Valuation of bonds are nothing more than an application of what you learned in the pre-assignment. Hence, they are our first topic. Introduction to bonds and bond markets Define what a bond is? Get everyone comfortable with their workings. Give some examples Zero coupon bonds Focus on the simplest kind of bond to start off with Talk about how to price the bond Talk about how the price of the bond reacts to changes in interest rates Coupon bonds Focus on a slightly more complex bond Talk about how to price the bond Talk about how the price of the bond reacts to changes in interest rates Spot and Forward Rates What do these terms mean and to what do they refer Nominal and real interest rates Discuss very topically the effect of inflation on interest rates and why you should care about it. Leave the more detailed issues for macro
A Bond is a loan <Illustrate this by borrowing money from someone in the audience.> Fixed Income – CF’s known in advance (fixed in many, but not all, cases) Most all bonds are characterized by: -Maturity: life of contract -Face (Par) Value: principal amount that gets repaid at maturity -Coupon rate: payments made over life of bond in addition to the principal Bonds differ -Repayment: no coupons, fixed-Coupon, floating coupon -Issuer (Debtor): Government, Federal Agency, Corporation, Municipality -Maturity: Range from a few months (T-bills) to forever (Consols) -Security: Collateral backed vs. good faith of issuer -Priority: Jr. vs. Sr. --Talk about each of these in more detail
Balloon Payment Bonds Zero -Sell at a discount // -Face – price = interest // -Pay bonds face value at maturity // Demand comes from ppl who don’t want intermediate cash flows (send kids to college) Coupon -Sell at a discount/par/premium depending on relation of c to r // -Pay bonds face value at maturity // Send kid to college and give him an allowance while getting there. Floating Rate -LIBOR (London Interbank Offered Rate, interbank lending rate) is one popular reference rate // -Prime rate (benchmark IR charged by banks) on corp loans // -Pay bonds face value at maturity No Balloon Payment Bonds Perpetual Bonds - no maturity // Rolled over across generations or repurchased for the war effort. // -Perpetuities or Consols, issued in Britain // -in US (used for Panama canal but all were repurchased// for tax purposes the IRS treats them as equity (many people think they are illegal in the US, but this is probably not exactly the case- the issue is the tax treatment) Annuity Bonds -each payment = Interest + fraction of principal // -Mortgages are an e.g. // Immediate Variable Annuity = spread $ over different investments (stocks, bonds, etc.) You get bigger initial income but your invests better perform. Can offer more bang for the buck.
Treasury Bills No coupons (zero coupon security) // Face value paid at maturity // Maturities up to one year (3, 6, 12 month) 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 . Treasury Strips Zero-coupon bond // Created by “stripping” the coupons and principal from Treasury bonds and notes. // Created by IB’s in 1982 (Salomon’s CATS, Merryl’s TIGRS) // e.g. 3 people want zero coupon bonds. I own a coupon bond. I sell each coupon to a different person for a slight premium. This is stripping. Remarks: -Mutual Funds, Institutional Investors, Private Investors (see web-site: bond-market assoc, bureau of public debt) // -No default risk (Backed by full faith & credit of US gov’t) // -Exempt from state and local taxes // -Traded in the OTC market // Highly liquid market in terms of $ size and # of holders.
