Fixed Income

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

• Par, Premium and Discount If YTM > %COUPON YTM = %COUPON Price Bond sold Price<1000 At discount Price =1000 At par YTM < %COUPON Price>1000 At premium • Bond prices and yield are inversely related • Bond prices and maturity are inversely related • Bond prices and coupon are positively related

The expected yield on the bond may be computed from the market price
P m
Where: 
t
2  
n
1 ( 1
C i

i
2 2 )
t
 ( 1 
i P p
2 ) 2
n
i = the discount rate that will discount the cash flows to equal the current market price of the bond

Yield Measure Purpose Nominal Yield Measures the coupon rate Current yield Measures current income rate Promised yield to maturity Promised yield to call Realized (horizon) yield Measures expected rate of return for bond held to maturity Measures expected rate of return for bond held to first call date 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.

Similar to dividend yield for stocks Important to income oriented investors CY = C i /P m where: CY = the current yield on a bond C i P m = the annual coupon payment of bond i = the current market price of the bond

• 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) FV=1000) 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,

• 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.

Current yield = Annual coupon pmt Current price Capital gains yield = 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%.
Current yield = $90 $887 = 10.15%.
YTM = Current yield + Capital gains yield.
Cap gains yield = YTM - Current yield = 10.91% - 10.15% = 0.76%.

• “Nominal” Rate =Risk-free rate =Real rate + Inflation  Production opportunities  Time preferences for consumption  Risk  Expected inflation + Risk Premium + Risk Premium DRP
= Default risk premium.
LP = Liquidity premium.
MRP = Maturity risk premium.
Treasury: IP, MRP Corporate: IP, DRP, MRP, LP

• 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)

Hypothetical Treasury Yield Curve Interest Rate (%) 15 Maturity risk premium 1 yr 8.0% 10 yr 11.4% 20 yr 12.65% 10 Inflation premium 5 0 1 10 Real risk-free rate 20 Years to Maturity

Actual Treasury Yield Curve Interest Rate (%) 15 1 yr 5 yr 10 yr 30 yr 6.3% 6.7% 6.5% 6.2% 10 Yield Curve (May 2000) 5 0 10 20 30 Years to Maturity

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 (%) 10 5 5.2% 5.9% BB-Rated AAA-Rated 6.0% Treasury yield curve 0 0 1 5 10 15 Years to 20 maturity

U.S. Yield Curve Inverts Before Last Five Recessions (5-year Treasury bond - 3-month Treasury bill) 8 Yield Curve 6 4 2 % Real annual GDP growth 0 Yield curve -2 Recession -4 -6 Recession Correct 2 Recessions Correct Recession Correct Recession Correct
?
Data though 12/20/00 M ar -69 M ar -71 M ar -73 M ar -75 M ar -77 M ar -79 M ar -81 M ar -83 M ar -85 M ar -87 M ar -89 M ar -91 M ar -93 M ar -95 M ar -97 M ar -99 M ar -01

• 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.

• 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?

• 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.

If
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(
C t
price ) 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
C t
= interest or principal payment that occurs in period t i = yield to maturity on the bond

• 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

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

• 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 P
 100  
D
mod  
i
 P = change in price for the bond P = beginning price for the bond
D
mod = 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?

• 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

The convexity is the measure of the curvature and is the second derivative of price with resect to yield (d 2
P/di
2 ) divided by price Convexity is the percentage change in dP/di for a given change in yield 2
d P
Convexity 
di
2
P
• Inverse relationship between coupon and convexity • Direct relationship between maturity and convexity • Inverse relationship between yield and convexity

• 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 
P P
 
D
mod  
i
 1 2 
C
 
i
2

• 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 PS
2 2
PS
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

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.

• 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 prices : lower coupons and play on an increase in bond • Q: What is the duration of a portfolio of bonds?
• A: The weighted average duration of each bond in a portfolio—I.e.,
D

i i
 
n
 1
W i

D i

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

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: 1% What happens if interest rates increase or decrease suddenly by

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
Increase of 1% or R=7%
=$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
Strategy 1: FV=PMT x FVIFA + 1000=60 x 7.153 +1000=$1,429.2
Strategy 2: FV=PMT x FVIFA + PMT x PVIFA +1000/FVIF =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!

R=6% R=5% Strategy 1 1,418.5
1408 Strategy 2 1,418.5
1417.6
R=7% %change (-1%) 1429.2
-0.07% 1419.9
0% %change (+1%) 0.08% 0% 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…)

• 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!