UFL Investments: Chapter 14-16
*Ch. 14 Bond Prices and Yields* Credit risk 469-470
bond default risk, which is usually measured by Moody's Investor Services, Standard & Poor's Coporation, and Fitch Investor Service. Rates BBB or above (S&P, Fitch) or Baa and above (Moody) are considered *investment-grade bonds*, whereas lower-rated bonds are classified as *speculative-grade, junk bonds, high-yield bonds* Half of the bonds rated CCC have defaulted within 10 years.
*Ch. 14 Bond Prices and Yields* Puttable bonds
extendable or put bonds give the option for an issue to extend the bond past the call date. If the bond's coupon rate exceeds current market yields, the bondholder will choose to extend the bond's life.
*Ch. 16 Managing Bond Portfolios* Duration and Convexity of Callable Bonds
negative convexity: When the price-yield curve lies below the tangency line, usually due to rates falling that cause the bond's price to compress (being "compressed" to the call price). Effective duration: (ΔP/P) / Δr The convention on Wall Street is to compute the effective duration of bonds with embedded options. ED is the proportional change in the bond price per unit change in the market interest rates --- *it's a calculation used to approximate the actual, modified duration of a callable bond.* Callable bond with call price of $1050 selling today for $980. If the yield curve shifts up by .5%,the bond price will fall to $930. If it shifts down by .5%, the bond price will rise to $1,010. Compute the effective duration: (ΔP/P) / Δr (ΔP/P) = 930 - 1,010 = -$80 Δr = .5 - (.5) = .01 or 1% (-80/980) / .01 = 8.16 years The bond price changes by 8.16% for a 1% point swing in rate around current values.
*Ch. 14 Bond Prices and Yields* nominal return on a bond: real rate of return on a bond:
nominal return: Interest + price appreciation / initial price real rate of return: (1 + nominal return / 1 + inflation) - 1
*Ch. 14 Bond Prices and Yields* After-tax returns
the IRS calculates a price appreciation schedule to impute taxable interest income for the built-in appreciation during a tax year, even if the asset is not sold or does not mature until a future year. Any additional gains or losses that arise from changes in market interest rates are treated as capital gains or losses if the OID bond is sold during that year.
*Ch. 14 Bond Prices and Yields* Reinvestment rate risk
As interest rates change, investors are subject to two offsetting risks: as rates rise, bond prices fall, which reduces value of the portfolio. And, reinvested coupon income will compound more readily at those higher interest rates. The reinvestment rate risk will offset the impact of price risk.
*Ch. 14 Bond Prices and Yields* zero-coupon bonds
Bonds are usually issued with coupon rates set just high enough to induce investors to pay par value to buy the bond; however, *zero-coupon bonds* are issued that make no coupon payments. The investor's return comes solely from the difference between issue price and the payment of par value at maturity.
*Ch. 16 Managing Bond Portfolios* Why do investors like convexity?
Bonds with greater curvature gain more in price when yields fall than they lose when yields rise (greater price increases, less price decreases when interest rates fluctuate by large amounts). It is worth it, to some investors, to pay higher prices and accept lower YTM on bonds with greater convexity, then.
*Ch. 14 Bond Prices and Yields* Horizon analysis
Forecasting the realized compound yield over various holding periods or investment horizons -- forecast of total return depends on forecasts of both the price of the bond when you sell it at the end of your horizon and the rate at which you're able to reinvest coupon income.
*Ch. 16 Managing Bond Portfolios* Show that the duration of the perpetuity increases as the interest rate decreases in accordance with rule 4.
y: .01, .02, .05, .10, .20, .25, .40 D: 101yrs, 51yrs, 21yrs, 11yrs, 6yrs, 5yrs, 3.5yrs duration of a level perpetuity is (1+y/y) which clearly falls as y increases.
*Ch. 16 Managing Bond Portfolios* Calculating the DURATION of two bonds: 8% coupon bond and zero-coupon bond
*A. 8% COUPON BOND* Period: 1, 2, 3, 4 t: 0.5, 1.0, 1.5, 2.0 Cash flow: 40, 40, 40, 1040 *PV of CF* (discount rate, or YTM, is 5% per period: *CF,t / (1 + y)^t* 40/(1+.05)^1 = 38.095 40/(1+.05)^2 = 36.281 40/(1+.05)^3 = 34.554 1040/(1+.05)^4 = 855.615 PV of CF = 965.545 *w,t = PV,CF/Bond price* 38.095/965.545 = .0395 36.281/965.545 = .0376 34.554/965.545 = .0358 855.615/965.545 = .8861 w,t = 1.0 *Duration = (w,t) x time until payment (yrs)* .0395 x .05 = .0197 .0376 x 1.0 = .0376 .0358 x 1.5 = .0537 .8861 x 2.0 = 1.7741 *Duration for 8% coupon bond = 1.8852* *B. ZERO COUPON BOND* Period: 1, 2, 3, 4 t: 0.5, 1.0, 1.5, 2.0 Cash flow: 0, 0, 0, 1000 *PV of CF* (discount rate, or YTM, is 5% per period: *CF,t / (1 + y)^t* 0/(1+.05)^1 = 0 0/(1+.05)^2 = 0 0/(1+.05)^3 = 0 1000/(1+.05)^4 = 822.702 PV of CF = 822.702 *w,t = PV,CF/Bond price* 0/822.702 = 0 0/822.702 = 0 0/822.702 = 0 822.702/822.702 = 1.0 w,t = 1.0 *Duration = (w,t) x time until payment (yrs)* 0 x .05 = 0 0 x 1.0 = 0 0 x 1.5 = 0 1.0 x 2.0 = 2.0 *Duration for zero coupon bond = 2.0* ^^ the duration of the zero-coupon bond is exactly equal to its time to its maturity of 2 years, which makes sense because with only one payment, the average time until payment must be the bond's maturity.