Agency Bonds GNMA, FNMA, Sallie Mae, Freddie Mac, others? // Maturity: FNMA – ST debt like T-bills & LT debt like // Sold through dealers // Not risk-free, backed only by the agency, not gov’t // How it works: FNMA buys mortgages from small banks. This gets loans off their balance sheet and allows them to continue making loans. In order to buy the loans, FNMA issues bonds. When FNMA receives mortgage payments, it turns around and passes them onto the bond holders. Expenses are covered since FNMA purchases mortgages at a slight discount. // No real default risk since that is not systematic. // Pre-payment is a big risk because when IR’s fall, everyone refinances. Thus, these bonds sell at a discount (I.e. offer a higher yield than Treasuries.) Fannie Mae and Freddie Mac do the same thing (buy home loans under $322.7k) but Fannie Mae is bigger. Both are publicly trade co.'s. Freddie Mac and Fannie Mae operate under gov’t charter quasigov’t entity. Similar to any publicly traded company(line of credit with the Treasury) Ginnie Mae is an agency of the dept. of housing and urban development that guarantees mortgages for other lenders. Sallie Mae, a privatized co., buys student loans Farmer Mac guarantees pools of agricultural loans Municipal Bonds Issued by states, cities, counties and other gov't entities to raise $ for schools, highways, hospitals, sewage systems etc. (I.e. projects for the public good) Many issues can be called (I.e. retired before the end of the loan) Some are insured to reduce investment risk Revenue bond’s coupon and principal payments are derived from tolls, charges or rents paid by users of the facility built w/ bond proceeds. General obligation bonds are voter approved and supported by the full faith & credit of the issuer Not all munis are tax free. Fed gov’t won’t subsidize certain activities that don’t benefit the public at large (e.g. local sports facilities, investor-led housing, borrowing to fund an under funded pension system, etc.) These munis are table at the fed level. Ownership (forms of issuance): primarily issued in registered form w/ your name appearing on the bond and issuers books. More common is book-entry form, like US gov’t bonds where ownership is recorded through a central clearinghouse (no need for physical transfer here). Before 1983 munis issued in certificate form w/ coupons attached (I.e. bearer bonds). No record of who owns the bond, you detach coupon and collect interest from issuing agent. Transfer of bonds requires physical delivery. munis denominations are integral multiples of $5k, w/ min = $5k. can sell munis in secondary markets (banks & security dealers Lower pre-tax yields, offset by taxing savings // Other examples of tax exempt bonds are issued by states, state univ’s and school districts. // risk—Orange county? Robert Citron in 1994 invested in inverse floaters (floating rate bonds whose CF’s fall as IR’s rise. They pay 17% - 2 times the short rate (if greater than 0).) These are 3 times more sensitive to changes in IR’s of fixed rate debt. He also leveraged his position (I.e. borrowed) through repurchase agreements making the position 3 times greater than the cash he had invested. Net position was 9 times more sensitive than unlevered fixed rate debt. When IR’s rose, end of story.
Co.'s issue bonds to raise money for investment, working capital, repayment of other debt, takeover attempts, etc. Bonds vs. Debentures -Bonds are secured by physical assets -Debentures backed by good faith of the firm -Important in BK, Explicit seniority (Bonds OVER debentures) Indenture or Trust Deed (took out of the slide itself) -this is bond agreement in public offerings -Specifies the details of the bond offering (complex) -contains covenants restricting the behavior of the firm --Asset: defines the bond type, Sr. vs. Jr. (subordinated), Secured --Dividend: so firm doesn’t liquidate via dividends --Financing: new debt is subordinated or credit risk is very low --Financial Ratio: net working capital = cur. Assets – cur. Liabilities, interest coverage = earnings/interest --Bonding mechanism: ensure borrower adheres to covenants Fixed vs. Floating Investment vs. Junk Additional Features (Optionality) Convertible=bonds that can be exchanged for stock (pay premium) typically called when IR’s fall. Callable=entitles debtor right to repay and retire early. (get a discount) Putable=entitles holder to demand early repayment. (pay premium) Additional Typically sold in multiples of $1k or $5k // semiannual coupons // no ownership rights // primarily traded in OTC markets (I.e. no central location. Broker & dealers trade via phone or electronically) - exchange traded bond markets are small // you have to pay tax each year on accrued interest on 0-coupon bonds // sinking fund provisions – periodically the firm must retire a portion of the outstanding bonds, which are selected by lottery // issued in 3 forms: 1-registered bonds have owner’s name on them, no attached coupons, 2 – bearer bonds have no printed name on them and have coupons that can be redeemed (like cash and as such valuable). Tax reform act of 1982 ended issuance of these bonds, 3 – book entry bonds – have only one master certificate and ownership is recorded in investor brokerage acct.
