Chapter 5
Understanding Fixed-Income Risk
and Return
Rate of Return
• Total realized holding period return
𝑟𝑇𝑜𝑡𝑎𝑙 𝐻𝑃𝑅
𝑇𝑜𝑡𝑎𝑙 𝑅𝑒𝑐𝑒𝑖𝑣𝑒𝑑 − 𝑃𝑎𝑖𝑑 𝑇𝑜𝑡𝑎𝑙 𝑅𝑒𝑐𝑒𝑖𝑣𝑒𝑑
=
=
−1
𝑃𝑎𝑖𝑑
𝑃𝑎𝑖𝑑
• Annualized realized holding period return
(1 + 𝑟𝐴𝑛𝑛. 𝐻𝑃𝑅 )𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑌𝑒𝑎𝑟𝑠 𝑖𝑛 𝐻𝑃𝑅 = (1 + 𝑟𝑇𝑜𝑡𝑎𝑙 𝐻𝑃𝑅 )
Sources of Return
1. Promised coupon and principal payments
on the scheduled dates
2. Reinvestment of coupon payments
3. Potential capital gains or losses on the
sale of the bond prior to maturity
Computing Bond Returns
• Let’s see how we can compute realized
bond returns
• An investor initially buys a 10-year, 8%
annual coupon payment bond at a price of
85.503075 per 100 of par value. This
bond’s yield to maturity is 10.40%.
Example 1: Buy & Hold, YTM
Does Not Change
A “buy-and-hold” investor purchases a 10-year, 8% annual
coupon payment bond at 85.503075 per 100 of par value and
holds it until maturity. The yield-to-maturity of this bond is
10.40%.
The investor receives the series of 10 coupon payments of 8
(per 100 of par value) plus the redemption of principal (100) at
maturity.
The investor reinvests the coupon payments at 10.40%.
What is the investor’s annualized realized return?
Example 1: Buy & Hold, YTM
Does Not Change
Step 1: Find the total value of the investment at the end of the
holding period
–
–
–
–
–
First payment reinvested:
Second payment reinvested:
…
Last payment:
Sum:
8x1.10409
8x1.10408
108
229.970678
Step 2: Find total holding period return
– Total Return = (229.970678 – 85.503075)/85.503075 = 168.96%
Step 3: Find annualized holding period return (r)
– (1+r)10 = (1+1.6896)  r = 10.40%
Example 2: Buy & Sell, YTM
Does Not Change
An investor purchases a 10-year, 8% annual coupon
payment bond at 85.503075 per 100 of par value.
The yield-to-maturity of this bond is 10.40%.
The investor sells the bond after four years. Because
the yield-to-maturity remains at 10.40%, the sale
price is 89.668770 and coupons are reinvested at
10.40%.
What is the investor’s annualized realized return?
Example 2: Buy & Sell, YTM
Does Not Change
Step 1: Find the total value of the investment at the end of the holding
period
–
–
–
–
–
–
First payment reinvested:
Second payment reinvested:
Third payment reinvested:
Fourth payment:
Sale price:
Sum:
8x1.10403
8x1.10402
8x1.10401
8
89.668770
127.015881
Step 2: Find total holding period return
– Total Return = (127.015881 – 85.503075)/85.503075 = 48.551%
Step 3: Find annualized holding period return (r)
– (1+r)4 = (1+0.48551)  r = 10.40%
Capital Gains & Losses
Example 3: Buy & Hold, YTM
Increases
A buy and hold investor purchases a 10-year, 8%
annual coupon payment bond at 85.503075 per 100
of par value. The yield-to-maturity of this bond is
10.40%.
After the bond is purchased and before the first
coupon is received, interest rates go up to 11.40%.
So, the investor reinvests coupons at 11.40%.
What is the investor’s annualized realized return?
Example 3: Buy & Hold, YTM
Increases
Step 1: Find the total value of the investment at the end of the holding
period
–
–
–
–
–
–
First payment reinvested:
Second payment reinvested:
Third payment reinvested:
…..
Last payment:
Sum:
8x1.11409
8x1.11408
8x1.11407
108
236.380195
Step 2: Find total holding period return
– Total Return = (236.380195 – 85.503075)/85.503075 = 76.458%
Step 3: Find annualized holding period return (r)
– (1+r)10 = (1+0.76458)  r = 10.70%
Example 4: Buy & Sell, YTM
Increases
An investor purchases a 10-year, 8% annual
coupon payment bond at 85.503075 per 100 of
par value. The yield-to-maturity of this bond is
10.40%.
After the bond is purchased, interest rates go
up to 11.40%. The investor reinvests coupon
payments at 11.40% and sells the bond after
four years.
What is the investor’s annualized realized
return?
Example 4: Buy & Sell, YTM
Increases
Step 1: Find the total value of the investment at the end of the holding period
–
–
–
–
–
–
First payment reinvested:
Second payment reinvested:
Third payment reinvested:
Fourth payment:
Sale price:
Sum:
8x1.11403
8x1.11402
8x1.11401
8
85.780408
123.680132
Step 2: Find total holding period return
–
Total Return = (123.680132 – 85.503075)/85.503075 = 44.6499%
Step 3: Find annualized holding period return (r)
–
(1+r)4 = (1+0.446499)  r = 9.67%
What is the source of gains & losses?
