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Fixed Income Securities - II
Mapping to Curriculum ead din ing g 56: 56: Und nder erst sta and ndin ing g Yield Spreads • Rea
• Reading 58: Yield Measures, pot Rates and Forward Rates • Reading 59: Introduction to M asurement of Interest Rate Risk
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Expect Exp ect aro aroun und d 15 que questi stions ons in the exam from today’s lecture
Key Concepts • • • • • • • •
Interest Rate Policy Yield Curve Shapes Theories Of Term Structure Of Interest Rates LIBOR Yield Measures Reinvestment Risk Bootstrapping Nomina Nom inall Spread, Spread, Zero-vo Zero-volat latili ilitt Spread, This files has expired at 30-Jun-13 Option-adjusted Spread • Forward Rates • Duration, Convexity, PVBP
Agenda • Features of Debt Securities • Risks Associated with Investin in Bonds • Overview of Bond Sectors and Instruments • Understanding Yield Spreads
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Key Issues In Understanding Yield Sp reads Interest Rate Policy Yield Curve Theories of Term structure of Interest Rates Spot Rate Yield Spread measures Credit S read
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Embedded options affect on yield spread Liquidity affect on yield spread After-tax Yield LIBOR
Interest Rate Policy To implement the Fed‘s monetary policy, the Fed ses the following four interest rate tools: – – – –
Discount rate: is the rate at which banks borrow from the Fed. Open Market Operations: refers to purchase and s le of Treasury Securities in the open market. Bank Reserve requirements: refers to the percentage of deposits the bank must keep with itself. Pursuation: refers to the Fed asking banks to alter heir lending policies.
Lowering the discount rate and/or engaging in open market operations decrease the overall interest rates in the market. This files has expired at 30-Jun-13
Imp
Yield Curve And Its Shapes ield Curve: Shows the relationship between Yield nd Maturity
It can be: – – – –
Upward Sloping - Normal Downward Sloping - Inverted Flat Humped
Rising
Declining
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Humped
Theories Of Term Structure Of Interes Rates
Imp
Pure Expectations Theory: – States that the future value of interest rates is eq al to the summation of market expectations. If short-term rates are expected to rise then the yield curve will be upward sloping Shape of Term Structure
Implication Ac ording to Pure Expectations Theory
Upward sloping (normal)
Rates expected to rise
Downward slo in Flat
inverted Rates ex ected to decline This files has expired at 30-Jun-13 Rates not exp cted to change
Liquidity Preference Theory: – States that investors are risk-averse and will dem nd a premium for securities with longer maturities – Yield curve can be normal, inverted or flat as long s yield premium for interest rate risk increases with maturity.
Theories Of Term Structure Of Interes Rates Market Segmentation Theory: – States that most investors have set preferences regarding the length of maturities they will invest in – Example: a bank having large amount of short ter
liabilities will prefer to invest in short term securities.
An offshoot to above theory is that an investor can be induced to invest outside their term of preference, if they are compensated for taking on that additional risk by moving out of their preferred This files has expired at 30-Jun-13 range. This is known as the Preferred Habitat Th ory
Spot Rate The discount rate of a zero coupon bond is called he spot rate for that maturity. In the case of a treasury security, its called the tre sury spot rate. The relationship between maturity an d treasury s ot rates is called the term structure of interest rates.
This is different from the treasury yield curve. This files has expired at 30-Jun-13
Yield Curve Spot rate: The rate of return earned on a zero-coupon bond, if held to maturity. Forward rate: The yield on a zero-coupon security issued at some point in the future. Since the
securities have not been issued yet, we can never observe a forward rate, we can only estimate it. In short, a graph of forward rates is a graph of intere t rates that are expected to be paid on short-term securities in the future. (Forward rates are typicall estimated for 6-month Treasury bills.) Yield curve: A graph that shows the yield earned on bonds of various maturities. In short, it shows
the relationshi between short-term andhas lon -term interest rates. This files expired at 30-Jun-13
Yield Spread Measures Yield Spread Measures: Yield Spread is the diffe ence between the yield on two bonds – Absolute Yield Spread = (Yield on the subject bond - Yield on benchmark bond) – Relative Yield Spread = (Absolute Yield Spread/Yi ld on benchmark bond) – Yield Ratio = (Subject Bond Yield/Benchmark Bon Yield)
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Credit Spread Credit Spread: It is the spread between non - Tre sury and Treasury securities that are identical in
all respects except for the credit rating In an expanding economy, credit spreads becom narrow In a contracting economy, credit spreads widen. – This is because in a contracting economy, companies experience decline in revenues and cash flows making it more difficult for corporate issuers to ser ice their debt obligations. Thus, credit quality This files has expired at 30-Jun-13 deteriorates, and investors sell corporates and buy treasuries. Thus, widening the spreads.