Enormous importance for bond prices and interest rates. Downgrades and upgrades have a large impact on not only bond prices (& yields) but stock prices, as well. All bonds are rated, gov’t, agencies, munis, and corp. NC’s AAA bond was downgrade on 8/20/2002 Find bond ratings simply by calling Moody’s (knowing CUSIP helps) or going to S&P web site-ratings inquiry Line Top 4 ratings = investment grade Below = junk bonds IBM vs. PG&E vs. State of Ca. Remarks -S&P has 23 ratings with + and – -Moody’s: A1, A2, Baa2,… -There are also Fitch and D&P -KMV rates all firms with continuous measure (recently purchased by Moody's) - Recent action: Ford & GM were just downgraded from A to BBB and are being -watched -Lowest for Ford since 84 -Junk bonds: exploded in 80’s (LBO’s, Milken), Busted in 90/91, came back thereafter -A lot of institutions are not allowed to invest in below-investment grade debt. When rating falls, institutions are often required to sell. -Downgrades result in falling stock prices cause higher prob of default. Empirically we see 7-8% fall in price after downgrade. Default Probabilities: 1y: 5y: 10y: AAA 0 0.1 0.1 A 0 0.2 0.6 BBB 0 1.6 2.8 B 1.5 22.0 33.0 CCC 2.3 35.4 47.5
without corp 30855.7 with corp 49540.2
Done with Intro - MENTION THAT BONDS ARE ACTIVELY TRADED. THEY’RE RARELY HELD UNTIL MATURITY BY ONE INDIVIDUAL -Start working Start Valuation with Zeros -Notation (Face = Par) -Already know who to do this from preassignment -discounting a single future cash flow Notation m = # of compounding period per year (semiannual m=2, monthly m = 12,…) i = IR we get for each compounding period (annual i = r) 2 Cash flows Purchase price of the bond, which we pay receive the principal or face at maturity interest comes in through difference between price and face Bond Price Remember equivalent way to write this B0 = F/(1+R/m)^Tm Some intuition: How much do you have to invest today in a risk-free security to get F in the future? This equation answers that question.
COVER Value 5-year Notation: T = 5, m = 1, R= 7.5% I =7.5%, N=5 m = 4 I = 1.875%, N = 20 Notice: freq of compounding goes up, price falls, return goes up Leave to you to do continuous compounding Annual Compounding = P = 1000/(1.075)^5 = $696.56 Quarterly Compounding = P = 1000/(1+0.075/4)^20 = $689.68 Find APR We also call R the yield to maturity (I.e. the discount rate that gets me the current price of the bond) More on this later Annual compounding = R = (1000/591.11)^(1/7) – 1 = 7.8% Semi-annual compounding = R = 2[(1000/591.11)^(1/14) – 1] = 7.65% Why is the higher compounding frequency have a lower rate? A comment: What we discount by corresponds to an opportunity cost. What rate of return can I earn on an another asset with equivalent risk characteristics.
Why should we care about IR sensitivity? Quantifies risk exposure of our holdings, personal, bank, institutional investors. How much money can I lose if IR’s change? understand limitations of measures Features of table: Compute prices using standard formula Note a couple of things: 1) relationship b/w price and interest rate 2) relationship b/w price and maturity 3) Longer bonds are more sensitive to changes in interest rates (both absolute and relative) 4) Convexity (the lower the interest rate the, the greater the change in price) See this on next picture 10-year bond 9-10% 36.87 9.56% 10-11% 33.36 9.47%
Graphical illustration of previous comments 1) relationship b/w price and interest rate: across all maturities 2) relationship b/w price and maturity: fix IR 3) Longer bonds are more sensitive to changes in interest rates (both absolute and relative) bigger price change for a change in IR, longer maturity is 4) Convexity (the lower the interest rate the, the greater the change in price)
Quantify sensitivity DV01 Obvious measure look at the difference for a small change in IR One is similar to a one-sided numerical derivative Three is similar to a two-sided derivative (closer to method 2) Method 2: We’re approximating the change in a function using: [f(a+h) – f(a)] ~ f’(a) x h as h gets bigger, this approximation gets worse! All we’re doing is measuring the change in a function for a change in the argument Tweak the IR a little and what happens to the price. -For method 2: the derivative gives us the change in the bond price for a unit change in the interest rate. If we are measuring interest rates in percentages, then this tells us that for a 1% change in interest rates (e.g. 5% to 6%), what the change in bond price is.To get the $ change for a 1 bp change, we need to change units by dividing by 10,000.
