CFA REVIEW Fixed Income

Bond Portfolio Management Yields and Term-structures Bond risk Duration Convexity Bond Portfolio Strategies Passive strategies Active strategies Protective strategies

Bond Properties Par, Premium and Discount Bond prices and yield are inversely related Bond prices and maturity are inversely related Bond prices and coupon are positively related

The Yield Model The expected yield on the bond may be computed from the market price Where: i = the discount rate that will discount the cash flows to equal the current market price of the bond

Computing Bond Yields Yield Measure Purpose Nominal Yield Measures the coupon rate Current yield Measures current income rate Promised yield to maturity Measures expected rate of return for bond held to maturity Promised yield to call Measures expected rate of return for bond held to first call date Realized (horizon) yield Measures expected rate of return for a bond likely to be sold prior to maturity. It considers specified reinvestment assumptions and an estimated sales price. It can also measure the actual rate of return on a bond during some past period of time.

Current Yield Similar to dividend yield for stocks Important to income oriented investors CY = Ci/Pm where: CY = the current yield on a bond Ci = the annual coupon payment of bond i Pm = the current market price of the bond

Promised and Realized Yield to Maturity PYTM Assumes that all the bond’s cash flow is reinvested at the computed yield to maturity (same as IRR) RYTM assumes that all the bond’s cash flow is reinvested at the computed yield to maturity Example: A bond yields 5%. It has 20 years to maturity and pays 20% coupon annually. What is the realized yield to maturity over a 6 years horizon and a reinvestment rate of 3%? RYTM=(FV/PV)1/6-1 Where FV= Future values of coupons re-invested at 3% over 6 years (i=3%, pmt=200, n=6) + present value of bond with 14 years left to maturity (i=5%, pmt=200, n=14, FV=1000) PV is the present value of the bond (i=5%, pmt=200, n=20, FV=1000)

Promised Yield to Call Callable bond pay the face value (1000) + one periodic coupon and expire prior to maturity Example: A 10-year, 10% semiannual coupon,$1,000 par value bond is selling for$1,135.90 with an 8% yield to maturity.It can be called after 5 years at $1,050. What’s the bond’s nominal yield to call (YTC)? Note: In general, if a bond sells at a premium, then coupon > kd, so a call is likely. Then, expect to earn: YTC on premium bonds.; YTM on par & discount bonds.

How do you make money on a bond? Annual coupon pmt Current price Current yield = Capital gains yield = = YTM = + Change in price Beginning price Exp total return Exp Curr yld Exp cap gains yld

Find current yield and capital gains yield for a 9%, 10-year bond when the bond sells for $887 and YTM = 10.91%. $90 $887 Current yield = = 10.15%. YTM = Current yield + Capital gains yield. Cap gains yield = YTM - Current yield = 10.91% - 10.15% = 0.76%.

What four factors affect the cost of money? “Nominal” Rate =Risk-free rate + Risk Premium =Real rate + Inflation + Risk Premium Production opportunities Time preferences for consumption Risk Expected inflation k = k* + IP + DRP + LP + MRP. DRP = Default risk premium. LP = Liquidity premium. MRP = Maturity risk premium. Treasury: IP, MRP Corporate: IP, DRP, MRP, LP

Term Structure of Interest Rates Relationship between term to maturity and yield to maturity for a sample of bonds at a fixed point of time. Referred to as the “yield curve.” Issues differ only in their maturities--Treasury instruments 3 shapes (Normal,Flat,Inverted) 3 underlying theories, relating to the different supply and demand pressures in different maturity sectors: Expectation (expected to earn on successive investments in ST bonds during the term to maturity of a LT bond) Liquidity (investors prefer the liquidity of ST bonds but will buy LT bonds if the yields are higher) Market segmentation (yields curve reflects the investment policies of financial institutions who have different maturity preferences) * The term structure of interest rates is the relationship between term to maturity and yield to maturity for a sample of bonds at a fixed point of time. * It is commonly referred to as the “yield curve”, which is the graphical representation of the term structure. It is a static function, it does not indicate what future rates will be. * Ideally, the issues used to construct a yield curve differ only with respect to maturity, i.e., all other factors, such as call, taxability, and default risk, should be held fixed. * Therefore, treasury instruments are the most suitable.

