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 

Investments 15 4

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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Investments 15 7

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

Investments 15 8

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

Investments 15 10

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

Investments 15 15

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…

Investments 15 22

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

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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?

Investments 15 29

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

Investments 15 31

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

Investments 15 32

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

Investments 15 33

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

Investments 15 36

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