Bond prices all computed in standard way The bold face represents original bond Look at 1-year bond Method 1 is 1-sided, 1 basis point down (9.990% - 10%) Method 2 is formula Method 3 is 2-sided, ½ basic point up and down (9.995% - 10.005%) Longer term zeros are more sensitive to interest rate changes. Method 2 B=F/[1+R/m]^Tm db/dR = -F/[m(1+R/m)]^Tm+1 All methods yield roughly the same result assuming the function is “smooth”. However, small differences can translate into big $.
Higher slope = greater sensitivity At lower interest rates, slope is larger (in magnitude) so small changes in IR’s = big changes in Price At higher interest rates, slope is smaller (in magnitude) so small changes in IR’s = small changes in Price
Consider Amort Bond Discuss each parameter How can we value? Brute force discount each cash flow and add Recognize that this example is just an annuity use formula: a x [1 – (1+R/m)^-Tm]/(R/m) Replication Can we reproduce the timing and cash flows of this security with a portfolio of existing assets in the market? If so, we have perfectly replicated the security and the price of our portfolio is the same as the security assuming NA. Draw the arbitrage table on board.
COVER first
Market price of the bond should be 3,545.95. Bond price is too low—buy it (make a loan) and short the portfolio (borrow the money). Erase previous arbitrage table and fill in the blanks. This is an arbitrage table. It is important!!! Intuition: -We’ve got a security -Match cash flows and timing -Figure out which security gives me those cash flows and that timing -compute prices -add ‘em up Don’t run out and execute because there are no transaction costs being accounted for here. Also, this is not a new idea. Quants run programs to look for these mispricings 24 hours a day. Arbitrage won’t last because of supply and demand.
Market price of the bond should be 3,545.95. Bond price is too low—buy it (make a loan) and short the portfolio (borrow the money). This is an arbitrage table. It is important!!! Intuition: -We’ve got a security -Match cash flows and timing -Figure out which security gives me those cash flows and that timing -compute prices -add ‘em up Don’t run out and execute because there are no transaction costs being accounted for here.
COVER Don’t want to use brute force method or replicating portfolio, too long. Use elegant approach: annuity + zero P = {a x [1-(1+R/m)^-Tm]/(R/m)} + {F/(1+R/m)^Tm} Walk through notation. T = 10 m = 2 N=20 C = 90 c=45 a = 45 R = 10% I = 5% F= 1000 B = annuity + zero A = (45/0.05)*(1-(1.05)^(-20)) = $560.80 Z = 1000/1.05^(20) = $376.89 B = A + Z = $937.69
What happens when compounding is monthly? Continuous? (1+I)^N e^RT Just change the discount factor: 1/(1+I)^N
We’ve already seen this concept. You see prices in the paper. Math Moment Analytic solution to a 5 th degree polynomial Reduce 5 th degree by solving for one root to 4 th degree = product of 2 quadratics Can only be computed using numerical procedures when number of compounding periods is greater than 5. Don’t get into the Term Structure of interest rates at this point, and differing rate just stick to the inferring interest rate from prices.
What happens to bond prices as IR’s change? Coupon bonds Thus a coupon bond should react in a similar manner to IR changes as a zero. Walk through some computations Reminder: F=1000 T=10 m=2 N=20 C=900 —>c=450 R=?