Embedded Options
Imp
mbedded Options Effect on Yield Spread:
Call Provision: – Grants the issuer the right to retire the debt, fully o partially, before the scheduled maturity date. – From an investors point of view, a non-callable bond is preferred against a Callable bond. – Investors require a higher yield on the Callable bond and the yield spread is also larger for such bonds.
Put Provision/Conversion Provision This files has expired at 30-Jun-13 – A putable-bond is more preferred to a plain vanilla bond from the investor‘s point of view and will have a lower yield spread
The higher spread on an MBS is due to prepayme t risk.
After-tax Yield The difference in yield between tax-exempt securities and treasury securities is typically measured not in terms of absolute yield spread but as a yield ratio. One should compare the after-tax yield to arrive at an investment decision
After Tax Yield Taxable Yield * 1 Marginal tax ate
Tax Exempt yield This files has expired at 30-Jun-13 1 Marginal tax rate
LIBOR IBOR: It stands for London Inter bank Offered Rate – Is the rate paid on Negotiable CDs by banks located in London – Determined by the British Bank Association (BBA) – It is quoted in many currencies: – Has become the most important reference rate over time – Is important because the fluctuations in LIBOR will impact the rate at which the funded investor (one who This files has expired at 30-Jun-13 borrows to make an investments) will be able to borrow funds
Questions 1. The pure expectation theory can be used to expl in any shape of the yield curve. This statement is most likely A. Incorrect; The market segmentation theory can be used to explain any shape of the yield curve B. Incorrect; The liquidity preference theory can be used to explain any shape of the yield curve C. Correct; The pure expectation theory explains any shape of the yield curve
2. With respect to the term structure of interest rates, the market segmentation theory holds that : A. An increase in demand for long term borrowings could lead to an inverted yield curve . This files has expired at 30-Jun-13 yield curve C. The yield curve reflects the maturity demands of financial institutions and investors
. The tool most commonly used by Fed is: A. Open Market Operations B. Bank reserve requirement C. Discount rate
Questions (Cont...) . As per the Liquidity Preference Theory : A. Investors will demand a premium for shorter mat rity securities. B. Investors will demand a premium for longer maturity securities. C. Investors will not demand any premium.
. As per the Preference habitat Theory : A. Investors are will not move out of their preferenc habitat B. Investors demand a premium to invest outside their preference range . This files has expired at 30-Jun-13
. The impact of an expanding economy on the yiel spread is: A. To increase the yield spread B. To decrease the yield spread C. Will not effect the yield spread
. Which of the following will have the least Yield Spread: A. Callable Bond B. Putable Bond C. A plain Fixed Coupon Bond
Solutions 1. A. The market segmentation theory asserts that t e supply and demand for funds within the different
maturity sectors of the yield curve determine the interest rate for that sector. . C. The correct answer is the yield curve reflects t e maturity demands of financial institutions and
investors. . A. Open Market Operations . B. Investors will demand a premium for longer m turity securities This files has expired at 30-Jun-13 . B. Investors demand a premium to invest outside their preference range . B. To decrease the yield spread . B. Puttable Bond
Agenda • Introduction to the Valuation of
ebt Securities
• Yield Measures, Spot Rates, an Forward Rates • Introduction to Measurement of Interest Rate Risk
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Key Issues In Yield Measures, Spot Rat s, And Forward Rates Returns from Investing in a Bond Traditional Yield Measures Reinvestment Income Bond Equivalent Yield and Annual-pay Yield Computing theoretical Treasury Spot rate Nominal s read Zero-volatilit s read O tion-ad sted s read This files has expired at 30-Jun-13 Option cost in a bond Forward Rates
Returns From Investing In A Bond A person realizes the following returns from a cou on paying security – Interest payment made by the issuer – Reinvestment income from reinvesting the intere t payments received – Recovery of the principal. includes the capital gai /loss on selling the security.
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Traditional Yield Measures Traditional Yield Measures Current Yield: the annnual interest income from the bond
Current Yield =
Annual Coup n interest received
Bond Price The current yield is simply the coupon payment (C) as a percentage of the ( current ) bond price (P). Current yield = C / P 0. This files has expired at 30-Jun-13 Drawbacks :
Only Considers coupon interest Capital Gains/Losses not taken into account No consideration for reinvestment income
Imp
Traditional Yield Measures
Yield Yield to to Maturi Maturity ty(YT (YTM): M): YTM YTM is the IRR IRR of of the bon bon . It is the annualised rate of return on the bond –
1 Yield Measure Relationships:
C YTM 2
C Y TM 1 2
2
.....