*Ch. 14 Bond Prices and Yields* Bond Pricing
*Bond Value* present value of coupons + present value of par value Bond value = Σ [coupon/(1+r)t] + par value/(1+r)t *to the power of "t" *price* coupon X Annuity factor (r,T) + par value X PV factor(r,T) .... coupon X 1/r[1-(1/1+r,power of T] + par value + 1/(1+r)power of T Page 453 n = #how many semiannual payments i = whtaever's given FV = one-time cash flow when bond matures PMT = each semiannual coupon payment amount PV = ????
*Ch. 14 Bond Prices and Yields* Other domestic issuers of stocks:
*Municipal bonds* (interest is tax free) *International Bonds* 1. foreign bonds (issued by a borrower form a country other than the one in which the bond was sold --- demoninated in the currency of the country in which it's marketed. Foreign bonds sold in the US are YANKEE BONDS and are registered with the SEC. British pound-denominated foreign bonds are called BULLDOG BONDS) 2. Eurobonds (denominated in one currency, usually of the issuer, but it's sold in other markets. NOT regulated by the SEC or fed agencies) *Inverse floater bonds*: similar to floating-rate bonds, except that the coupon rate here FALLS when the general level of interest rates rise. Investors in these bonds suffer doubly when rates rise, but benefit extra when rates fall. *Asset-backed Bonds* A financial security collateralized by a pool of assets such as loans, leases, credit card debt, royalties or receivables. For investors, asset-backed securities are an alternative to investing in corporate debt. (ie. Dominos issuing bonds with payments backed by revenues from pizza franchises) *Catastrophe Bonds* These bonds are a way to transfer catastrophe risk from the firm to capital markets. Investors in these bonds receive compensation for taking on the risk in the form of higher coupon rates. In the event of trouble, the bondholders give up all or part of their investments. *Indexed Bonds* Make payments that are tied to a general price index or price of a particular commodity. The interest rates on these bonds is the risk-free real rate.
*Ch. 14 Bond Prices and Yields* Determinants of Bonds Safety
1. Coverage ratios compare earning to fixed costs 2. Leverage ratio, debt to equity ratio 3. Liquidity ratio current and quick ratio 4. Profitability ratio return on assets, return on equity 5. Cash flow to debt ratio total cash flow to outstanding debt (Edward Altman) tested whether financial ratios can in fact be used to predict default risk. Altman used a discriminant analysis to predict bankruptcy. Using this analysis, Altman claims that score below 1.23 indicate vulnerability, scores between 1.23-2.90 are gray area, and scores above 2.90 are considered safe.
*Ch. 16 Managing Bond Portfolios* Why is duration important to fixed-income portfolio management?
1. It's a simple summary statistic for the effective average maturity of the portfolio 2. It's an essential tool to immunizing portfolios from interest risk 3. Duration is a measure of the interest rate sensitivity of a portfolio.
*Ch. 16 Managing Bond Portfolios* Rules for Duration Pg 522-523
1. The duration of a zero coupon bond = its time to maturity 2. Holding maturity constant, a bond's duration is lower when the coupon rate is higher (malkiel's 5th) 3. Holding the coupon rate constant, a bond's duration generally increases with its time to maturity. Duration always increases with maturity for bonds selling at par or at premium. (Malkiel's 3rd) 4. Holding other factors constant, the duration of a coupon bond is higher when the bond's yield to maturity YTM is lower. (Homer and Liebowitz 6th) 5. The duration of a level perpetuity is: 1+y/y (10% yield, duration of perpetuity that pays $100 annually, is 1.10/.10 = 11 years. But an 8% = 1.08/.08 or 13.5 years.
