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Part1 Bond Valuation What is a bond? –Different issuers –Different maturities –Different styles –Bond rating –Terminology Bond pricing –Zero coupon bond –Ordinary coupon bond –Examples –Converse question Some concepts –Yield to maturity –Term structure Future spot rate Forward rate

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What is a bond? 1.Bonds are loans that investors make to corporations and governments. The borrowers get the cash they need while the lenders earn interest. (The Wall Street Journal – Guide to understanding money & investing) a.Different issuers – Government (Treasury securities: T-bills, T-notes, T-bonds, Foreign-targeted T-notes), states, cities, counties, and towns (Municipal bonds), corporations (corporate bonds)

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What is a bond? b.Different maturities – short term (less than 1 years, e.g. T-bills) vs. long term (T-notes, T- bonds, etc) c.Different styles – Coupon bonds (most of corporate bonds) vs. zero-coupon bonds (e.g. STRIPS – Separate Trading of Registered Interest and Principal of Securities) → Why are STRIPS popular especially for an agent who can defer tax?

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What is a bond? d. Bond rating – Hard to evaluate the default risk in a bond 1.Agencies: S&P (Standard and Poor’s) and Moody’s 2.Investment grade: At least BBB (S&P) or Baa(Moody’s) 3.Junk bonds: At most CC (S&P) or Ca (Moody’s)

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What is a bond? Moody’s Long-Term Rating Definitions - Example Aaa Obligations rated Aaa are judged to be of the highest quality, with minimal credit risk. Aa Obligations rated Aa are judged to be of high quality and are subject to very low credit risk. A Obligations rated A are considered upper-medium grade and are subject to low credit risk. Baa Obligations rated Baa are subject to moderate credit risk. They are considered medium-grade and as such may possess certain speculative characteristics.

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What is a bond? Ba Obligations rated Ba are judged to have speculative elements and are subject to substantial credit risk. B Obligations rated B are considered speculative and are subject to high credit risk. Caa Obligations rated Caa are judged to be of poor standing and are subject to very high credit risk. Ca Obligations rated Ca are highly speculative and are likely in, or very near, default, with some prospect of recovery of principal and interest. C Obligations rated C are the lowest rated class of bonds and are typically in default, with little prospect for recovery of principal or interest.

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What is a bond? e. Terminology 1.Par value (face value): Dollar amount of the bond at the time it is issued. It appears several times on the face of bond. Usually, multiplies of $1, Interest rate (coupon rate): Percentage of par value that is paid to the bond-holder on a regular basis.

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Bond pricing a.Zero coupon bond Price = PV (Face value) = b.Ordinary coupon bond Price = PV (Coupons) + PV (Face value) =

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Bond pricing c.Examples Suppose you now own a T-bond that has a coupon rate of 5%, face value of $1,000, and maturity of 5 years. A comparable investment has rate of return of 6%. Assume annual coupons

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Bond pricing Assume semi-annual coupons Why is there difference of bonds prices between annual coupons and semi-annual coupons? 2 conflicting factors: More immediate payment and higher discount rate. → cannot be determined simply. higher discount rate. → cannot be determined simply.

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Bond pricing d. Converse question Since the market decides the bond price, then we can back out “r”. So, this “r” and the price of a bond have one-to-one relationship. Example: Suppose a bond with face value of $1,000, coupon rate of 11.5%, and maturity of 5 years is being traded at $ Then, what is “r” for this bond? = 115/(1+r) + 115/(1+r) 2 + … + 115/(1+r) /(1+r) 5. r = 0.075, but usually through trial and error method.

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Some concept a. Yield to maturity (YTM) – “r” in our notation. IRR for a bond. Financial newspapers report this number. IRR: Internal rate of return

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Some concept Example: WSJ 5/15/02 Maturity ASK Maturity ASK RATE MO/YR BID ASKED CHG YLD May :6 107:

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Some concept “6.500” means coupon rate (%) “May 05” means maturity “107:6” is bid price and it is not $107.6, but (107 and 6/32) % of FV (usually, $1,000) → $1, “107:7” is ask price → (107 and 7/32) % of FV, $1, Notice: Spread is really small, 1/32 (about 0.03) % of FV, Spread of stock price – around 1-2%

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Some concept “6”means that the bond price is increased by $6/32 from the previous day. “3.90” is YTM: This is backed out by using = 65/(1+r) + 65/(1+r) /(1+r) /(1+r) 3 → r ≈ → r ≈ Problems of YTM – The assumption of the same discount rate from t=1 to T. It is very unusual. → Think the job of Mr. Greenspan. → Think the job of Mr. Greenspan.

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Some concept b. Change of interest rate and bond price In our example (price is $ ), if interest rate (not coupon rate) is increased into 15%, then the bond price will be decreased. PV at 15% = 65/(1.15) + 65/(1.15) /(1.15) /(1.15) 3 = = So, price is decreased with higher interest rate.

