Chapter 9 Debt Instruments Quantitative Issues. Pricing a Bond where P 0 = price of bond today T =...
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Transcript of Chapter 9 Debt Instruments Quantitative Issues. Pricing a Bond where P 0 = price of bond today T =...
![Page 1: Chapter 9 Debt Instruments Quantitative Issues. Pricing a Bond where P 0 = price of bond today T = maturity of the bond Y = appropriate discount rate.](https://reader031.fdocuments.us/reader031/viewer/2022013101/56649e7e5503460f94b8212e/html5/thumbnails/1.jpg)
Chapter 9
Debt Instruments
Quantitative Issues
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Pricing a Bond
where P0 = price of bond today
T = maturity of the bond
Y = appropriate discount rate
PAR = par or face value of the bond
T
T
1tt0
Y1
PAR
Y1
coupon P
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Bond Prices with Semiannual Payments
• Divide coupon payment by two • Multiply maturity of bond by two.• Divide discount rate by two
T2
2T
1tt0
2
Y 1
PAR
2
Y 1
2
coupon
P
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Bond Yields & Rates
• Coupon rate (nominal yield)• Current yield (coupon / price)• Yield to maturity (YTM = IRR)• Realized compound yield to maturity (RCYTM)• Yield to First (earliest) Call• Realized return
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ABC Example
• Coupon: $40 per year
• Par Value: $1,000
• Maturity: 6 years
• Callable: in 3 years @ $1040
• Price: $950
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Coupon Rate
• Stated dollar return of fixed-income investment
• Equals annual interest payments divided by par value
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Current Yield
• Bond’s coupon rate divided by current market price
OR
• Stock’s indicated dividend rate divided by per-share price
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Yield to Maturity• Measure of bond yield that takes into account capital gain
or loss, as well as coupon payments
• Discount rate that would make present value of bond’s cash flows (payments plus face value at maturity) equal purchase price of bond
where C = the coupon payment
T
1tTt0
Y 1
PAR
Y 1
C P
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Yield Relationships
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Yield to Call
where Tc = time to earliest call
Yc = yield to first call
• Almost identical to YTM, except– Call price replaces par value
– Time to call replaces term to maturity
cT
1tT
ct
c
0Y 1
price call
Y 1
coupon P
3c
3c
2cc Y 1
1040
Y 1
40
Y 1
40
Y 1
40 950
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Realized Rate (Yield)
where TH = holding period
YH = realized rate of return
• Ex post rate of return or yield from investment (internal rate of return)
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Bond Price Volatility
• Bond prices and interest rates inversely related• Maturity effect: longer a bond’s term to maturity, greater
percentage change in price for given change in interest rates• Coupon effect: lower a bond’s coupon rate, greater
percentage change in price for given change in interest rates• Yield-to-maturity effect: For given change in interest rates,
bonds with lower YTMs have greater percentage price changes than bonds with higher YTMs – all other things equal
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Which Bond’s Price Is Most Volatile?