C
P ar
YTM 1 2
Bond Selling at:
Relationship
Par
Coupon rate = Current Yield = Yield to Maturity
2N
Discount
Coupon rate < Current Yield < Yield to Maturity at >30-Jun-13 Premium This files Coupohas n rateexpired > Current Yield Yield to Maturity dvantages: Consid Considers ers both both coupo coupon n incom income e and and capit capital al gain gain/lo /lo s if held to maturity. Considers the timing of cashflows Limitations It co considers the re reinvestment income; th the in interim coupon payments are reinvested at a rate equal to the YTM.
Traditional Yield Measures YTM of Annual Annual Coupon Bond:
A 10 year, $1000 par value bond has a coupon of 7%. If it is priced at $920 what is the YTM? PV = -920; N=10; N=10; FV=1000; FV=1000; PMT=70 PMT=70 I/Y = 8.20%
YTM for zero cou onThis bond: bond: files has expired at 30-Jun-13
The The pric price e of of a 5-y 5-year ear Tre Treas asur ury y bond bond is $804 $804.. Cal Calc c late the semiannual-pay YTM and annual-pay YTM. Semiannual-pay YTM =
1000 110 1 * 2 4.41 804
1 0 0 0 Annual-pay YTM = 8 0 4
1 5
1
4 . 46 %
Traditional Yield Measures Bond Equivalent Yield: Doubling the semiannual yield to maturity. Yield to Call: yield on callable bonds (bonds can e called before maturity) that are selling at a
premiu premium. m. The The calcu calculat lation ion is is the same same as as for norm norm l bonds. The par value is substitued with the call pric price e and and the the tota totall peri period od is subs substi titu tute ted d wit with h the the period upto the call date Yield to Put: Put: yiel yield d on on put putta tabl ble e bon bonds ds that that are are sel sellili g at a discount Yield to Worst: Worst: A yield can be calculated for ever possible call date and put date. The lowest of This files has expired at 30-Jun-13
these YTM‘s is called Yield to Worst.
Cash Flow Yield: used for Amortisinfg Securities. The limitation with this measure is that the actual
prepayment rates may differ from those assumed or calculation purposes. Yield to maturity maturity (YTM): (YTM): most po popular yi yield me measure of all the above. The limitation with this
measure is that it assumes that cash flows are rei vested vested at the YTM and the bond is held till maturity maturity
Calculate And Compare Yield Spread Absolute yield spread : – It is Simply the difference between yields or two bonds.
( Yield on higher yield bond - yield on lower yield bond )
Relative yield Spread :
– It is the Absolute yield spread expressed as perce tage of the yield on benchmark bond. This files expired at 30-Jun-13 so ute has y e spre
Relative yield spread
Yield on the benchmark bond
Yield Ratio : It is the ratio of yield on the subject bon to the yield on the benchmark bond
Yield Ratio
Subject bond yield benchmark bond yield
Imp
Reinvestment Income
If the reinvestment rate is less than the YTM then he actual yield realised will be less than YTM How to calculate the Reinvestment Income earned??? 20-year Treasury bond purchased at par, 7% cou on rate, how much reinvestment income should be generated to earn a YTM of 7%? Total Value generated in 20 years = 100(1.035)40
395.9260
Reinvestment income required = 395.9260 – 100 40*3.50 = 155.9260 This files has expired at 30-Jun-13 Factors Affecting: – Higher the coupon rate higher the reinvestment ris – Longer the maturity higher the reinvestment risk
If the above problem was for a 10 year bond with
coupon of 5%, the reinvestment income required
would have been $13.8616 as compared to $ 155.9260
Bond Equivalent Yield And Annual-p y Yield The following formula identifies the relationship be ween the two.