Reminder: F=1000 T=10 m=2 N=20 C=90 —>c=45 R=? Price B = (45/0.04)*(1-(1.04)^(-20)) + 1000/1.04^20 = 1067.95 What do we see? IR goes up, P goes down When R<c, bond sells at premium When R>c, bond sells at discount When R=c, bond sells at par
Inversely related -Coupon bonds are just portfolio bonds. When IR’s fall, all zero prices fall coupon prices fall These results hold when settlement occurs on a coupon date. When bond is between coupon dates, quoted price < par when coupon = yield Caveat Flat price = quoted price Full price = flat price + accrued interest (amount of int. owed between settle dates) t t+1/2 t+1 |-----------------------|--------------------| I own bond ------><------You own bond I deserve the first fraction of the coupon = accrued interest = period coupon x (days elapsed / days in coupon period)
2 bonds only differ in their time to maturity. Bigger price change (absolute and relative) for longer term bond greater sensitivity These results follow from a coupon bond as a portfolio of zeros. Note convexity: At lower interest rates, the change in price is greater for a fixed change in interest rate.
Similar picture before: Note: the long-term bond is steeper and more convex The point of intersection is when the interest rate = 10% (coupon rate) When R>c, they both sell at a discount When R<c, they both sell at a premium Larger premium for the long-term bond when R<c because we’re not discounting the CF’s in the future that hard. This reverses when R gets big so the far off CF’s get discounted really hard.
DV01 is the wrong measure because of failure to account for timing of CF’s. Graph %change = [P(7.995) – P(8.005)]/P(8.005) = (1080.06 – 1079.64) / 1079.64 Different sensitivities has to do with coupon rate, cause all else equal! The problem is in the timing of the cash flows. Note: Coupon = 6% < APR = 8% discount price Coupon = 10% > APR = 8% premium price
Point Maturity doesn’t cut it cause it doesn’t take account of timing of CF’s Need a measure that will account for this Duration Remark: Look at supplemental material for more detail on duration
With the adjustment we can interpret duration nicely T n /m is the time (in years) to the nth cash flow. c(1+I)^-n is the PV of the nth cash flow Since the sum of the PV’s = B, we have a weighted average. Remark: We can use the same calculation for a portfolio of bonds. Just aggregate the CF’s Points The sum of the present values of the cash flows is B, thus the PV(c) are the weights. Duration is a weighted average of time to maturity where the weights are the present value of the cash flows. Think of it as an expected value (rv is T(I) with probs PV(c(I))/B.) Higher coupon rate implies greater cash flows sooner and thus a lower duration and less price sensitivity.
Remarks See class notes and special slides for duration on the web. How might any of this be used? Brealy & Myers: Firm has pension liability worth $1mil in PV terms It puts $1mil into pension fund and invest in gov’t bonds today As IR changes value of pension fund changes To ensure value of pension can meet pension payments, match duration. value of portfolio moves with pension payments. We can link duration and DV01 so that both are equally good for summarizing and hedging IR risk. (See Grinblatt & Titman) DV01 = DUR * B/(1+R/m) * 0.0001 Immunization Lock in value of a portfolio at end of planning horizon. Turn a portfolio in a 0-coupon bond. Remark: -Duration assumes that all interest rates move in the same direction. That is that a change in IR occurs for all periods (I.e. the yield curve shifts up or down). In reality this is not the case. -See website for more details. In BM they call modified duration volatility
Term Structure -Expectations says the shape of the term structure reflects investors expectations about future interest rates. -Liquidity preference says that a rate premium should be attached to longer term bonds because of a dearth of long term investors. These bonds are less liquid and a premium should be attached. This suggests an upward sloping term structure. This argument can also work in reverse if there are a dearth of short term investors, a premium may need to be attached to short term bonds suggesting a downward sloping term structure. -Preferred habitat says that different rates across different maturities reflect different demands by investors. Other Term Structures -I defined it as spot term structure, using yields of zeros of dif. Maturity -Par Yield Curve: use yields of on-the-run (most recently issued) coupon paying Treasuries of different maturities. -Annuity term structure: yields of riskless bonds with level payments of different maturities. 1999 – Term structure was increasing 2000 – Inverted Term Structure 2001 – S-shaped term structure. Assumption in duration was parallel shifts in the term structure.