Hypothetical Treasury Yield Curve Interest Rate (%) 1 yr 8.0% 10 yr 11.4% 20 yr 12.65% 15 Maturity risk premium 10 Inflation premium 5 Real risk-free rate Years to Maturity 1 10 20

Actual Treasury Yield Curve Interest Rate (%) 1 yr 6.3% 5 yr 6.7% 10 yr 6.5% 30 yr 6.2% 15 10 Yield Curve (May 2000) 5 Years to Maturity 10 20 30

Corporate yield curves are higher than for Treasury bond Corporate yield curves are higher than for Treasury bond. However, corporate yield curves are not necessarily parallel to the Treasury curve. The spread between a corporate yield curve and the Treasury curve widens as the corporate bond rating decreases. 15 Interest Rate (%) BB-Rated 10 AAA-Rated Treasury yield curve 6.0% 5 5.9% 5.2% Years to maturity 1 5 10 15 20

U.S. Yield Curve Inverts Before Last Five Recessions (5-year Treasury bond - 3-month Treasury bill) % GDP Growth/ Yield Curve % Real annual GDP growth Yield curve ? Recession Correct Recession Correct Recession Correct Recession Correct 2 Recessions Correct Data though 12/20/00

Expected yields

Bond Risks Interest rate risk dichotomy: Price risk or price volatility Reinvestment risk or “ending wealth” volatility If interest rates are expected to increase; bond price will decrease and ending wealth will increase. interest rates are expected to decrease; bond price will increase and ending wealth will decrease.

Bond risk… As Coupon is greater, Price sensitivity to yield decreases. As Maturity gets greater, Price sensitivity to yield increases. A bond with high yield is less sensitive to a change in interests than a bond with low yield. Bond risk = Price risk and reinvestment risk Q: with an expected change interest rates, which bond would you pick?

Some Trading Strategies… If market rates are expected to decline, bond prices will rise you want bonds with maximum price volatility. Maximum price increase (capital gain) results from long term, low coupon bonds, low yield If market rates are expected to rise, bond prices will fall you want bonds with minimum price volatility. Invest in short term, high coupon bonds to minimize price volatility and capital loss, high yield. Some trading strategies * If an investor expects rates to decline, then bond prices are expected to rise. The maximum price increase results from long term and from low coupons. For practical purposes a 30 year zero coupon bond will give the investor the maximum price gain. * If prices are expected to rise, the investor should do just the opposite, invest in short term, high coupon bonds. This will minimize the price volatility and expected capital loss.

Evidence of reinvestment risk (8% coupon, 25 years, 8% yield, semi-annual). How does the ending wealth change if interest rates increase by 1%ANS: ≈+15%) * This figure illustrates the impact of interest on interest for an 8%, 25 year bond bought at par to yield 8%. If you invested $1000 today at 8% for 25 years, you would have $7100 at the end of the period. $1000 in principal $2000 in coupon payments $4100 in interest on interest If you didn’t reinvest, you would have $3000 at the end of 25 years.

Approximating Price Risk - Duration * Bond price volatility is directly related to term to maturity and inversely related to coupon. Duration is a measure of volatility that considers both of these opposing factors. The Macaulay measure of duration is the weighted average time to full recovery of principal and interest payments. Convexity

The Duration Measure Duration: the weighted average time to full recovery of principal and interest payments. Developed by Frederick R. Macaulay, 1938 Where: t = time period in which the coupon or principal payment occurs Ct = interest or principal payment that occurs in period t i = yield to maturity on the bond

Characteristics of Duration Duration of a bond with coupons is always less than its term to maturity because duration gives weight to these interim payments A zero-coupon bond’s duration equals its maturity There is an inverse relation between duration and coupon There is a positive relation between term to maturity and duration, but duration increases at a decreasing rate with maturity There is an inverse relation between YTM and duration Sinking funds and call provisions can have a dramatic effect on a bond’s duration

Modified Duration and Bond Price Volatility An adjusted measure of duration can be used to approximate the price volatility of a bond Where: m = number of payments a year YTM = nominal YTM

Duration and Bond Price Volatility Bond price movements will vary proportionally with modified duration for small changes in yields An estimate of the percentage change in bond prices equals the change in yield time modified duration Where: P = change in price for the bond P = beginning price for the bond Dmod = the modified duration of the bond i = yield change in basis points divided by 100 Example: Given a bond that pays semi-annual coupons with a duration of 6 years and a yield of 8%, what will the percentage change in price be if market rates are expected to rise by 50 basis points?