Bond Equivalent Yield(BEY) of an Annual-pay BEY
ond
2 * 1 Annual Y M
1
2
1
This files has expired at 30-Jun-13 Yield on an annual pay basis
BE Y 1 1 YTM 2 2
Computing Theoretical Treasury Spot Rate Bootstrapping: It is the method of calculating the spot rates using the prices of coupon bonds. One
spot rate is used to calculate the spot rate for the next period. The two consecutive spot rates are used for calculating the next spot rate
Spot Rate Curve:
Theoretical Spot Rate Curve This files has expired at 30-Jun-13 (Term Structure of interest rates) 7% 6% 5% 4% Rate
3% 2% 1% 0% 0
0.5
1
1.5
2
2.5
3
Bootstrapping Summarizing the curves
Yield Curve
po
urve
Forward Curve
Yields are bond-specific; given a bon 's market price and coupons, the yield is the rate that all cash flows are discounted at to make present and future values the same. The spot curve diagrams what pure discount rate the market app es This o any files has expired at 30-Jun-13 cash flow at each maturity point. It is not bond specific. Also called the zero curve. This is a plot of what the market charges to borrow money for a 6 month period starting at certain future dates. Note that forward curves could be m de for any borrowing term (i.e. 1 year forwards, 3 month forwar s, etc.)
Bootstrapping Example: Consider 3 treasury securities with their maturities and market rates given in the table below:
Maturity
Market Rate
6 months
3%
12 months
4%
18 months
5%
Using the method of bootstrapping, find the theoretical Treasury spot rates. This files has expired at 30-Jun-13 Solution: – The bond with six months left to maturity has a se iannual discount rate of 0.03/2 = 0.015 or 3.0% on an annual bond equivalent yield (BEY) basis. – Since the bond will only make a single payment of 101.50 in six months, the market rate is the spot rate for cash flows to be received six months from now.
Solution The one-year bond will make two payments, one in six months of 2 and one in one year of 102. We can solve for the one-year spot rate in the equation: 2 1.015
102 100 S 1 2 (1 ) 2
where S1.0 is the annualized 1-year spot rate. Solvi g we get: S1.0 = 4.01 %. Using the 6-month and 1-year spot rates, we can use the same approach to find the 18-month spot rate from the equation .
.This files has expired at 30-Jun-13 100 S 1.5 3 1.015 (1.02) 2 (1 ) 2 where S1.0 is the annualized 18-month spot rate. Solving we get: S1.5 = 5.03%.
.
Nominal Spread, Zero-volatility Sprea , Option-adjusted Spread
Imp
Nominal Spread: is the YTM of a bond minus the YTM of a Treasury security of similar maturity
N om in a l
Spread
Y M
Bond
Y TM
Treasury
Zero-Volatility Spread: is the constant spread that is to be added to the spot rate yield at EACH
POINT on the Treasury curve where a cash flow is received that will make the price of a security equal to the present value of its cash flows. Each ash flow of the security is discounted at the appropriate Treasury spot rate plus the Z-spread. It is also known as "static spread" This files has expired at 30-Jun-13 PV of Bond(for a two year annual pay security)
Z-spread Vs Nominal:
C ou pon C oupo n Price 1 2 1 1 yr Spo t rate ZS 1 2 yr Spot rate ZS
A nominal spread uses one point on the Treasury yield curve to determine the spread at a single point that will equal the present value of the security's cash flows to its price
Option Adjusted Spread: is the spread without the affect of the option for a bond with embedded
options.
Option Adjusted S pread Z - Spread – Option Cost
Option Cost In A Bond Option Cost in % = Z-spread – Option Adjusted S read(OAS) In case of a callble bond the OAS < Z-spread as one needs to be compensated for the call feature In case of putable options the OAS > Z-spread
Spread Measure
Benchmark
Nominal
Treasury Yield Credit Risk, Liquidity Risk, Curve tion Risk This filesO has expired at 30-Jun-13 Treasury Spot Credit Risk, Liquidity Risk, Rate Curve Option Risk
Zero-Volatility Option Adjusted
Treasury Spot Rate Curve
Reflects Co pensation for
Credit Risk, Liquidity Risk
Forward Rates Forward Rates: rates of interest implied by the current zero rates for a period of time in the future
For example, 6-Month Forward in 6 Months is e uivalent to borrowing or lending the Notional
Amount for 6 month after 6 months from today
The same is represented as: S = f = Current SThis ot ratefiles has expired at 30-Jun-13 1f 1
= is the rate for a 1-year loan to be made on year from now
1f 2
= is the rate for a 1-year loan to be made tw years from now
Relating the above terminology:
1 S 1 f 1 f 1 3
3
1 0
1 1
1
f 2
