maturity when the bonds have different spot rates / coupons. • Term structure – describes the relationship of spot rates with different maturities o Exists for only a moment in time, as interest rates are likely to change. o When one-‐year and two-‐year spot rates are known at date 0, you can calculate the forward rate that an investor would lock in § Forward rate – future interest rates, and is equal to the spot rate at year 1. • For Bond B, the spot rate of 10% was inclusive of both yr 1 and yr 2. But we know yr 1 interst rate was 8%, therefore we would solve for the forward rate at yr 2 by: ! = 1 ∗ 1.10 ! = 1.21 1.21 = 1 ∗ 1.08 + (1 ∗ (1 + )) Ch. 8: Interest Rates and Bond Valuation 3/04/2015 = 0.12037 ( + ) = + ∗ ( + ) r1, r2 = spot rates f2 = forward rate Forward rate + = + − Calculated at date 0 for any future interest rate: + = + ! − o Estimating the price of a bond at a future date § When the initial bond sells at par, the payment at maturity is > $1,000 Date 0 Year 1 Date 1 Year 2 Date 2 $1,000 $1,080 initial payment Bond A 8% purchase at price maturity $1,000 $1,210 initial payment Bond B 10% purchase at price maturity 1-‐yr spot rate from date 1 to ?% date 2 is unknown as of date 0 § Steps to solve • Date 0 PV of each bond given payment at maturities (be mindful of the 2-‐yr bond as it gives the interest rate over the 2 yrs, not the indiv interest rates of the 2 yrs, so may be off.) o Bond A: $1,000 Bond B: $1,000 • Determine price of each bond at Date 1 o Bond A: $1,080 Bond B: If you find out the spot rate Date 1 – Date 2 is 6%, 6% is the 1-‐yr spot rate over yr 2. (Invest $1,000 at date 1 and receive $1,060 at Date 2. $1,141.51 = $1,210 1.06 o Note: Even if the forward rate is known at Date 0, the 1-‐yr spot rate beginning at Date 1 is unknown ahead of time – can only speak of expected price / value = $ + § Jensen’s Inequality $ + > $ + Ch. 8: Interest Rates and Bond Valuation § § 3/04/2015 If the forward rate = expected spot rate, one would earn the same amount whether (1) invest in a 1-‐yr bond, or (2) invest in a 2-‐yr bond but sold after 1 yr. Expectations Hypothesis = If individuals are risk-‐neutral, individuals always buy the 1-‐yr bond and not the 2-‐yr bond, bec they make more on the 1-‐yr bond than they would selling after 1 yr on the 2-‐yr bond. Liquidity Preference Hypothesis > If indiv are risk-‐averse, they would invest in the safe option Option 1: 1-‐yr bond (safer because know return = r1) Option 2: 2-‐yr bond but sell after 1 yr (riskier bec depends on interest rate) Forward rate over the 2nd yr must be greater than spot rate expected over 2nd yr Option 3: 2-‐yr zero coupon bond (safer) Option 4: 1 yr bond, then immediately purchase another upon maturity (riskier) ...
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