Bond Price, Yield, Duration
Pricing and Yield
Yield Curve
Duration
Immunization
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General Bond Characteristics Price Face or par value Coupon rate Compounding and payment
frequency Indenture, i.e. attached options,
covenants, etc.
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Example from July 1, 2004 WSJ U.S. Treasury Notes and Bonds are typically issued with
face value of $10,000, and pay semi-annual coupons The following bond quoted in the July 1, 2004 WSJ:
Matures in February 2026 (2/15/2026) Coupon rate is 6%. Semi-annual coupon payments are made on 2/15 and
8/15 of each year in the amount of (0.06 x $10,000)/2 = $300 At maturity (2/15/2026) the payment is the coupon of $300 plus the
principal of $10,000 Quoted decimal price per $100 par is $108 8/32 = $108.25 Quoted bond price is $10,825.00
RateMaturity
Bid Asked ChgAsked
Mo/Yr Yield
6 Feb 26 108:07 108:08 +10 5.35
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Bond Price and Yield (YTM) Bond Price, P
C: Coupon per period N: Number of periods F: Face (par) value y: Yield per period
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Prices and Yields
Price
Yield
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Bond Equivalent Yield (BEY) Bond Equivalent Yield (BEY) is the interest rate that
makes the present value of a bond’s payments equal to its price assuming semi-annual compounding convention
Example: What’s YTM of the following bond F = $1,000, C = $40, N = 60, P = $1,276.76
Notice the difference among y, yBEY, and yEAY
BEY is the yield quoted in financial press BEY is just annualized YTM, and we will use them interchangeably
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Term Structure of Interest Rates (Yield Curve) Is there a single interest rate?
US Treasury Yield Curve – Nov 24, 2008
Source: U.S. Treasury at www.ustreas.gov
0%
1%
2%
3%
4%
5%
0 5 10 15 20 25 30
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Yield Curve and Interest Rate Risk On one hand, yield curve rates reflect today’s
expectations of interest rates in the future and inflation in coming years
If either inflation of the real interest rate are expected to change in the future, then long term rates will differ from short term rates
On the other hand, yield curve rates also reflect the risk premium over longer maturities, since holding long-term bonds could be risky
Typically, forward rates are higher than expected actual rates, reflecting the risk premium
Investments 15 9
The Deep End of the Yield Curve It is typical that the yields on the longest available
maturities decrease, since U.S. Treasury bonds do not have close substitutes in longest maturities
Who can guarantee what happens to any corporate bond in 30 years? Few alternatives in other countries’ bonds
e.g. no big Latin American government has ever fully repaid a 30-year bond It is impossible to immunize a 30 year U.S. Treasury bond (will see later…)
Nov 24, 2008 Yield Curve
0%
1%
2%
3%
4%
5%
0 5 10 15 20 25 30
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Bond Terminology Flat Price is quoted in financial press Accrued Interest is not accounted for in the
Flat Price Invoice Price is the actual price a buyer pays
for the bond
Invoice Price = Flat Price + Accrued Interest Current Yield = Annual Coupon / Bond Price Discount Bond sells below par value Premium Bond sells above par value
Investments 15 11
Day Count Conventions for Accrued Interest Actual/Actual - Actual number of days between two dates is used.
AI = C x days/actual days in the year Actual/365 - Actual number of days between two dates is used as
the numerator. All years are assumed to have 365 days. AI = C x days/365
Actual/360 – Actual number of days between two dates is used as the numerator. All years are assumed to have 360 days.
AI = C x days/360 30/360 - All months are assumed to have 30 days.
If the first date falls on the 31st, it is changed to the 30th. If the second date falls on the 31th, it is changed to the 30th, but
only if the first date falls on the 30th or the 31st. 30E/360 - All months are assumed to have 30 days.
If the first date falls on the 31st, it is changed to the 30th. If the second date falls on the 31th, it is changed to the 30th
Investments 15 12
Example30 year U.S. Treasury bond Issued on 5/15/75 Coupon rate = 12% Semi-annual coupon payments on
5/15 and 11/15 Par value = $10,000 Flat (Quoted) Price on January 23, 2003 =
$12303.125 Next day settlement (January 24, 2003)
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Example Objectives
Find: Accrued Interest Invoice Price Bond Equivalent Yield (BEY) Current Yield
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Example Continued Semi-annual coupon =
(0.12 x $10,000)/2 = $600 Days between coupon payments
on 11/15/2002 and 5/15/2003 = 181 Days past since last coupon payment on 11/15/2002
until the settlement date on 1/24/2003 = 70 Accrued interest (January 23, 2003) =
(70/181)*$600 = $232.044 Invoice price =
$12303.125 + $232.044 = $12,535.17
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Example Continued
BEY = 1.76%
IRR = BEY 1.76% Payment Date 1/24/2003 5/15/2003 11/15/2003 5/15/2004 11/15/2004 5/15/2005
12% Bond Cash Flow -12535.17 600.00 600.00 600.00 600.00 10600.00
Time to Receipt in 6m Units 0 0.62 1.62 2.62 3.62 4.62
Discount Factor 1 0.9946 0.9859 0.9773 0.9688 0.9603
PV of the Cash Flow -12535.17 596.76 591.55 586.38 581.26 10179.22
Sum of PVs 0.00
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Example Continued
Current yield = $1200 / $12,303.125 = 9.75% Recall BEY = 1.76% Current yield is high, but BEY is low !!!
This is because investors expect capital loss!!!