Trading Strategies Using Duration Longest-duration security provides the maximum price variation If you expect a decline in interest rates, increase the average duration of your bond portfolio to experience maximum price volatility If you expect an increase in interest rates, reduce the average duration to minimize your price decline Note that the duration of your portfolio is the market-value-weighted average of the duration of the individual bonds in the portfolio

Convexity The convexity is the measure of the curvature and is the second derivative of price with resect to yield (d2P/di2) divided by price Convexity is the percentage change in dP/di for a given change in yield Inverse relationship between coupon and convexity Direct relationship between maturity and convexity Inverse relationship between yield and convexity

Modified Duration-Convexity Effects Changes in a bond’s price resulting from a change in yield are due to: Bond’s modified duration Bond’s convexity Relative effect of these two factors depends on the characteristics of the bond (its convexity) and the size of the yield change Convexity is desirable

Effective Duration Measure of the interest rate sensitivity of an asset Use a pricing model to estimate the market prices surrounding a change in interest rates Effective Duration Effective Convexity P- = the estimated price after a downward shift in interest rates P+ = the estimated price after a upward shift in interest rates P = the current price S = the assumed shift in the term structure

Passive Bond Portfolio Strategies Buy-and-Hold Strategy Investor selection based on quality, coupon and maturity Match maturity with investment horizon Modified buy and hold Indexing Strategy Money managers can’t beat the market“If you can’t beat them, join them.” Difficulties: Tracking error - difference between the portfolio’s return and the return for the index. You must know characteristics and composition of the various indexesIndexes change over time. The most common and easiest to implement bond management strategy is the buy-and-hold strategy. 8 The investor looks to find issues with the desired attributes of 8 quality 8 coupon, 8 term to maturity, etc. 8 Investors do not want to actively trade. Instead they attempt to set the maturity (or duration) of the bonds in the portfolio equal to their time horizon. 8 Selection is important, in order to identify the right issues.

Active Bond Strategies Active management strategies » Interest Rate Anticipation (Valuation Analysis, Credit Analysis, Yield Spread Analysis, and Bond Swaps) Riskiest If i is expected to increase, preserve capital If i is expected to decrease, make capital gains Objectives are achieved by adjusting the portfolio’s duration (maturity). Shorten duration if rates are expected to ­ Play the Reinvestment advantage card and get Cash flow ASAP (liquidity) Lengthen duration if rates are expected to ¯ Play the Interest rate card : lower coupons and play on an increase in bond prices Q: What is the duration of a portfolio of bonds? A: The weighted average duration of each bond in a portfolio—I.e., Five active management strategies are available. These are: 8 Interest Rate Anticipation 8 Valuation Analysis 8 Credit Analysis 8 Yield Spread Analysis 8 Bond Swaps

Matched Funding Techniques: Dedicated Portfolios What are they? Bond portfolio management technique used to service a specific set of liabilities Pure-Cash Matched Dedicated Portfolio Cash flows from all sources exactly match up in timing and size with the liability schedule. Can be achieved by buying a series of zero coupon Treasury securities. Total passive strategy Dedication with Reinvestment Cash flows don’t exactly match the liability schedule, also cash flows received earlier are reinvested at a relatively low interest rate. Advantages: (1) Allows for wider set of bonds to be considered; (2) Lower net cost of the portfolio; (3) Safety equivalent to with pure cash-matching. Potential problem: Early redemption 8 A cash-matched dedicated portfolio is the most conservative matched-funding technique. The objective of a pure-cash matched dedicated portfolio is to establish a bond portfolio whose cash flows from coupons, sinking funds, principal, and all other sources exactly match up in timing and size with a specific liability schedule. 8 This can be achieved one way by buying a number of zero coupon Treasury securities. This is referred to as a total passive strategy because it is designed so that any prior receipts would not have to be reinvested.