• Bond X: 25 years to maturity, 10% coupon rate, and a 6% YTM
• Bond Y: 10 years to maturity, 2% coupon rate, and a 6% YTM
• Bond Z: 17.5 years to maturity, 6% coupon rate, and a 4% YTM
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Answer
• Based on maturity effect, it would be X
• Based on coupon effect, it would be Y
• Based on yield-to-maturity effect, it would be Z
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Duration
• Weighted average amount of time until present value of bond’s purchase price repaid to the investor
• Based on time-weighted present value of bond’s principal and interest payments divided by the bond’s price
• Used as measure of bond’s sensitivity to interest rate changes
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Formula for Duration
Where P0 = price of the bond today
Y = yield to maturity
Ct = cash flow in period t (coupon, principal or both)
T = term to maturity
0
T
1tt
t
P
Y 1
Ct x
D
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Equation 9-6
• Insert Equation
• WhereY = yield to maturity
C = coupon rate
T = term to maturity
T
1 Y T C Y1 YD
Y C 1 Y 1 Y
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Uses of Duration
• Price volatility index– Larger duration statistic, more volatile price of
bond
• Immunization– Interest rate risk minimized on bond portfolio
by maintaining portfolio with duration equal to investor’s planning horizon
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Major Characteristics of Duration
• Duration of zero-coupon bond equal to term to maturity
• Duration of coupon bond always less than term to maturity
• Inverse relationship between coupon rate and duration
(continued)
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Major Characteristics of Duration (continued)
• Inverse relationship between yield to maturity and duration
• Direct relationship between maturity and duration
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Modified Duration
• Adjusted measure of duration used to estimate a bond’s interest rate sensitivity
D* = D (1 + YTM)
% Chg in price of bond = –D x % Chg in YTM
% Chg in price of bond = – D* x [Chg in YTM]
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Convexity
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Portfolio Duration
• Market value weighted average of durations of individual securities in the portfolio
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Components of InterestRate Risk
• Price Risk
• Reinvestment Rate Risk
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Price Risk
• Risk of existing bond’s price changing in response to unknown future interest rate changes– If rates increase, bond’s price decreases– If rates decrease, bond’s price increases
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Reinvestment Rate Risk
• Risk associated with reinvesting coupon payments at unknown future interest rates– If rates increase, coupons are reinvested at
higher rates than previously expected– If rates decrease, coupons are reinvested at
lower rates than previously expected
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Immunizing a Portfolio
• If a single time horizon goal, purchasing zero-coupon bond whose maturity corresponds with planning horizon
• If multiple goals, purchasing series of zero-coupon bonds whose maturities correspond with multiple planning horizons
(continued)
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Immunizing a Portfolio (continued)
• 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
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Bond Swaps
• Technique for managing bond portfolio by selling some bonds and buying others
• Possible benefits achieved:– tax treatment
– yields
– maturity structure– trading profits
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Types of Swaps
• Substitution swap– Tax swap
• Intermarket spread swap
• Pure-yield pick-up swap
• Rate anticipation swap
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Strategies for Managing a Bond Portfolio
• Bullet Portfolio
– Entire portfolio is placed in one maturity
• Bond ladders– Equally distributed dollar allocations over time
• Barbells– Majority of dollar allocations in shortest-term
and longest-term holdings
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Yield Curve or Term Structure
• Vertical axis: yield to maturity
• Horizontal axis: term to maturity
• Bonds of like quality
• Always based on Treasuries
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Shapes of Yield Curve
• Rising: Most common (used to be only one observed)
• Falling: Next most common
• Humped
• Flat: Rare
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Types of Yield Curves
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Theories of the Yield Curve
• Unbiased expectations– Long-term rates reflect market’s expectation of
current and future short-term rates.
• Preferred habitat– Significantly more attractive rates can induce
investors and borrowers out of their preferred maturity structures
(continued)
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Theories of Yield Curve(continued)
• Market Segmentation:– Yields reflect supply and demand for each
maturity class.
• Liquidity Preference:– Borrowers are risk averse and demand premium
for buying long-term securities– Yield curves tend to be upward sloping.
(continued)
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Theories of the Yield Curve (continued)
• Preferred habitat– Significantly more attractive rates can induce
investors and borrowers out of their preferred maturity structures
• Unbiased expectations– Long-term rates reflect market’s expectation of
current and future short-term rates.
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Factors Affecting Bond Yields
• General credit conditions: Credit conditions affect all yields to one degree or another.
• Default risk: Riskier issues require higher promised yields.
• Term structure: Yields vary with maturity• Duration: Weighted average amount of time until
present value of purchase price is recouped.• Coupon effect: Low-coupon issues offer yields
that are partially taxed as capital gains.(continued)
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• Seasonings: Newly issued bonds may sell at slight discount to otherwise-equivalent established issues.
• Marketability: Actively traded issues tend to be worth more than similar issues less actively traded.
• Call protection: Protection from early call tends to enhance bond’s value.
• Sinking fund provisions: Sinking funds reduce probability of default, thereby tending to enhance bond’s value.
• Me-first rules: Bonds protected from diluting effect of additional borrowings are generally worth more than otherwise-equivalent unprotected issues.
Factors Affecting Bond Yields (continued)