Forward Rates For example: if we have the zero rates for year 4 and year 5 then the forward rate for the period of time between year 4 and year 5 would be known as the forward rate for that time period of 1 year. Year 4 F4= 4%
Year 5 F4,5
F5= 5%
at 30-Jun-13 The 5-year spot rate isThis 10.50%files and the has 4-yearexpired spot rate is 11.25%. What is the one year forward rate four years from no? – 7.02% – 7.55% – 8.35%
Solution: (1+z5)5 = (1+z4)4*(1+f 1) = (1.105)5 = (1.1125)4*(1+f 1) (1+f 1) = 1.0755 f 1 = 7.55%
Questions 1. Karen invests in an 8% 5-year semi-annual calla le bond on 5th January 2010. The Z-spread for the callable bond is 150bps. The option cost is 56 bp . The OAS is closest to A. 100 bps B. 94 bps C. 206 bps . Reinvestment income is least effected by: A. The time to maturity. This B. The size of the debt issue. files has expired at 30-Jun-13 C. The Coupon rate. . The z-spread of a callable bond is 340 basis poin s. The OAS of the bond is most likely to be: A. Greater than 340 basis points B. Lesser than 340 basis points C. Equal to 340 basis points
Questions (Cont....) . For a 7% 3-year semi-annual option-free bond. T e Treasury spot rates are given below. The bond is at par. Calculate the no-arbitrage price for the bo d. If the market price is $104.5 the BEY is closest to Maturity (months)
Yield
6
5.2%
12
5.5%
18
5.8%
24 30 36
No-Arbitrage Price
BEY
A
102.34
5.45%
B
101.48
5.36%
6.0% This files has expired at104.50 30-Jun-135.25% C 6.2% 6.5%
5. The yield on a Bond Equivalent basis of an annu l-pay 8.50% coupon bond prices at par is: A. 4.16% B. 8.33% C. 6.43% 6. The annual-pay yield to maturity of a 8.50% coupon semi-annual pay bond is: A. 17.72% B. 8.68% C. 13.43%
Solutions 1. B. OAS = Z-spread – option cost = 150 -56 = 94 bps . B. The size of the debt issue . B. Lesser than 340 basis points . B. No-arbitrage price is calculated by discounting all the cash flows by the spot rates M o n th s
Y ie ld
P V F a c to r
C a s h F lo w
PV of CF
6
5 .2 0 %
0 .9 7 4 7
3 .5
3 .4 1 1 3
12
5 .5 0 %
0 .9 4 7 2
3 .5
3 .3 1 5 2
5 .8 0 %
0 .9 1 7 8
3 .5
3 .2 1 2 3
6 .0 0 %
0 .8 8 8 5
3 .5
3 .1 0 9 7
30
6 .2 0 %
0 .8 5 8 4
3 .5
3 .0 0 4 5
36
6 .5 0 %
0 .8 2 5 4
1 0 3 .5
8 5 .4 2 8 0
18 24
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101.4809809
he bond equivalent yield can be calculated by using the CF function Input 6 cash flows for coupon payment and one principal payment cash flow. CF0 = 104.5 CPT IRR. IRR = 2.68% BEY = 2* IRR = 5.36% . B. 8.33% . B. 8.68%
Agenda • Introduction to the Valuation of Debt ecurities • Yield Measures, Spot Rates, and For ard Rates • Introduction to Measurement of Interest Rate Risk
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Key Issues In Introduction To The Me surement Of Interest Rate Risk Measuring Interest Rate Risk Price Volatility Convexity Effective Duration Alternative definitions of Duration Duration of a ortfolio This files has expired at 30-Jun-13 Convexity measure of a bond Modified and Effective Convexity Price Value of a Basis Point(PVBP)
Measuring Interest Rate Risk Interest rate risk can be measured by two method : – Full Valuation Method: • This is referred to as scenario analysis. • Under this method the normal pricing technique are used to value a bond or a bond with embedded options • When the interest rates change the entire portfolio is re-evaluated by the same method This files has expired at 30-Jun-13 • The two values are compared to arrive at the impact of change in interest rate • Calculation gets complicated when there are a l rge number of bonds in the portfolio.
– Duration/Convexity Method: • This gives an approximate result of the sensitivity of the bond. • It is much simpler compared to the full valuation method.
Disadvantages Of A Callable Bond From the investor‘s perspective the disadvantages of an embedded call option is: – Cash flow pattern is not known with certainity – Investor exposed to reinvestment risk – Price appreciation potential will be decreased relative to an otherwise comparable option-free bond. • Negative convexity
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Price Volatility And Convexity We have already seen that the price-yield curve is a negatively sloped and is a curve. This is referred to as convex. Price
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Properties concerning the price volatility of an option free bond: Percentage price change per change in interest ra es is not the same for all bonds For either small increases or decreases in yield, p rcentage change in price for given bond is roughly the same. For a given large change in yield, the percentage rice increase is greater than the percentage price decrease.
