Chapter 9

Debt InstrumentsQuantitative Issues

Learning Objectives

A. Bond ValuationB. Yield MeasuresC. DurationD. Managing Bond PortfoliosE. Term StructureF. Factors affecting Prices/Yields

Five Bond Pricing Theorems

A. Bond prices move inversely to changes in interest ratesB. The longer the maturity of a bond, the more price

sensitive the bondC. The price sensitivity of bonds to changes in interest rates

increases as maturity increases, but at a decreasing rateD. Bonds with lower coupons are more price sensitiveE. Yield decreases have a greater impact on bond prices

than similar yield increases

TI BA II Plus (Pro) - BOND PRICES

A. The price of a bond (PB) is a combination of a present value of an annuity (the present value of the coupons to be received) and the present value of the face (par) value of the bond.

B. PB = $Coupon * PVIFA + $Face * PVIF

N ..., 3, 2, 1, nfor )k(1

CFP

N

1nn

b

nB

COMPUTING BOND PRICES using the BA II PlusBA II Plus

A. Example: Suppose we have a bond paying a 12% coupon rate ($120), paid semi-annually. The bond matures in 20 years and has a face value of $1,000. If the current market rate (YTM) is 9%, how much should this bond sell for (price)?

B. In this type of problem we will use all five TVM keys;[ N ] [ I/Y ] [ PV ] [ PMT ] [ FV ]

COMPUTING BOND PRICES using the BA II PlusBA II Plus

C. The bond pays coupons (interest) twice a year (semi-annual): We set the periods per year (P/Y) and (C/Y) to 2.

D. The 12% coupon ratecoupon rate ( $120 per year) is paid in two [PMT=] $60 installments.

E. The bond will have a maturity (face, par) valuematurity (face, par) value [FV] of $1000.00.

F. The current market ratecurrent market rate is [I/Y] 9% (the required YTMYTM for bonds in this risk class).

COMPUTING BOND PRICES using the BA II PlusBA II Plus

G. BA II PLUS Solution1. ENTER 20 [2nd] [N], [N] N = 40.002. ENTER 9 [I/Y] I/Y = 9.003. ENTER 60 [PMT] PMT = 60.004. ENTER 1000 [FV] FV = 1,000.005. PRESS [CPT] [PV] PV = -1,276.02We would have to pay $1,276.02 to buy this bond

today.

COMPUTING BOND PRICES using the BA II PlusBA II Plus

H. What if the current YTM is 8.5%?1. Enter 8.5, press [I/Y]2. Press [CPT] [PV]: PV = - 1,333.853. Clearly, a lower YTM results in a higher price.

I. What if the current YTM is 9.5%?1. Enter 9.5, press [I/Y]2. Press [CPT] [PV]: PV = - 1,222.043. Clearly, a lower YTM results in a higher price.

Bond prices move inversely to changes in interest rates

Yield Measures

A. Coupon Rate = Annual Coupon ÷ $1000B. Current Yield = Annual Coupon ÷ PriceC. Yield To Maturity = actual rate earned on bond if

held to maturity (same concept as IRR)1. To compute YTM: CLR TVM, then

a. Set P/Y value (2 or 4 are typical)b. Enter N, [-]Price (PV), interest PMT, FV ($1000 typical)c. Compute I/Y

2. Text Example (p9.9): P/Y = 2, N = 12, PV = -950, PMT = 20, FV = 1000. CPT I/Y = 4.9741

Yield to Maturity

A. Yield to maturity is the rate at which a bond’s cash flows are discounted

B. Changes as market interest rates changeC. Yield to maturity and coupon rate (CR)

1. If P(b) < F, then YTM > CR (discount bond)2. If P(b) = F, then YTM = CR 3. If P(b) > F, then YTM < CR (premium bond)

Actual Return & Yield to Maturity

A. If you buy a bond and hold it until maturity, will your actual return equal the bond’s yield to maturity?

1. No, unless you can reinvest the coupons at the yield to maturity rate

2. If reinvestment rate is less than YTM, actual return will be less than YTM

3. If reinvestment rate is greater than YTM, actual return will be greater than YTM

Yield Relationships (Fig 9-2)

Computing Yield to [First] Call

A. Bonds may be issued “Callable” during periods of high interest rates. The “callable” feature allows the issuer to recall the bonds and reissue the debt at lower rates.

1. Recalls typically require a premium to be paid.2. Example: Suppose you are considering buying a 10%

coupon bond (paid quarterly) recallable in 3 years at 107.5 (a premium of $75 in addition to any accrued interest). The FV = 1075 and the N value would be 3 * P/Y.