Term Structure -Expectations says the shape of the term structure reflects investors expectations about future interest rates. -Liquidity preference says that a rate premium should be attached to longer term bonds because of a dearth of long term investors. These bonds are less liquid and a premium should be attached. This suggests an upward sloping term structure. This argument can also work in reverse if there are a dearth of short term investors, a premium may need to be attached to short term bonds suggesting a downward sloping term structure. -Preferred habitat says that different rates across different maturities reflect different demands by investors. Other Term Structures -I defined it as spot term structure, using yields of zeros of dif. Maturity -Par Yield Curve: use yields of on-the-run (most recently issued) coupon paying Treasuries of different maturities. -Annuity term structure: yields of riskless bonds with level payments of different maturities. 1999 – Term structure was increasing 2000 – Inverted Term Structure 2001 – S-shaped term structure. Assumption in duration was parallel shifts in the term structure.
1) Federal funds rate is the interest rate at which a depository institution lends immediately available funds (balances at the Federal Reserve) to another depository institution overnight. This is what news reports are referring to when they talk about the Fed changing interest rates . In fact, the FOMC sets a target for this rate, but not the actual rate itself (because it is determined by the open market). 2) The discount rate is the interest rate charged to commercial banks and other depository institutions on loans they receive from their regional Federal Reserve Bank's lending facility--the discount window. The Federal Reserve Banks offer three discount window programs to depository institutions: primary credit, secondary credit, and seasonal credit, each with its own interest rate. All discount window loans are fully secured. Under the primary credit program, loans are extended for a very short term (usually overnight) to depository institutions in generally sound financial condition. Depository institutions that are not eligible for primary credit may apply for secondary credit to meet short-term liquidity needs or to resolve severe financial difficulties. Seasonal credit is extended to relatively small depository institutions that have recurring intra-year fluctuations in funding needs, such as banks in agricultural or seasonal resort communities. The discount rate charged for primary credit (the primary credit rate) is set above the usual level of short-term market interest rates. (Because primary credit is the Federal Reserve's main discount window program, the Federal Reserve at times uses the term &quot;discount rate&quot; to mean the primary credit rate.) The discount rate on secondary credit is above the rate on primary credit. The discount rate for seasonal credit is an average of selected market rates. Discount rates are established by each Reserve Bank's board of directors, subject to the review and determination of the Board of Governors of the Federal Reserve System. The discount rates for the three lending programs are the same across all Reserve Banks except on days around a change in the rate. This type of borrowing from the Fed is fairly limited. Institutions will often seek other means of meeting short-term liquidity needs. The Federal funds discount rate is one of two interest rates the Fed sets, the other being the overnight lending rate, or the Fed funds rate.
Alon: where do you download this from??
point out that the bills are zeros. remember the pricing convention in terms of pricing at a discount.
COVER Consider Amort Bond Discuss each parameter How can we value? Brute force discount each cash flow and add Recognize that this example is just an annuity use formula: a x [1 – (1+R/m)^-Tm]/(R/m) Replication Can we reproduce the timing and cash flows of this security with a portfolio of existing assets in the market? If so, we have perfectly replicated the security and the price of our portfolio is the same as the security assuming NA.
Intuition Bigger duration means that the cash flows occur later in life. These are more sensitive to changes in IR cause of discounting Higher duration more sensitive bonds. Zeros’ are more sensitive to IR changes cause all CF’s occur at end (duration = T) Coupons spread CF’s so that some CF’s are only minimally affected. Think of extreme (all but $1 is paid tomorrow). Change IR and nothing really happens. The adjustment is for a reason - dividing by B gives us a percent change - minus is to get a positive term (since the derivative is negative) - divide by m to get in annual terms
With the adjustment we can interpret duration nicely T n /m is the time (in years) to the nth cash flow. c(1+I)^-n is the PV of the nth cash flow Since the sum of the PV’s = B, we have a weighted average. Remark: We can use the same calculation for a portfolio of bonds. Just aggregate the CF’s Points The sum of the present values of the cash flows is B, thus the PV(c) are the weights. Duration is a weighted average of time to maturity where the weights are the present value of the cash flows. Think of it as an expected value (rv is T(I) with probs PV(c(I))/B.) Higher coupon rate implies greater cash flows sooner and thus a lower duration and less price sensitivity.