Price Volatility And Convexity The curve of a Callable bond exhibits Negative C nvexity. This is because the increase in the price of a security as a result of fall in the yield is cappe at the call price. See the below graph: Value of call
Option-free bond
e ci r
n
d
P
This files has expired at 30-Jun-13 b
o el b al l a C
Coupon
Yield
Price Volatility And Convexity The curve of a Puttable bond exhibits Positive Convexity. This is because the decrease in the price of a security as a result of increase in the yield is limited to the put price. See the below graph:
Putable Bond
e c i r P
This files hasValue expired at 30-Jun-13 of Put
Coupon
Yield
Imp
Effective Duration
Duration is the measure of how long on an average the holder of the bond has to wait before he receives his payments on the bond. A coupon paying bond’s duration would be lower than “n” as the holder gets some of his payments in the form of coupons before “n” years In simple words, duration of a bond is sensitivity of ond price to change in its interest rate Effective duration is calculated as: Effective Duration
(Bon price when yield falls – Bond price when yield rises) 2 * (Initial Price) * (Change in yield in decimals)
This expired 30-Jun-13 ercentage change in Bond Price =files -Effectivehas Duration * Change at in yield in percent. (Δy) Example: Consider a bond trading at 96.54 with duration of 4.5 years. In this case ΔB = - 96.54* 4.5 Δy ΔB = -434.43 Δy If there is 10 basis points increase ( + Δy) in the yield then the bond price would change by: ΔB = -434.43 * ( 0.001) = -.43443 Hence, B = 96.54- .43443 = 96.10
Percentage Change In Price Using Du ration
Approximate percentage price change = - Duratio * y * 100 For example, you hold a bond that has a duration f 7.8 years. The interest rates fell by 25 bps. Calculate the approximate percentage price chan e. Answer: Approximate percentage price change = - Duration * y * 100 = -7.8 *(- .0025) * 100 This files has expired at 30-Jun-13 = 1.95% For large changes in yield, convexity should also be used. Percentage change in price becomes inaccurate with only taking duration into account.
Alternative Definitions Of Duration Macaulay Duration: is the weighted average of he times when the payments are made. And the
weights are a ratio of the coupon paid at time “t” to the present bond price Macaulay duration is also used to measure how s nsitive a bond or a bond portfolio's price is to n t* C n * M changes in interest rates. M acaulay
Duration
t1
(1 y)
C u rrent
t
Bond
(1 y)
n
Price
where: t = Respective time period C= Periodic Coupon payments ; y =Periodic yield : n = Total number of periods This files has expired at 30-Jun-13 -964.54 0 40 40 Calculating Macaulay Duration: 0
40
D
1.05
1
40
1 .05
2
2
40
1 .05
964.54
3
3
1
1040
1.05
4
4
3636.76 964.54
2
3.77
Note that this is 3.77 six-month periods, which is about 1.89 years
3
, 40
4
Change In Bond Price With Change In Discount Rate Modified Duration M odD
-
1 V V
y
V V . M od D
. y
– The modified duration is equal to the percentage c ange in price for a given change in yield.
Example: The current price of a bond is 98.75. Its mo ified duration is 5.2 years. The YTM of the bond is . . This files has expired at 30-Jun-13 Solution: V = -98.75 * 5.2 * 0.005 = -2.57 The new price of the bond is 96.18
Alternative Definitions Of Duration Modified Duration: is derived from Macaulay Dur tion. It is better than Macaulay Duration as it takes into account the current YTM. Modified Duration (1
Macaulay Duration YTM
) no of interest payments per year
Effective Duration calculations explicitly take into account the a bond‘s option provisions such as This files has expired at 30-Jun-13 embedded options. The other methods of calculation ignore the option provision
In summary duration is, – The first derivative of the price-yield function – The slope of the price-yield curve. – A weighted average of the time till the cash flows willl be received.(Macaulay Duration) – The approximate percentage change in price fo a 1% change in yield.(Effective Duration)
Duration Of A Portfolio Duration of a portfolio is the weighted average of t e duration of the individual securities in the portfolio. Portfolio Duration =
W1D1 W2 D 2 ......... W N D N
The problem with the above equation is that it hol s good only for a parallel shift in the yield curve. This is because securities with different maturities may have different changes in yield. This files has expired at 30-Jun-13
Convexity Measure Of A Bond onvexity is the measure of the curvature of a price- ield cuve. d n o B
) $ ( e c i r P
Curvature effect not i incorporated by Duration Act al Price – Yield Cur e
P
This files has expired at 30-Jun-13 . Y
Convexity
(Bond price when yield falls Bond price when yield rises - 2 * Initial Bond Price) 2 * (Initial Price) * (Change in yield in decimals) 2
Duration is an appropriate measure for small changes in the yield. For larger changes in yield convexity should also be used. Percentage Change in Price = Duration Effect + Convexity Effect =[(-Duration * Δy) + (Convexity * Δy2) ] * 100 Note: In this formula all the values are used as nu bers. E.g. 1% must be written as 0.01. This is also the reason to multiply it by 100
Price Value Of A Basis Point (PVBP) This is a measure of interest rate risk. This is also known as the dollar value of an 01 (D 01) PVBP – It is the absolute value of the change in the price of a bond for a 1 basis point change in yield.