*Ch. 16 Managing Bond Portfolios* Passive Bond Management
2 broad classes of passive management: 1. *Indexing strategy* (attempts to replication the performance of a given bond index) 2. *Immunization* techniques (attempt to shield the overall financial status of the institution from exposure to interest rate fluctuations. These are similar in that they accept market prices as correctly set, but they are different in terms of risk exposure. A bond index portfolio will have the same risk-reward as the bond market index to which it's tied. But immunization seeks to establish a zero-risk profile, in which interest rates movements have no impact on the value of the firm. Three major indexes of broad bond market: 1. Barclays Capital US 2. Salomon Broad Investment Grade (BIG) Index 3. Merrill Lynch Domestic Master Index Fixed income investors faces two offsetting types of interest rate risk: 1. Price risk 2. Reinvestment rate risk (increase in interest rates cause capital losses but at the same time increase the rate at which reinvested income will grow)
*Ch. 16 Managing Bond Portfolios* Modern Duration
A natural measure of the bond's exposure to changes in interest rates. D* = D/(1+y) or ΔP/P = -D*Δy The percentage change in bond price is just the product of modified duration and the change in the bond's YTM.
*Ch. 16 Managing Bond Portfolios* Suppose the interest rate decreases to 9% as an annual percentage rate. What will happen to the prices and duration of the two bonds in the previous scenario? Seeing as how these bonds use a semiannual factor, you would change the previous scenario's y from .05 (semiannual of 10%) and make it .045 (semiannual of 9%)
8% coupon rate: - Prices will increase PV of CF = 982.062 and duration will increase 1.8864. zero-coupon rate: - Prices will increase PV of CF = 838.561 and duration will remain the same = 2.
*Ch. 15 Term Structure of i rates* The Yield Curve and Future Interest Rates
A *spot rate* refers to the yield to maturity on zero-coupon bonds, aka. the rate that prevails today for a time period corresponding to the zero's maturity. In contrast to the spot rate, the *short rate* for a given time interval refers to the interest rate for that interval available at different points in time. A 2 year spot rate is the average of today's short rate and next year's short rate; but it is the geometric average, not the arithmetic. That is, we multiply the rates and take the "nth" root. (1+Y,2)^2 = (1+r,1) X (1 + r,2) 1 + Y,2 = [(1 + r,1) X (1 + r,2)]^1/2 So, when next year's short rate, r2, is greater than this year's short rate, r1, the average of the two rates is higher than today's, so y2>r1 and the yield curve slopes upwards. *HPR* (holding period returns) - Regardless of changing yields to maturity on the bonds, the Holding Period Return will remain the same; All bonds must offer identical returns, or investors will flock to the higher-return securities, bidding up their prices and thus reducing their returns. *Forward Rates;* Forward interest rates refer to interest rates for investments we agree on today, but take place in the future. In the absence of arbitrage opportunities,the term structure determines all forward rates. Forward rates exceed expected short term rates, as the interest here needs to prevail in a 2nd year to make the long and short term investments equally attractive (ignoring risk). (1 + Yn)^n = (1 + Yn-1)^n-1 x (1 + r,n) n=period in question Yn = YTM of a zero coupon bond with an n-period maturity Forward rates can be high for 2 reasons: investors expect rising interest rates (+E,rn) or they require a large premium for holding longer-term bonds. *Liquidity premium* compensates short term investors for the uncertainty about the price at which they will be able to sell their long-term bonds at the end of the year.
*Ch. 14 Bond Prices and Yields* Characteristics of Bonds
A bond is a security that's issued in connection with a borrowing arrangement. The borrower "issues/sells" a bond to the "lender/borrower" for some amount of cash; the bond is the "IOU" of the borrower. Issuer makes payments of interest to the bondholder on specified dates.
*Ch. 14 Bond Prices and Yields* Debt security
A claim on a specified periodic stream of income - often called fixed-income securities because they promise either a fixed stream of income or one that is determined according to a specified formula.
*Ch. 14 Bond Prices and Yields* Sinking fund 473
A fund that helps spread the payment (commitments for the issuer) burden over several years by: 1. Repurchasing a fraction of the outstanding bonds in the open market each year. 2. Purchase a fraction of the outstanding bonds at a special call price associated with the sinking fund provision.
*Ch. 15 Term Structure of i rates* Yield Curve
A graph of yield to maturity as a function of time to maturity. A yield curve is a line that plots the interest rates, at a set point in time, of bonds having equal credit quality but differing maturity dates. Yield Curves: 1. A key concern of fixed-income investors 2. Central to bond valuation 3. Allows investors to gauge their expectations for future interest rates against those of the market
*Ch. 15 Term Structure of i rates* Synthetic forward loan
A synthetic forward contract, or synthetic long forward contract, is a position in which the investor buys a call option and sells a put option at the same time. Both options must have the same strike price and expiration date Its purpose is to mimic a regular forward contract, and is also called a synthetic futures contract. The investor will typically pay a net option premium when executing a synthetic forward contract as not all the premium paid for the long position is offset by sale of the short position. For a synthetic short forward contract, the investor buys a put and sells a call, again with the same strike price and expiration date.