-What is F? What is i? What is N? <F=100, i=R/m=7.5%/1, N=Tm=5 B=100/(1.075)^5=$69.66> <Quarterly: B=100/(1+0.075/4)^20=68.97> <Continuous B=100e^-RT=100exp(-0.075x5)=68.73 <Note the inverse relationship between the bond price and the compounding frequency. What does this mean when you put money in a savings account?>
-What is F? What is i? What is N? <F=5000, i=R/m=3.2%/4=0.8%, N=Tm=6 B=5000/(1.008)^6=$4766.58> Bank B: i = 3.2%/2=1.6%, N=1.5*2=3 B=5000/(1.016)^3=4767.48 * Do not switch!
This is just an annuity that will pay $2000 per month for the next 20 years (30 years). The present value for 20-year term is: Annuity = a*(1-(1+i)^-N)/i =2000*(1-(1+0.00658)^240)/0.00658 =240,969.89 For 30 year term it is: =275,282.06
The formula is: B = cp/(1+i) + cp/(1+i)^2 1876 = 1000/(1+i) + 1000/(1+i)^2 i = 4.38% *-Pick only positive, real roots. Example 2: <IRR function in Excel will find this. A sophisticated trial and error program>
Intuition: In case 1, for bond A you receive most of the money at the end of the bond term when interest rates are high. The higher yield to maturity for bond A is just reflecting the fact that longer term interest rates are higher than short term interest rates, and bond A has a relatively high cash flow during that high return period. In case 2, the situation is reversed and as such bond B shows a higher yield since it takes advantage of higher short term rates.
The perfect hedge makes the new portfolio, including the hedge investment, have a duration of 0. To get this, the ratio of the duration of the unhedged bond to the duration of its hedge must be Inversely proportional to the ratio of the respective market values of the bonds. Each bond has duration equal to maturity (they’re zeros) The ratio of durations is: 10/5 = 2 The ratio of their market values should be ½. Market value of the 10-year is 1000/1.04^20 = 456.39 so The market value of the 5 should be: ?/1.04^10 Depending on the face. Solve for this by: 456.39 = ½ x (?/1.04^10) ? = 1351.13 So, short sell 1,351.13 of the 5-year.
Term Structure -Expectations says the shape of the term structure reflects investors expectations about future interest rates. -Liquidity preference says that a rate premium should be attached to longer term bonds because of a dearth of long term investors. These bonds are less liquid and a premium should be attached. This suggests an upward sloping term structure. This argument can also work in reverse if there are a dearth of short term investors, a premium may need to be attached to short term bonds suggesting a downward sloping term structure. -Preferred habitat says that different rates across different maturities reflect different demands by investors. Other Term Structures -I defined it as spot term structure, using yields of zeros of dif. Maturity -Par Yield Curve: use yields of on-the-run (most recently issued) coupon paying Treasuries of different maturities. -Annuity term structure: yields of riskless bonds with level payments of different maturities. 1999 – Term structure was increasing 2000 – Inverted Term Structure 2001 – S-shaped term structure. Assumption in duration was parallel shifts in the term structure.
-Money is of value because of its ability to purchase goods. -Saving gives us cash in the future but this is pointless if the price of goods increases at a rate greater than the rate our savings earn. -Could have bought that new Yugo today but can’t tomorrow because of the price increase. -CPI is typically measured as a percentage of some base year. -e.g. Base year 1990 = 100, 1991 CPI = 110 means that inflation over the year was 10%. You’ve got $1000. Invest it at R=10% for 1 year and you got $1100 next year. If prices go up by 6%, then our $1100 is really only worth 1100/1.06 =$1038. That is we can only buy $1038 worth of goods because of the price increase. Thus we didn’t really earn 10% interest, but rather (1038-1000)/1000 = 3.8% So, (1 + rr) = (1 + R) / (1 + PI)
Value a 5 year, U.S. Treasury strip with face value of $1,000. The APR is R=7.5% with annual compounding? What about quarterly compounding?