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Some concept c. Rate of return What is the difference between rate of return and YTM for a bond? From the above example, after 1 year, the bond price is Then, what would be rate of return and YTM? First, rate of return is ($65 + $1, $1,072.19)/$1, = 6.52%

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Some concept For YTM, (PV at year 1) = 65/(1+r) + 65/(1+r) /(1+r) 2 Then, YTM = 2.5% What is the relationship between YTM and the rate of return? → If interest rates do not change, the bond price changes with time so that the total return on the bond is equal to the YTM.

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Some concept d. Spot rate and forward rate 1 Year r 1 = 0 f 1 r 2 = 0 r 2 represented by 1 year rate 1 f 2 at t=0

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Some concept i. Future spot rate The actual interest rate in the future, e.g. the actual interest rate from year 1 to year 2 at year 1. YTM on zero coupon bond. YTM on zero coupon bond. ii. Forward rate Since a future spot rate is unknown now, we have to infer it from various bonds prices. In general, (1 + t-1 f t )(1 + 1 r t-1 ) (t-1) = (1 + 1 r t ) t

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Some concept Example for spot rate and forward rate: Consider following 4 bonds with face value of $100 and annual coupons. MaturityCouponPriceYTM

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Some concept What is 2 year spot rate (r2)? = 6.5/(1.06) + 6.5/(1+r 2 ) /(1+r 2 ) /(1+r 2 ) 2 So, r2 = 6.2%

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Some concept Then, r 3 ? = 6.0/(1.06) + 6.0/( ) /(1+ r 3 ) 3 So, r 3 = 6.4%

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Some concept What are 1 f 2 ? (1+r 2 ) 2 = (1+r 1 )*(1+ 1 f 2 ) In the above problem, we know that r 2 is 6.2% and r 1 is 6%. So, 1 f 2 is 6.4% And 2 f 3 ? With the same logic, 2 f 3 = 6.8%

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Some concept e. Term structure The relationship between short- and long-term interest rates at a particular time on bonds that are fundamentally similar except maturity. The relationship can be presented graphically as a yield curve. Usually, the yield curve is upward sloping. In our example, r 1 < r 2 < r 3. Advanced points: Term structure can be represented by the relation between spot rate (e.g. r 2 ) and forward rate ( 1 f 2 ). If r 2 > r 1, then 1 f 2 > r 2, and if r 2 r 1, then 1 f 2 > r 2, and if r 2 < r 1, then 1 f 2 < r 2.

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Part2 BOND RISK Default Risk Call Risk Market Risk

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Default Risk (1) Default Risk: Uncertainty that the realized return will deviate from the expected return because the issuer will fail to meet the contractual obligations specified in the indenture. The major concern is failure to meet interest and principal payments.

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Default Risk(2) Most investors do not directly access a bond’s default risk, but instead use the quality ratings provided by Moody’s, S&P, and Fetch to evaluate the degree of risk. –AAA, AA, A, BBB, ……, C, …… Default Rates: –.12% per year for all bonds since WWII. –1980s: 3.27% per year on junk bonds. – : 9% on junk bonds

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Default Risk(3) Empirical studies that looked at the relation between default risk premium (RP) and the state of the economy.

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Default Risk(4) Study looked at the yield curves for different quality bonds. Found that the YC for lower quality bonds tended on average to be negatively sloped. Reason: Greater concern over the repayment of principal on low quality bonds.

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Default Risk(5) Applied the Evans and Archer methodology to bond portfolios grouped in terms of their quality ratings. Found that lower quality bond portfolios had less risk because of their lower correlations

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Call Risk (1) Call Risk: Uncertainty that the realized return will deviate from the expected return because the issuer calls the bond, forcing the investor to reinvest in a market with lower rates. Note: When a bond is called the holder receives the call price (CP). Since the CP usually exceeds the principal, the return the investor receives over the call period is often greater than the initial YTM. The investor, though, usually has to reinvest in a market with lower rates which often causes his return for the investment period to be less than the initial YTM.

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Call Risk (2) Example: Compare the ARR for the call period with the ARR for the investment period for a bond that is called. Buy: – 10-year, 10% annual coupon bond at par ($1000); –callable at 110: CP = $1100. Assume: –HD = 10 years. –Flat YC at 10%. –YC stays at 10% until the end of year 3. –Year 3, the YC shifts down to a flat 8% and the bond is called. –Investor reinvest at 8% for the next 7 years.

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Call Risk (3)

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Call Risk (4) Call Risk Premium Price Compression: Call features put limitations on the price-yield curve. At the rate where the bond could be called (YTM*), the YC flattens, with the price equal to the CP.

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Call Risk (5) Points: Need for a valuation model for callable bonds different from the PV model. –A 10-year callable bond may be more like a 3-year bond. –Binomial Tree Model or option pricing model.

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Market Risk (1) Market Risk: Uncertainty that the realized return will deviate from the expected return because interest rates change. Recall, the return on a bond comes from coupons, the interest earned from reinvesting coupons (interest on interest), and capital gains. A change in rates affects interest on interest and capital gains or losses.

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Market Risk (2) Note: One way to eliminate market risk is to buy a PDB with M = HD.