*Ch. 14 Bond Prices and Yields* Coupon payment
A typical coupon bond obligates the issuer to make semiannual payments of interest to the bondholder for the life of the bond -- these are called coupon payments because in precomputer days most bonds had coupons that investors would clip off and present to claim the interest payment.
*Ch. 16 Managing Bond Portfolios* Duration and Convexity of Mortgage-backed securities
Biggest market for call provisions is mortgage-backed securities See pg 532
*Ch. 16 Managing Bond Portfolios* Determinants of interest rate risk
Bond prices decrease when yields rise. The price curve is convex, meaning that decreases in yields have bigger impacts on price than increases in yields of equal magnitude. So,... *See figure 16.1 on page 516* A - coupon 12%, maturity 5 years, YTM 10% B - coupon 12%, maturity 30 years, YTM 10% C - coupon 3%, maturity 30 years, YTM 10% D - coupon 3%, maturity 30 years, YTM 6% 1. Bond prices and yields are inversely related: as yields increase, bond prices fall; as yields fall, bond prices rise. 2. An increase in a bond's YTM results in a smaller price change than a decrease in yield of equal magnitude. 3. Prices of long-term bonds tend to be more sensitive to interest rate changes than prices of short-term bonds. (interest rate sensitivity between A and B) 4. The sensitivity of bond prices to changes in yields increases at a decreasing rate as maturity increases. In other words, interest rate risk is less than proportional to bond maturity. (between A and B) 5. Interest rate risk is inversely related to the bond's coupon rate. Prices of low-coupon bonds are more sensitive to changes in interest rates than prices of high-coupon bonds. (between C and D) - aka. price sensitivity falls with coupon rates: Higher-coupon bonds have a higher fraction of value tied to coupons rather than final payment of part value, and so the "portfolio of coupons" is more heavily weighted toward the earlier, short-maturity payments, which gives it lower "effective maturity". 6. The sensitivity of a bond's price to change in its yield is inversely related to the YTM at which the bond currently is selling. (between C and D) - or, price sensitivity falls with yield to maturity (YTM). At a higher yield, a higher fraction of the bond's value is due to its earlier payments, which have lower effective maturity and interest rate sensitivity. *Malkiel's bond pricing relationships*: #1-5 Homer and Liebowitz came up with #6
*Ch. 16 Managing Bond Portfolios* Dedication Strategy
Cash flow matching on a multi-period basis. An asset management method by which the anticipated returns on an investment portfolio are matched with estimated future liabilities. A dedication strategy is frequently used in pension funds and insurance company portfolios to ensure that future liabilities can be met. Dedication strategy is also called portfolio dedication, cash flow matching and structured portfolio strategy.
*Ch. 14 Bond Prices and Yields* Collateral vs. debenture
Collateral is a particular asset that bondholder receive if the firm defaults on the bond (mortgage bonds, collateral trust bonds, equipment obligation bonds, etc.); these are safer than debenture bonds, which are unsecured, and do not provide for specific collateral.
*Ch. 14 Bond Prices and Yields* Corporate bonds
Corporations borrow money by issuing bonds, like the government, over-the-counter in a network of bond dealers linked by a computer quotation system. Safer bonds with higher ratings promise lower yields than other bonds. *call provisions*: Allow the issuer to repurchase the bond at specified call prices before the maturity date. "Refunding" is when a company issues a bond with a high coupon rate when market interests are high, and interest rates later fall, the firm might like to retire the high-coupon debt and issue new bonds at a lower coupon rate to reduce interest payments. *Callable bonds*: A callable bond is a bond that can be redeemed by the issuer prior to its maturity. If interest rates have declined since the company first issued the bond, the company is likely to want to refinance this debt at the lower rate of interest. In this case, the company "calls" its current bonds and reissues them at a lower interest rate. Typically issued at par value., so underwriters of the bond must choose a coupon rate that very closely approximates market yields.
*Ch. 16 Managing Bond Portfolios* What happens to duration when you change the coupon rate of the bond? YTM? The maturity? What happens to duration is the bond pays its coupons annually rather than semiannually?
Coupon rate change: duration is shorter when coupon rate and/or YTM is higher. Duration increases with maturity for most bonds. Duration is shorter when coupons are paid semiannually rather than annually because, on average payments come earlier.
*Ch. 14 Bond Prices and Yields* Convertible bonds
Give bondholders an option to exchange each bond for a specific number of shares of common stock of the firm. Conversion ratio: # of shares for which each bond may be exchanged market conversion value: the current value of the shares for which the bonds may be exchanged conversion premium: the excess of the bond value over its conversion value.
*Ch. 15 Term Structure of i rates* What factors can account for a rising yield curve?
If the yield curve is rising, fn+1 must exceed Yn. Or, the yield curve is upward sloping at a maturity date, n, for which the forward rate for the coming period is greater than the yield at the maturity.
*Ch. 15 Term Structure of i rates* What does the upward sloping yield curve represent?