PVBP Initial Price - Price when yield changes by 1 basis point This files has expired at 30-Jun-13 The PVBP is the same for both increase and decr ase (because change in yield is small)
The PVBP is a special case of dollar duration.
PVBP Duration * 0.01% * Bond Value
Yield Volatility Price Yield Relationship
As seen in the graph, the when the yield level is high, a change in interest rates does not produce a large change in price. This files has expired at 30-Jun-13 However, when yields are low, changes in interest rates produces a large change in price. Interest Rate Risk can be decomposed into: – Duration risk – Yield Volatility
Yield volatility explains why junk bonds have higher interest rate risk than treasuries. Yield Volatility is given by the standard deviation of yield changes
Questions 1. A 5 year bond paying 8% annual pay coupon is currently trading for $1023.56 and having YTM of 7.42%, calculate the effective duration of the bond given 25 asis point in YTM. Given : V- = 1033.88, V+ = $1013.29 A. 5.03% B. 4.02% C. 4.56%
. Calculate the duration of the portfolio of two bonds A and B having weights of 60% and 40% respectively. Duration of bond A is 7.9 and duration of bond B is 6.7. A. 7.64 B. 7.42 C. 7.24
This files has expired at 30-Jun-13
. A bond has a convexity of 63.80. The convexity effect if the yield decreases by 80 basis points is: A. 0.41% B. 0.35% C. 0.54%
. A bond has a duration of 9.75 and a convexity of 105.80. What is the change in the price of the bond for a 100 basis fall in the yield: A. 10.25% B. 9.75% C. 10.80%
Questions (Cont...) . The most accurate measure for arriving at the effect of duration is? A. Duration Approach B. Full valuation approach C. PVBP
.
A bond manager has collected the following information regarding a portfolio of fixed income investments which have a par value of $10mn. T e current market price is $11.25mn. If the duration is 5.2 the most likely estimate of the price change for the bond issue for a 25 bps change is A. 1.3% of $10mn . . . This files has expired at 30-Jun-13 C. 2.1% of $11.25mn . A portfolio manager notices the following in his p rtfolio has a portfolio duration of 4.35. How much will be the change in the portfolio if the interest ra e declines by 25 bps A. $ 28,280 Issue Maturity Market Value B. $ 14,250 C. $ 27,100 A 2 $8.5mn B
5
$4.6mn
C
10
$12.9mn
Solutions 1. B. V = $1023.56, V- = 1033.88, V+ = $1013.29, hange in yield is = 25 bps = 0.0025
So effective duration is = ($1033.88 - $1013.29)/ * $1023.88 *0.0025 = 4.02 . B. The portfolio duration is =0.6 * 7.9 + 0.4 *6.7 = 7.42 . A. 0.41% . C. 10.80% . B. Full valuation a
roach This files has expired at 30-Jun-13
. B. The estimated change = 5.2*0.25 = 1.3%. (Th par value of $10mn is given to confuse the
candidate. Par value never changes. Current val e of $11.25mn is more important) . A. 26mn * 4.35 * (0.25)% = $ 28,280
Extra-Quiz Questions 1. What is least likely to be true regarding Macaulay and modified duration A. Both are calculated from the bond’s expected cash flows with no adjustments for embedded options on cash flows B. For bonds with no options, modified duration is si ilar to effective duration C. Macaulay duration takes into consideration embe ded options in the bond . A fixed income analyst makes the following two state ents: Statement 1 – Statement 1: YTM assumes that coupon payment are A Correct reinvested at the rateThis e ual tofiles the cash has flow ield. expired at 30-Jun-13 orrec – Statement 2: The bond is assumed to be held C Incorrect till maturity. . Consider the following two statements: – Statement 1: The static spread is the spread over the Treasury spot rate that makes the PV of all the cash flows from a non-Treasury security equal to its price. – Statement 2: The Z-spread ignores the interest rate volatility and assumes it to be zero.