3. The bond currently sells for $1464.07. What is the YTFC (yield to first call)? YTM = 6%

Computing Yield to [First] Call

3. N = 3 * 4 = 124. PV = - 1464.075. PMT = 256. FV = 10757. [CPT] [I/Y] = 2.31338. This is obviously not a good deal.

1. Right off the bat – you’re taking a $389.07 capital loss.2. The $75 early call premium is insufficient to cover the

expected capital loss.3. The YTFC (2.3133%) is less than the current YTM (6%).4. You’re better off buying a new issue bond.

Assessing Interest Rate Risk

A. Bond Price Volatility1. Maturity effect: longer a bond’s term to maturity,

greater percentage change in price for given change in interest rates

2. Coupon effect: lower a bond’s coupon rate, greater percentage change in price for given change in interest rates

3. Yield-to-maturity effect: For given change in interest rates, bonds with lower YTM have greater percentage price changes than bonds with higher YTM – all other things equal.

Assessing Interest Rate Risk

A. A bond’s interest rate risk is defined as the sensitivity of price to a change in YTM.

B. Which bond is more price sensitive?1. Bond A: 10% coupon, 10 year maturity2. Bond B: 5% coupon, 5 year maturity

C. We can’t say without some sort of summary measure of interest rate risk

D. Such a measure is called duration

Duration

A. Duration measures the amount of time before the investor receives the “average” dollar from a bond

B. Duration is a function of a bond’s coupon rate, time to maturity and yield to maturity

C. Duration:1. Increases as the coupon rate decreases2. Increases as the time to maturity increases3. Increases as yield to maturity decreases

D. The longer the duration of a bond, the more sensitive its price to a given change in interest rates.

Duration

E. Formulas for Duration

T

tt

t

T

tt

t

YCF

YtCF

D

1

1

)1(

)1(*

T

tt

t

Y

CFP

10 )1(

Note: Exponents in Eq. 9-5 should be t not T

Denominator above is also equal to the current price (DCF)

Duration

F. Uses of Duration1. Price volatility index

a. Larger duration statistic, more volatile price of bond

2. Immunizationa. Interest rate risk minimized on bond portfolio by maintaining portfolio

with duration equal to investor’s planning horizon

G. Principal Characteristics1. Duration of zero-coupon bond equal to term to maturity2. Duration of coupon bond always less than term to maturity3. Inverse relationship between coupon rate and duration4. Direct relationship between maturity and duration

Duration

A. Modified Duration1. Adjusted measure of duration used to estimate a

bond’s interest rate sensitivity 2. D* = D (1 + YTM)

% Chg in price of bond = –D x % Chg in YTM

% Chg in price of bond = – D* x [Chg in YTM]

Interest Rate Risk

A. Price Risk1. Risk of existing bond’s price changing in response to

unknown future interest rate changesa. If rates increase, bond’s price decreasesb. If rates decrease, bond’s price increases

B. Reinvestment Rate Risk1. Risk associated with reinvesting coupon payments at

unknown future interest ratesa. If rates increase, coupons are reinvested at higher rates than

previously expectedb. If rates decrease, coupons are reinvested at lower rates than

previously expected

Bond Portfolio Immunization

A. Strategies a Function of Needs1. If a single time horizon goal, purchasing zero-coupon

bond whose maturity corresponds with planning horizon

2. If multiple goals, purchasing series of zero-coupon bonds whose maturities correspond with multiple planning horizons

3. Assembling and managing bond portfolio whose duration is kept equal to planning horizon

Note: this strategy involves regular adjustment of portfolio because duration of portfolio will change at SLOWER rate than will time itself.

Managing Bond Portfolios

A. Bond Swaps1. Technique for managing bond portfolio by selling

some bonds and buying others2. Possible benefits achieved:

a. tax treatmentb. yieldsc. maturity structured. trading profits

Managing Bond Portfolios

A. Types of Swaps1. Substitution swap (tax loss issues)2. Inter-market spread swap (transports vs. utilities)3. Pure-yield pick-up swap4. Rate anticipation swap: Yield expectations

B. Portfolio Structure1. Bullet Portfolio (one maturity date)2. Bond ladders (equally distributed dollar allocations

over time)3. Barbells (varying maturities)

a. Allocations to shortest-term and longest-term holdings

Term Structure of Interest Rates

A. Typical: rising to right (see Fig 9-5, pp 9.35)1. Normal2. Inverted3. Flat

B. Theories of Term Structure1. Expectations (spot vs. forward rates – see pp 9.38ff)2. Liquidity Preference (premiums for longer terms)3. Market Segmentation (effects of supply & demand)4. Preferred Habitat (maturity preference)

Factors Affecting Bond Yields

A. General credit conditions: Credit conditions affect all yields to one degree or another.

B. Default risk: Riskier issues require higher promised yields.

C. Coupon effect: Low-coupon issues offer yields that are partially taxed as capital gains.

D. Marketability: Actively traded issues tend to be worth more than similar issues less actively traded.

E. Call protection: Protection from early call tends to enhance bond’s value.

F. Sinking Fund Requirements: reduce probability of default