What is the APR on a U.S. Treasury strip that pays $1,000 in exactly 7 years and is currently selling for $591.11 under annual compounding? Semi-annual compounding?
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 interest rate is 10% compounded semiannually?
0 6 12 18 24 ... 120 Months
45 45 45 45 1045
Valuation of Coupon Bonds: Example 2: Straight Bonds
What is the market price of a bond that has an annual coupon C , face value F and matures exactly T years from today if the required rate of return is R, with m-periodic compounding?
So far we have valued bonds by using a given interest rate, then discounted all payments to the bond.
Prices are usually given from trade prices
need to infer interest rate that has been used
Definition: The yield to maturity is that interest rate that equates the present discounted value of all future payments to bondholders to the market price:
The duration of a bond is less than its time to maturity (except for zero coupon bonds).
The duration of the bond decreases the greater the coupon rate. This is because more weight (present value weight) is being given to the coupon payments.
As market interest rate increases, the duration of the bond decreases. This is a direct result of discounting. Discounting at a higher rate means lower weight on payments in the far future. Hence, the weighting of the cash flows will be more heavily placed on the early cash flows -- decreasing the duration.
Modified Duration = Duration / (1+yield)
37.
A Few Bond Markets Statistics U.S. Treasuries, May 20th 2007. Bills MATURITY DISCOUNT/YIELD DISCOUNT/YIELD TIME DATE CHANGE 3-Month 08/16/2007 4.72 / 4.84 0.01 / .010 13:41 6-Month 11/15/2007 4.78 / 4.98 0.01 / .015 13:41 Notes/Bonds COUPON MATURITY CURRENT PRICE/YIELD TIME DATE PRICE/YIELD CHANGE 2-Year 4.500 04/30/2009 99-121⁄4 / 4.84 -0-02 / .035 14:08 3-Year 4.500 05/15/2010 99-081⁄2 / 4.77 -0-031⁄2 / .040 14:06 5-Year 4.500 04/30/2012 98-281⁄2 / 4.75 -0-06 / .043 14:07 10-Year 4.500 05/15/2017 97-15 / 4.82 -0-091⁄2 / .038 14:07 30-Year 4.750 02/15/2037 96-17+ / 4.97 -0-17 / .035 14:07
The logical way to measure sensitivity of the bond price to changes in interest rates is to take the derivative of the price B with respect to effective rate i:
We adjust the derivative by dividing by minus the bond price and the number of periods per year m, and multiply by one plus the effective rate.
The measure obtained is often called Macaulay Duration .
If we replace n/m with T n -- which will be the time (in years) until the nth cash flow, the formula is:
Duration is a weighted average term to maturity where the cash flows are in terms of their present value. We can rewrite the above equation in a simpler format:
One and half years from today, you want to buy a new car that requires a down payment of $5,000. How much do you have to save today assuming Bank A is offering a nominal interest rate of 3.2% with quarterly compounding?
Bank B is advertising a special rate 3.5% with semiannual compounding. Should you switch your account to bank B?
You are buying a new home and you can only afford monthly payments of $2000. The current APR is 7.9%. How large of a loan can you afford if the term is 20 years? 30 years?
Yield to maturity is simply the internal rate of return with a different name. As such, this measure experiences the same problems when used for comparing investments.
Example (from Brealey & Myers)
Consider 2 bonds, both with $1000 face value, annual coupon payments and maturing in 5 years.
Bond A: 5% coupon; 8.78% yield; market price = $852.11
Bond B: 10% coupon; 8.62% yield; market price = $1,054.29
Is bond A a better buy than bond B? That is, has the market erred in pricing these two bonds at different yields?
I own a $1,000, 10 year zero coupon bond with 8% yield compounded semiannually. How can I perfectly hedge this position with a short position in a 5-year zero with yield to maturity of 8% (semiannual compounding)