It's evidence that short-term rates are going to be higher than they are now.
*Ch. 14 Bond Prices and Yields* Floating-Rate bonds
Make interest payments that are tied to some measure of current market rates. major risk with floaters: if the firm's financial strength changes. the yield spread is fixed over the life of the security, which may be many years, so if the financial health of the firm deteriorates, then investors will demand a greater yield premium than is offered by the security. coupon rate adjusts to the general level of the market interest rates, not the financial conditions of a firm
*Ch. 14 Bond Prices and Yields* Bond Yields
Most bonds don't sell for par value but they will mature at par value. Investors must use the bond price, maturity date and coupon payments to infer the return offered by the bond over its life. *Yield to maturity (YTM)* The interest rate that makes the present value of a bond's payments equal to its price. This is calculated by solving for the bond price equation for the interest rate given the bond's price --- based on the assumption that the bond will be held until maturity. *Current yield* The bond's annual coupon payment divided by the bond price *premium bonds* Bonds selling above par value with a coupon rate that's greater than the current yield, which in turn is greater than yield to maturity. *discount bonds* bonds selling below par value with an opposite relationship to the YTD than premium bonds.
*Ch. 14 Bond Prices and Yields* Treasury bonds & notes
Notes are issued with original maturities within 1-10 years, while bonds are issued with maturities between 10-30 years. Only purchased in denominations of $100 and both may semiannual coupon payments. *ask yield / yield to maturity*: is based on the ask price. The yield to maturity is a measure of the average rate of return to an investor who purchases the bond for the ask price and holds until its maturity date. *Accrued interest & quoted bond prices* Accrued interest = (annual coupon payment/2) X (days since last coupon payment/days separating coupon payments) ex. $80 annual coupon, 30 days since last payment, semiannual payment (182 days). Quoted price of bond is $990. = $40 X (30/182) = $6.59 (accrued interest) = $990 + $6.59 = $996.59 (invoice price = flat price+accrued interest)
*Ch. 15 Term Structure of i rates* Bond pricing
Noteworthy definitions: *Stripped Treasuries*: Zero-coupon bonds created by selling each coupon or principal payment from a whole Treasury bond as a separate cash flow; where an investor's return is determined by the difference between the purchase price and the bond's trading value, or face value if held to maturity. *If each cash flow can be sold off as a separate security, then the value of the whole bond should b the same as the value of its cash flows bought piece by piece in the STRIPS market. *Bond Stripping*: A stripped bond is a bond that has had its coupon payments and principal repayment stripped into two separate components and sold individually. So, this is the selling of bond cash flows (either coupon or principal payments) as a stand-alone zero-coupon securities. *Bond Reconstruction*: Combining stripped Treasury securities to re-create the original cash flows of a Treasury bond.
*Ch. 14 Bond Prices and Yields* Preferred stock
Preferred stock is generally considered equity, it often is included in the fixed-income universe. Preferred stock commonly pays a fixed dividend, so it's like a perpetuity, providing a level of cash flow indefinitely. floating rate preferred stock is like floating rate bonds, where the dividend rate is linked to a measure of current market interest rates and is adjusted at regular invtervals Dividends are not considered tax-deductible expenses to the firm here, unlike interest payments on bonds. offsetting tax advantage: when one corp buys preferred stock of another corp, it pays taxes on only 30% of the dividends it receives. so, your company tax rate is 35% and you have $10,000 of another's stock. You only pay tax on $3000. .35x3000 = $1,050 total tax due. .... so, the firm's effective tax rate on preferred dividends here is only .30 X .35 = 10.5% Rarely are shareholders given voting rights with preferred stock.
*Ch. 15 Term Structure of i rates* Pure yield curve
Refers to the curve for stripped, zero-coupon, Treasuries.
*Ch. 16 Managing Bond Portfolios* What's the diff between Macaulay duration, modified duration and effective duration?
See response #5 page 555-556
*Ch. 15 Term Structure of i rates* Expectations Hypthoesis
Simplest theory of the term structure. The expectations theory attempts to predict what short-term interest rates will be in the future based on current long-term interest rates. The theory suggests that an investor earns the same amount of interest by investing in a one-year bond now and then another one-year bond after the first bond matures, as compared to purchasing a two-year bond in the present. So, f2 = E(r2) forward rate of a 2 year bond = expected future short interest rate *(1 + y2)^2 = (1 + r1) x [1 + E(r2)]* There are three primary theories associated with the expectations theory: 1. *preferred habitat theory* 2. *pure expectations theory* 3. *liquidity preference theory* (short term investors dominate the market so the forward rate will generally exceed the expected short rate. The excess of f2 over E(r2), the liquid premium, is predicted to be positive.