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Important Takeaways
For premium bonds
(like in the Example) Current Yield > BEY Investors expect capital loss
For discount bonds Current Yield < BEY Investors expect capital gain
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Price Sensitivity to Interest Rates Although 1-yr and 30-yr interest rates are closely correlated…
0%
2%
4%
6%
8%
10%
12%
14%
16%
18%
1Year Rate
30 Year Rate
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Price Sensitivity to Interest Rates1-yr and 30-yr bond prices display drastically different interest rate sensitivity!
0
20
40
60
80
100
120
140
160
180 1 Year Price ($100 par)
30 Year Price ($1000 par)
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Pricing Bond price higher …
If coupon rate is higher If interest rate (yield) is lowerC = $40 F = $1,000
Year T 4% 6% 8% 10% 12% 14%
1 2 1038.83 1019.13 1000.00 981.41 963.33 945.76
2 4 1076.15 1037.17 1000.00 964.54 930.70 898.38
5 10 1179.65 1085.30 1000.00 922.78 852.80 789.29
10 20 1327.03 1148.77 1000.00 875.38 770.60 682.18
30 60 1695.22 1276.76 1000.00 810.71 676.77 578.82
Annual Interest Rate (APR)Maturity
Premium Bond P > par value
Discount Bond P < par value
Par Bond P=par value
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Duration – Measure of Sensitivity Duration is a measure of bond price
sensitivity to interest rate changes It is a characteristic of a security or a portfolio at a
particular point in time, which changes over time along with changes in maturity, yield, and coupon payments
It provides a quantitative measure that can be used in risk management, hedging, immunization...
There are more than one duration measure, i.e. Macaulay, Modified, Dollar, etc…
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Duration - Is There a Single Maturity? Macaulay Duration ( D ) is the weighted average of the
times to each coupon or principal payment made by the bond. The weights are given by discounted values of coupon or principal payments.
D – Macaulay duration PVCi – present value of cash flow at time i P – current bond price
Macaulay duration is the most intuitive duration measure, and gives explanation as to why the name Duration came into being
P
PVCT
P
PVC
P
PVCD T ...21 21
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Macaulay Duration - Example 10% annual coupon 5 years to maturity par bond Par value at the time of issue gives the Yield of 10%
Time Cash Flow PV Time*PV
1 10 9.09 9.09
2 10 8.26 16.53
3 10 7.51 22.54
4 10 6.83 27.32
5 110 68.30 341.51
Total 100.00 416.99
Macaulay Duration 4.17
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Modified Duration Modified Duration ( D* )
D – Macaulay duration
y – YTM
k – number of compounding periods per year Modified duration describes a percentage change in
bond price with respect to the yield change
ky
DD*
1
yDP
P
*
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Using Modified Duration Example
20 year, 6% coupon (semiannual payments) $100 face value bond Currently yields 8%, and is priced at $80.21 Macaulay Duration D = 10.92 years Modified Duration D* = 10.92/(1.04) = 10.5
Suppose the yield increases from 8% to 8.1% Predicted price change = -10.5 × .001 = -1.05% Actual price change = -1.04%
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Using Modified Duration - Continued
Suppose the yield increases from 8% to 10% Predicted price change = -10.5 × .02 = -21% Actual price change = -18.11%
Duration approach to estimating price changes is only accurate for small yield changes!
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Duration Takeaways Duration provides an answer the question
“What happens to the value of my bond portfolio when interest rates change”…
Duration Limitations Accurate only for small yield changes Assumes a flat yield curve and parallel
shifts Bonds are assumed option-free
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Concepts Check How does Duration vary with maturity?
How does Duration vary with coupon?
How does Duration vary with yield?
How does Callability affect previous answers?
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Duration – Graphic Interpretation
Yield
Price
Yield-to-Price Curve
Current Price
Current Yield
Tangent Line
New Yield
Duration Prediction ErrorNew Price
Predicted Price
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Convexity Convexity measures the curvature of
the bond Yield-to-Price curve Positive convexity implies that duration
underestimates the price increase when yields drop, and overestimates the price decrease when yields increase It means that a long position benefits from
positive convexity All non-callable bonds have positive
convexity
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Immunization Suppose you need some pattern of cash flows
in the future To meet these cash needs requires holding a
suitable portfolio of bonds Ideally one would like to hold a portfolio of zero
coupon bonds, or Strips Such approach is known as “cash flow matching” Zero coupon bonds may not be the best because of
possible unattractive relative pricing It may be necessary to use a portfolio of
coupon bonds
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Immunization Procedure Choose an initial immunization portfolio with the
modified duration that equals the modified duration of a set of liabilities
Fund the immunization portfolio so that its present value matches the present value of the set of liabilities, discounting at the rate given by the yield of the immunization portfolio
Rebalance the investment portfolio to adjust for interest rate changes and liabilities payments
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Immunization Rebalancing How often do you need to rebalance the
immunization portfolio?
You need to rebalance as soon as a significant discrepancy in durations between liabilities and the immunization portfolio occurs due to
changes in interest rates payments made by immunization securities liabilities been paid off
There is no one-fits-all answer to determine the size of a significant discrepancy – it depends on your objectives and risk tolerance
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Immunization Limitations Immunization matches duration, which
assumes a flat yield curve
Immunization only protects against parallel yield curve shifts
Immunization is not a risk-free strategy
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Immunization Takeaways Immunization is a dynamic portfolio managing
strategy that allows to meet a set of liabilities out of proceeds from a self-financing bond portfolio
Immunization allows to meet future liabilities without having to use a zero coupon bond portfolio
Major Users of Immunization Policies Pension Funds Life Insurance Companies Banks
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Wrap-up How to evaluate a bond? What’s the meaning of yield? Yield Curve concept Interest rate risk measures the bond price
reaction to the change in interest rate Duration is a simple measure for interest rate
risk Immunization is a passive but dynamic
strategy to limit interest rate risk
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