A B C
Statement 1 Correct Correct Incorrect
Statement 2 Correct ncorrec Correct
Statement 2 Correct Incorrect Correct
Extra-Quiz Questions . Sally states that there are a number of yield measure that are used traditionally in the bond market. The least likely yield measure that is used A. Yield to call B. Yield to worst C. Yield to settlement . Duration is not a good measure for large changes in ield. Duration also assumes that the yield curve will shift in a parallel fashion. The statements are most lik ely A. Both statements are correct. This files has expired at 30-Jun-13 B. Only one statement is correct. C. Both the statements are incorrect. . An 8% coupon bond is valued at 104.35. When the yi ld increases by 20 bps the price of the bond declines to 103.44. The PVBP for the bond is closest to A. $0.0455 B. $0.0512 C. $0.0519
Extra-Quiz Questions . Which of the following 10-year fixed-coupon bonds has the least price volatility? All else equal, the bond with a coupon rate of: A. 6.50% B. 5.00% C. 8.00% . Carl manages the following portfolio he value for the portfolio duration is losest to
A. 5.833 B. 4.351 C. 4.555
Market Value 8% 5 years $ 5 mn $ 4 mn This files has expired at 30-Jun-13 11% 7 years $ 10 mn $ 11.4 mn $ 14.5 9.75% 10 years $ 15 mn mn $ 21.2 10.25% 5 years $ 20 mn mn Coupon
Maturity
Par Value
Duration 4.87 5.72 8.50 4.25
Solutions 1. C. . A. . A. – The Z-spread is also known as the static spread and it is the spreads that should be added on top of spot rates to calculate the PV of cash flows of a b nd. It also assumes the volatility of interest rates is zero hence it is also known as the zero-volatility AS. . C. – Yield to settlement is not a traditional measure of ield. The yield measures that are generally used are a ield to maturit b ield to call c ield to ut d ield to worst e current ield f cash flow ield. This files has expired at 30-Jun-13 . A. – As the duration measure is not useful for measuri g changes in price when there are large changes in yield. The duration also assumes that yields chan e is parallel across the entire yield curve. . A. – The PVBP = 104.35 – 103.44 / 20 = 0.0455
Solutions . C. – If bonds are identical except for the coupon rate, the one with the lowest coupon will exhibit the most price volatility. This is because a bond’s price is determi ed by discounting the cash flows. A lower-coupon bond pays more of its cash flows later (more of the cash flow is comprised of principal at maturity) than a higher-coupon bond does. Longer-term cash flows are discounted more heavily in the present value calculation. Another way to think about this: The relationship between the coupon rate and price volatility (all else equal) is inverse – a greater coupon results in less price volatility. Examination tip: If you get confused on the examination, remember that a zer -coupon bond has the highest interest rate risk because it delivers all its cash flows at maturity. Si ce a zero-coupon bond has a 0.00% coupon, a low coupon equates to high price volatility. This files has expired at 30-Jun-13 . A. Issue
Market Value
MV % of Portfolio Value
Duration
MV% * Duration
A
$ 4 mn
7.83%
4.87
0.3813
B
$ 11.4 mn
22.31%
5.72
1.2761
C
$ 14.5 mn
28.38%
8.50
2.4123
D
$ 21.2 mn
41.49%
4.25
1.7633
Total
$ 51.1 mn
100%
5.8330
Five Minute Recap C C PAR Value of a bond ...... 2 3 (1 YTM) (1 YTM) (1 YTM) 1 YTM) N C
C
BE Y
2 * 1 Annual YTM
1
2
1
BEY 2 YTM 1 1 2 Bond Selling at:
Relationship
Par
Coupon rate = Current Yield = Yield to Maturi y
Discount
Coupon rate < Current Yield < Yield to Maturi y
Premium
Coupon rate > Current Yield > Yield to Maturi y
Value of e call
Option-free bond ci r P
Callable bond
This files has expired at 30-Jun-13
Absolute Yield Spread
Relative yield spread
Yield Ratio
Coup on
Yield on Bond - Yield on Benchmark Bond
Yield
Absolute yield spread Yield on the benchmark bond
i r e P c
Putable Bond
Subject bond yield benchmark bond yield
Value of Put Coupon
Yield