*Ch. 16 Managing Bond Portfolios* Active Bond Management
Sources of potential profit: 1. Substitution swap Exchange of one bond for a nearly identical substitute. 2. Intermarket spread swap When the yield spread between two sectors of the bond market is temporarily out of line 3. Rate anticipation swap Pegged to interest rate forecasting, so if investors believe rates will fall, they'll swap into bonds of longer duration (or vice-versa). 4. Pure yield pickup swap Pursued as a means of increasing return by holding higher-yield bonds. 5. Tax swap swap to exploit some tax advantage, such as from one bond that has decreased in price to another if realization of capital losses is advantageous for tax purposes.
*Ch. 15 Term Structure of i rates* Find the Forward Rate
Suppose the bond trader uses the following. The forward rate for year 4 would be computed as >>> on stripped Treasuries with $1000 face values are as given... Years 1,2,3,4 -- YTM 5%,6%,7%,8% 1 + f4 = (1 + y4)^4 / (1 + y3)^3 = 1.08^4 / 1.07^3 = 1.1106 Therefore, the forward rate is f4 = 11.06%
*Ch. 15 Term Structure of i rates* Forward Interest Rate Contract (example)
Suppose the price of 1-year maturity zero coupon bonds with face value of $1000 is $952.38 and the price of 2-year zeros with $1000 face value is $890. The yield to maturity on the 1-year bond is therefore 5%, while that on the 2 year bond is 6%. The forward rate for the second year is thus: f2 = [(1 + y2)^2 / (1 + y1)] - 1 f2 = [(1 + .06)^2 / (1 + .05)] - 1 f2 = 1.1236/1.05 - 1 f2 = 1.0701 - 1 f2 = 7.01%
*Ch. 15 Term Structure of i rates* Calculate the price and yield to maturity of a 3 year bond with a coupon rate of 4%, making annual coupon payments.
Suppose the yields on stripped Treasuries with $1000 face values are as given... Years 1,2,3 -- YTM 5%,6%,4% ....and we wish to value a 4% coupon bond with a maturity of 3 years. Payments are made annually. Here, the first cash flow with $40 paid at the end of the first year, is discounted at 5%; the second cash flow, the $40 coupon is discounted at 6%; the final cash flow consisting of the final coupon plus par value, or $1,040, is discounted at 7%. The value of the coupon bond is therefore: = (40/1.05) + [40/(1.06^2)] + [1040/(1.07^3)] = 38.10 + 35.60 + 848.99 = $922.69 YTM: 6.88% ^^*YTM Calculation* FV = 1,000 PMT = 40 n = 3 PV = -922.69 *i = 6.94*
*Ch. 15 Term Structure of i rates* Valuing Coupon Bonds
Suppose the yields on stripped Treasuries with $1000 face values are as given... Years 1,2,3 -- YTM 5%,6%,7% ....and we wish to value a 10% coupon bond with a maturity of 3 years. Payments are made annually. Here, the first cash flow with $100 paid at the end of the first year, is discounted at 5%; the second cash flow, the $100 coupon is discounted at 6%; the final cash flow consisting of the final coupon plus par value, or $1,100, is discounted at 7%. The value of the coupon bond is therefore: = (100/1.05) + [100/(1.06^2)] + [1100/(1.07^3)] = 95.2381 + 88.9996 + 897.959 = $1,082.19 YTM: 6.88% ^^*YTM Calculation* FV = 1,000 PMT = 100 n = 3 PV = -1082.19 *i = 6.875* So, while its maturity matches that of the 3-year zero, based off the original info, its yield is a bit lower. Even further, if their coupon rates differ, *bonds of the same maturity generally will not have the same yield to maturity*.
*Ch. 15 Term Structure of i rates* Finding a Future Short Rate
Suppose the yields on stripped Treasuries with $1000 face values are as given... Years 1,2,3 -- YTM 5%,6%,7% Now compare two 3-year strategies. One is to buy a 3-year zero, with a YTM of *7%*, and hold it until maturity. The other is to buy a 2-year zero yielding 6%, and roll the proceeds into a 1-year bond in year 3, at the short rate r3. The growth factor for the invested funds under each policy will be: (1 + Y3)^3 = (1 + y2)^2 x (1 + r3) (1.07)^3 = (1.06)^2 x (1 + r3) 1.07^3 / 1.06^2 - 1 = r3 1.225 / 1.1236 - 1 = 9.02% now, the geometric average of the discount for the next 3 years: 1 + Y3 = [(1 + r1) x (1 + r2) x (1 + r3)]^1/3 *1.07* = [1.05 x 1.0701 x 1.0902]^1/3 So, the yield or spot rate on a long-term bond reflects the path of short rates anticipated by the market over the life of the bond.