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Market Risk (3) Example: HD = 3.5 years Buy Noncallable 10-year, 10% annual bond at par (F=$1000) at YTM of 10%. YC flat at 10%. If there is no change in the YC, your ARR for 3.5 years will be 10%:

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Market Risk (4) Example: If the YC shifted up to a flat 12% shortly after you bought the bond and remained there until your HD, then your ARR for 3.5 years would be 8.23%:

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Market Risk (5) Example: If the YC shifted down to a flat 8% shortly after you bought the bond and remained there until your HD, then your ARR for 3.5 years would be 11.95%:

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Market Risk (6) Relation: Rate change have two opposite effects: –Inverse price effect –Direct interest on interest effect Whether the ARR varies directly or inversely to rate changes depends on which effect dominates. In this case (10-year, 10% coupon bond with HD = 3.5 yrs), the inverse price effect dominates. This causes the ARR to vary inversely with rate changes.

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Market Risk (7) With an HD of 3.5 years, a 4 year, 20% bond has an interest on interest effect which dominates the price effect. This bond’s ARR would vary directly with rate changes.

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Market Risk (8) Note: It is possible to buy a bond in which the price and interest on interest effect exactly offset each other. In this case, the ARR will be invariant to changes in rates. For example, a 4-year, 9% coupon bond purchased when the YC is at 10% (Po = ) would yield an HD value at year 3.5 of 1351 and an ARR of 10%, regardless of rates.

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Market Risk (9) Bond Immunization: What is distinctive about a 4-year, 9% coupon bond when the YTM is 10% is that it has a duration (D) equal to 3.5 years -- duration = HD. Duration can be defined as the weighted average of the time periods. Bond immunization is bond strategy of minimizing market risk by buying a bond or a portfolio of bonds with a duration equal to the HD.

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Market Risk (9) Duration of 4-year, 9% coupon Bond:

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Duration (1) Duration is also a measure of a bond’s price sensitivity to interest rate changes. This measure of duration is known as the modified duration; the measure of duration as a weighted average of the time periods is known as Macauley’s duration.

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Duration (2)

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Duration (3) A modified duration measure for a bond which pays coupons each period and its principal at maturity (note: let y = YTM):

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Duration (4) Annualized Duration: Duration is defined in terms of the length of the period between payments. If the CFs are distributed annually, then duration is in years; if CFs are semi-annual, then duration is measured in half years. The convention is to expressed duration as an annual measure. The annualized duration is obtained by dividing duration by the number of payments per year (n):

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Duration (5) Example: The duration in half-years for a 10- year, 9% coupon bond selling at par (F= 100) and paying coupons semiannually is -13 and its annualized duration is -6.5:

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Duration (6) Descriptive Parameter: Measure of a bond’s price sensitivity to interest rate changes -- a measure of a bond’s volatility. Note, duration is consistent with our bond price relation

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Duration (7) Define Strategies: Use duration to define active (speculative) and passive bond strategies. Examples: –Rate-Anticipation Swap: Rates expected to decrease across all maturities, go long in high duration bonds. Rates expected to increase across all maturities, change bond portfolio composition so that it has lower duration bonds. –Bond Immunization Strategy.

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Duration (8) Estimate the percentage change in a bond’s price for a given change in rates: For 10-year, 9% bond, an increase in the annualized yield by 10BP (.09 to.0910) would lead to a.65% decrease in price (the actual is.6476%). A 200BP increase (9% to 11%) would lead to an estimated price decrease of 13%; the actual decrease, though, is only 12%.

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Convexity (1) Duration is a measure of the slope of the price-yield curve at a given point -- first- order derivative. Convexity is a measure of the change in the slope of the price-yield curve -- second- order derivative. Convexity measures how bowed-shaped the price-yield curve is.

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Convexity (2) Property: The greater a bond’s convexity, the greater its capital gains and the smaller its capital losses for given absolute changes in yields.

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Convexity (3) Measure:

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Convexity (4) The convexity measure for a bond which pays coupons each period and its principal at maturity:

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Convexity (5) Example: The convexity in half-years for a 10- year, 9% coupon bond selling at par (F= 100) and paying coupons semiannually is and its annualized convexity is 56.36:

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Convexity (6) Descriptive Parameter: Greater k-gains and smaller k-losses the greater a bond’s convexity. Estimation of : Using Taylor Expansion, a better estimate of price changes to discrete changes in yield than the duration measure can be obtained by combining duration and convexity measures.

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Convexity (7) Taylor Expansion: For 10-year, 9% bond, an increase in the annualized yield by 200 BP (9% to 11%) would lead to an estimated 11.87% decrease in price using Taylor Expansion (the actual is 12%):

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Convexity (8) Note: Using Taylor Expansion the percentage increases in price are not symmetrical with the percentage decreases for given absolute changes in yields.

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Derivation of Duration and Convexity (1) Duration: Take derivative with respect to y:

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Derivation of Duration and Convexity (2)

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Derivation of Duration and Convexity (3)

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Derivation of Duration and Convexity (4) Convexity: Take the derivative of

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