Matched Funding Techniques: Immunization Strategies Immunization: Attempt to generate a specified rate of return regardless of what happens to market rates during an investment horizon. Immunization is a process intended to eliminate interest risk; it is achieved if the ending wealth of a bond portfolio is the same regardless of whether interest rates change Example: Assume a 6 year strategic asset allocation horizon and market rates on 6% coupon bonds is 6%. Strategy one: Maturity (cash) Matching Strategy A manager has a portfolio of bonds with an average maturity of 6 years. The average coupon rate of the portfolio is 6%. Strategy two: Duration Matching strategy=portfolio immunization A manager has a portfolio of bonds with an average maturity of 7 years. The average coupon rate of the portfolio is 6%. The average duration is about 6 years. Q: What happens if interest rates increase or decrease suddenly by 1% 8 Classical immunization is the simplest method of simultaneously dealing with the two types of interest rate risk. 8 Changes in interest rates have two effects on the ending wealth position of portfolio, which work in opposite directions. 81. An increase in rates lead sto a drop in price but more interest on interest. 82. A decrease in rates, however, leads to a rise in the price but a drop in interest on interest. 8 The process of eliminating these effects is called immunization.

Interest rates unchanged or R=6% Example…continued Interest rates unchanged or R=6% Strategy 1: FV=PMT x FVIFA + 1000=60 x 6.975 +1000=$1,418.5 Strategy 2: FV=PMT x FVIFA + PMT x PVIFA +1000/FVIF= =60 x 6.975 +60 x .943 + 1000/1.06=$1,418.5 Decrease of 1% or R=5% Strategy 1: FV=PMT x FVIFA + 1000=60 x 6.802 +1000 =$1,408 Strategy 2: FV=PMT x FVIFA + PMT x PVIFA +1000/FVIF =60 x 6.802 +60 x .952 + 1000/1.05=$1,417.6 Increase of 1% or R=7% Strategy 1: FV=PMT x FVIFA + 1000=60 x 7.153 +1000=$1,429.2 =60 x 7.153 +60 x .935 + 1000/1.07=$1,419.9 For strategy 2: At t=6 years, bonds have 1 year left of life!

Conclusion: Strategy 1 Strategy 2 R=6% 1,418.5 R=5% 1408 1417.6 R=7% 1429.2 1419.9 %change (-1%) -0.07% 0% %change (+1%) 0.08% Applications: immunize the bond portion of your strategic allocation Immunize a future cash outflows (pension funds, insurance companies) Not as easy as it sounds (rebalancing, duration drift, unavailability…)

Questions You immunize a 4-year investment by purchasing a coupon bond with a duration of 4 years. If interest rates do not change, is your bond still immunized one year after? What if you purchased a 4-year zero coupon bond?

Matched Funding Techniques: Horizon Matching and contingent immunization Horizon matching combines cash matching and immunization to Provide protection against unequal interest rate changes » Short term end is set up as a cash matching portfolio » Longer term end is duration immunized » Roll out occurs when the time horizon is pushed out one year further into the long term time horizon Contingent immunization allows for active portfolio management while assuring a minimal return by creating a Cushion spread ( difference between market rates and the minimum that investors are willing to accept.)Immunize a specific return; play with the cushion! 8 Horizon matching combines cash matching dedication and immunization. 8 A time horizon is chosen to differentiate the two strategies. The exact choice will be a tradeoff between the safety of the cash match and the flexibility and savings of the duration strategy. 8 The short term end is set up as a cash matching portfolio. 8 The longer term end is duration immunized. 8 This provides protection against nonparallel shifts in the yield curve. Most of these types of problems occur in the short term end of the yield curve. The long term end of the yield curve does in fact show parallel shifts. 8 A roll out occurs when the time horizon is pushed out one year further into the long term time horizon every year. Therefore, the cash matching strategy will always be applied to roughly the same time frame.