*Ch. 15 Term Structure of i rates* Interest Rate Uncertainty and Forward Rates
Supposing that the current short rate to year 2 is 5%, but the expected short rate is 6%. However, if a short-term investor were to invest for only a year, they would be exposed to the riskless 5% return. Investing in another 2-year zero coupon bond would also have an expected rate of return of 5%. However, this rate of return is risky; if next year's interest rate is above expectations, then the bond price will be below expectations, and if it is lower, the bondprice will exceed expectations. Therefore there is a certain risk in the 2-year zero coupon bond applying the short rate, rather than the expected return. This means that the 2-year bond must sell at a price lower than its actual price to incorporate risk. Investors will require a risk premium in order to convince them to hold a longer-term bond. This liquidity premium will compensate short-term investors for the uncertainty about the price at which they will be able to sell their long-term bonds at the end of the year; Offered to induce investors to hold bonds whose maturities do not match their investment horizons. Reinvestment Risk arises because the investment horizon is not matched to our cashflows. Liquidity Risk arises because ******
*Ch. 15 Term Structure of i rates* Forward Rate and synthetic forward loan example. Suppose that the price of a 3-year zero coupon bond is $816.30. What is the forward rate for the third year? How would you construct a synthetic 1-year forward loan that commences at t=2 and matures at t=3?
The 3-year YTM (yield to maturity) is (1000/816.30)^1/3 - 1 = 7% The forward rate for the 3rd year is therefore: f3 = [(1 + y3)^3 / (1 + y2)^2] - 1 f3 = (1 + .07)^3 / (1 + .06)^2 - 1 f3 = (1.07)^3 / (1.06)^2 - 1 f3 = .0903 or 9.03% Alternatively, note that the ratio of the price of the 2-year zero to the price of the 3-year zero is 1 + f3 = 1.0903. To construct the synthetic loan, buy one 2-year maturity zero, and sell 1.0903 3-year maturity zeros. Your initial cash flow is zero, your cash flow at time 2 is +$1000, and your cash flow at time 3 is -$1,090.30, which corresponds to the cash flows on a 1 year forward loan commencing at time 2 with an interest rate of 9.03%.
*Ch. 14 Bond Prices and Yields* Realized compound return
The actual return earned during the holding period for an investment, and may include dividends, interest payments and other cash distributions. The term may be applied to a bond sold prior to its maturity date or a dividend-paying security. Generally speaking, the realized yield on bonds includes the coupon payments received during the holding period, plus or minus the change in the value of the original investment, calculated on an annual basis.
*Ch. 14 Bond Prices and Yields* Indenture
The contact between the issuer and the bondholder (lays out a set of restrictions that protect the right of the bondholders, such as provisions related to collateral, sinking funds, dividend policy and further borrowing).
*Ch. 14 Bond Prices and Yields* Bond indenture
The contract between the issuer and bondholder. *Maturity date*: the date on which the principal amount of a note, draft, acceptance bond or another debt instrument becomes due and is repaid to the investor and interest payments stop // when it needs to be paid off by. *Par (face) Value*: The face value of a bond (bonds will sell at par value when its coupon rate equals the market interest rate) *Coupon Rate*: a bond's interest payments per dollar of par value Annual payment = coupon rate X bond's par value
*Ch. 14 Bond Prices and Yields* Zero-Coupon bonds and treasury strips
Treasury STRIPS are fixed-income securities sold at a significant discount to face value and offer no interest payments because they mature at par value. STRIPS is an acronym for Separate Trading of Registered Interest and Principal of Securities. These zero-coupon bonds come about when the bond's coupons are separated from the bond or note; an investor's return is determined by the difference between the purchase price and the bond's trading value, or face value if held to maturity. All of zero-coupon returns are in price appreciation and provide only one cash flow to their owners on the maturity date of the bond. (ie. US TBills, are examples of short term zero-coupons)
*Ch. 16 Managing Bond Portfolios* Convexity
The curvature of the price-yield is called the convexity of the bond, which is the rate of change of the slope of the price-yield curve. It allows us to improve the duration approximation for bond price changes. ΔP/P = -D*Δy + 1/2 x Convexity x (Δy)^2 The duration approximation (straight line on a graph) always underestimates the value of the bond; it underestimates the increase in bond price when the yield falls and overestimates the decline in price when the yield rises. This is due to the curvature of the true price-yield relationship. If a 30-year bond has an initial yield of 11.26 years and its convexity is 212.4. If its yield increases from 8% to 10%, the bond price will fall and decline 18.85%. Using this info, the predicted price decline would be: ΔP/P = -D*Δy = -11.26 x (.10-.08) = -.2252 or -22.52% but, using the convexity rule, you'd more accurately get: ΔP/P = -D*Δy + 1/2 x Convexity x (Δy)^2 = -11.26 x .02 + .5 x 212.4 x (.02)^2 = -.1827 or -18.27% Clearly, the duration-with-convexity rule provides a far more accurate prediction. -18.27% ~ 18.85%
*Ch. 15 Term Structure of i rates* On-the-run Treasury yield
The on-the-run Treasury yield curve plots the yields of bonds, of similar quality, against their maturities. It is the primary benchmark used in pricing fixed-income securities. The on-the-run Treasury yield curve is typically used to price fixed-income securities. However, its shape is sometimes distorted by up to several basis points if an on-the-run Treasury goes "on special." A Treasury goes "on special" when its price is temporarily bid up. On-the-run Treasuries have the greatest liquidity, so traders have a keen interest in their yield curve. This yield indicates that there are two important factors that complicate the relationship between maturity and yield. A. is that the yield for on-the-run issues is distorted since these securities can be financed at cheaper rates, and therefore offer a lower yield than they would without this financing advantage. B. The second is that on-the-run Treasury issues and off-the-run issues have different interest rate reinvestment risks.