• Capital loss = 89.668770 - 85.780408 = 3.888362
• Interest rate gains = 37.899724 - 37.347111 = 0.552613
• Total Loss (per 100 face value) = 3.335749
So What?
• Investment horizon is at the heart of understanding a
bond’s interest rate risk and return
• There are two offsetting types of interest rate risk that
affect bondholders:
– Coupon reinvestment risk
– Market price risk
• Which one is more important for a bond investor?
– Can two investors holding the same bond have different
exposures to interest rate risk?
• See example 7 on page 161 of the textbook
Interest Rate Risk on FixedRate Bonds
• Interest rate risk: Sensitivity of a bond’s full price change with
respect to small changes in interest rates (yield-to-maturity)
• What determines interest rate risk?
• Relationship between the bond price and bond characteristics
– Bond price is inversely related to the market discount rate
– For the same coupon rate and time-to-maturity, the percentage price
change is greater when market discount rate goes down than when it
goes up
– For the same time-to-maturity, a lower-coupon bond has a greater
percentage prices change than a higher coupon bond when their market
discount rates change by the same amount
– Generally, for the same coupon rate, a longer-term bond has a greater
percentage price change than a shorter-term bond when their market
discount rates change by the same amount
– Bond prices are more sensitive to interest rate changes when interest
rates are lower.
Which bond has the highest
interest rate risk?
BOND
Coupon
Maturity
Yield
A
10%
14 years
8%
B
12%
14.5 years
6%
C
14%
15 years
7%
D
16%
15.5 years
10%
Too many moving parts, can we summarize
these characteristics in a simpler way and
quantify the interest rate risk?
Duration – Plain Vanilla Bond
Convexity – Plain Vanilla Bond
Measuring Duration
• Yield Duration: Sensitivity of the bond price with
respect to the bond’s own yield-to maturity
–
–
–
–
Macaulay duration
Modified duration
Money duration
Price value of a basis point
• Curve Duration: Sensitivity of the bond price with
respect to a benchmark yield curve
– Effective duration
– Key rate duration
Mod(ified) Duration
• ModDur = MacDur/(1+r)
• %ΔPVFull ≈ -ModDur x ΔYield
• So, what is MacDur?
ModDur
when yield-to-maturity is 10.40%
ModDur = MacDur/(1+r) = 7.0029/(1+0.1040) = 6.34
ModDur – Within Coupon PMT
Maturity date is 02/14/2022. Settlement date is 04/11/2014.
MacDur = 12.621268/2 = 6.310634
ModDur = MacDur/(1+r) = 6.31/(1+0.03) = 6.13
So What?
• Modified duration
– Bond A: 6.34
– Bond B: 6.13
• %ΔPVFull ≈ -ModDur x ΔYield
• If interest rates increase by 1%, the price of
– Bond A will approximately decline by 6.34%
– Bond A will approximately decline by 6.13%
Approximate Modified Duration
ApproxMacDur = ApproxModDur x (1 + r)
Effective Duration
•
Measures the percentage in price given a change in a benchmark yield
curve (e.g., spot curve)
•
Does EffDur look familiar to you?
•
It is similar to Approximate Modified Duration
– The only difference is the term: ΔCurve
– ΔCurve: A parallel shift in the benchmark curve
•
EffDur is also knows as the curve duration because it measures interest
rate risk in terms of a parallel shift in the benchmarked curve
Effective Duration - Example
A callable bond is trading at a full price of 101.060489 per 100 of
par value.
Using a binomial option pricing model, you predict that when the
government par spot curve is raised and lowered by 25 bps, the
new full prices are 99.050120 and 102.890738, respectively.
What is the effective duration of this bond?
(102.890738 – 99.050120)/(2x0.0025x101.060489) = 7.6006
Price-YTM Function for a
Callable Bond
Price-YTM Function for a
Putable Bond
Duration of a Bond Portfolio
Duration of a Bond Portfolio ≈ Weighted Average Duration of the Bonds
Note: All bonds make semiannual coupon payments.
Portfolio MacDur = 0.2581x4.761 + 0.2825x5.633 + 0.4594x7.652 = 6.336
Portfolio ModDur = 0.2581x4.761/(1+0.091/2) +
0.2825x5.633/(1+0.0938/2) +
0.4594x7.652/(1+0.0962/2) = 6.05
If interest rates increase by 100 bps, the market value of the bond portfolio
is expected to decline by approximately 6.05%.
Other Duration Measures
• Key rate duration
• Money duration
• Price value of a basis point
Convexity Matters
Convexity Adjustment
Example
Bond
Market Value
ModDur
Convexity
A
$
683.66
9.71
98.97
B
$
368.22
19.37
384.60
C
$
738.36
29.07
859.14
• Portfolio ModDur = 19.68
• Portolio Convexity = 471.24
• If interest rates increase by 100 bps, market value
of the bond portfolio will approximately change by:
– Using ModDur: -19.68x0.01 = -19.68%
– Using ModDur + Convexity:
-19.68x0.01+0.5x471.24x0.012 = -17.33%
Example
Example