*Ch. 15 Term Structure of i rates* Term structure interest rates
The structure of interest rates for discounting cash flows of different securities. The term structure of interest rates is the relationship between interest rates or bond yields and different terms or maturities. When graphed, the term structure of interest rates is known as a yield curve, and it plays a central role in an economy. The term structure reflects expectations of market participants about future changes in interest rates and their assessment of monetary policy conditions. In general terms, yields increase in line with maturity, giving rise to an upward-sloping yield curve or a normal yield curve. This US treasury yield curve is considered the benchmark for the credit market, as it reports the yields of risk-free fixed income investments across a range of maturities.
*Ch. 15 Term Structure of i rates* Term Structure Theories
The theory that the liquidity premiums determine the shape of the term structure is called the "liquidity preference theory" or the "preferred habitat theory". The liquidity premium is the excess of the forward rate, over the Expected short rate. Market Expectations Hypothesis - Market expectations on future interest rates will shape the term structure
*Ch. 16 Managing Bond Portfolios* Bond-Index Funds and Immunization (Extra info)
Three major indexes of broad bond market: 1. Barclays Capital US 2. Salomon Broad Investment Grade (BIG) Index 3. Merrill Lynch Domestic Master Index Problems with immunization: 1. duration matching will only immunize portfolios only for parallel shifts in the yield curve, which is unrealistic. 2. Can be inappropriate goal in an inflationary environment, as it's essentially a nominal notion and makes sense only for nominal liabilities.
*Ch. 16 Managing Bond Portfolios* Macaulay's Duration
To deal with the ambiguity of the "maturity" of a bond making many payments, we need a measure of the average maturity of the bond's promised cash flows. Macaulay's duration equals the weighted average (w,t) of the times to each coupon or principle payment. Duration is the weighted average number of years an investor must maintain a position in the bond until the present value of the bond's cash flows equals the amount paid for the bond *Duration: w,t = [ CF,t / (1 + y)^t ] / Bond price* w,t = weighted average of the times (t) to each coupon/principle payment CF,t = cash flow made at time 't' y = bond's yield to maturity (YTM) *although long-maturity bonds generally will be high-duration bonds, duration is a better measure of the long-term nature of the bond because it also accounts for coupon payments.
*Ch. 14 Bond Prices and Yields* Relationship between YTM vs. holding-period returns
When yields fluctuate, so will a bond's rate of return. Unanticipated changes in market rates will result in unanticipated changes in bond returns, and, after the fact, a bond's holding period return can be better or worse than the yield at which it initially sells. Increase in bond yield = reduces its price, which reduces the holding period in return. In this event, the holding period is likely to be less than the initial YTM. holding period return = cost of annual coupon (ie. $80) + [price change value (say, price increases to $1050) - par value (say it was $1000)] / Par value ($1000) HPR = .13 or 13% (pre-tax holding period) *YTM depends only on the bond's coupon, CURRENT price, and par value at maturity and measures the AVERAGE rate of return if the investment bond is held until maturity. In contrast, HPR is the rate of return over a particular investment period and depends on the market price of the bond at the end of the holding period. HPR can, at most, be a forecast.
*Ch. 14 Bond Prices and Yields* Yield to Call Pf 460
YTM assumes the bond will be held until maturity but what happens if it's callable or retired prior to the maturity date? At high interest rates, the risk of call is negligible because the PV of scheduled payments is less than the price; therefore the values of the straight and callable bonds coverage. At lower interest rates, values of the bonds begin to diverge, with the difference reflecting the value of the firm's option to reclaim the callable bond at the call price. At very low rates, the PV of scheduled payments exceeds the call price, so the bond's called and value is simply the call price. This suggests bond market analysts might be more interested in a bond's yield to call rather the YTM, especially if it's likely to be called.
*Ch. 14 Bond Prices and Yields* What will be the relationship among coupon rate, current yield, and YTM for bonds selling at discounts from par? Illustrate using the 8% (semianual payment) coupon bond, assuming it's selling at a yield to maturity of 10%
YTM exceeds current yield, which exceeds the coupon rate. Take as an example the 8% coupon bond with a yield to maturity of 10% per year (5% per half year). Its price is $810.71, and therefore its current yield is 80/810.71 = .0987 or 9.87%, which is higher than the coupon rate but lower than the YTM.
