Description
It explains Duration is a measurement of how long in years it takes for the price of a bond to be repaid by internal cash flows
Concept of Duration
Introduction
• Duration is a measurement of how long, in years, it
takes for the price of a bond to be repaid by its internal cash flows • Source of interest rate risk is measured by duration. Interest rate risk refers to the effect of changes in the prevailing market rate of interest on bond values
Market Value
YTM
•Duration
= - % change in bond price/ %change in yield
For each of the two basic types of bonds the duration is the following
?
1. Zero-Coupon Bond: Duration is equal to its time to maturity
?
2. Vanilla Bond : Duration will always be less than its time to maturity
Determinants of duration
•
Coupon rate • Yield • Time to maturity
•
Holding coupon rate and maturity constant –
Increases in market yield rates cause a decrease in the present value factors of each cash flow. Since Duration is a product of the present value of each cash flow and time, higher yield rates also lower Duration. Therefore Duration varies inversely with yield rates.
Determinants of duration
•Holding yield rate and maturity constant –
Increases in coupon rates raise internal cash flows. Thus it takes less time for the price of a bond to be repaid Therefore Duration varies inversely to coupon rate. • Holding
yield rate and coupon rate constant
An increase in maturity increases Duration and cause the bond to be more sensitive to changes in market yields. Decreases in maturity decrease Duration and render the bond less sensitive to changes in market yield. Therefore Duration varies directly with time-to-maturity (t) • Presence of Embedded presence of embedded options.
options: Duration varies inversely to
DV01 (Dollar value of a basis point)
•
It is the absolute change in bond price from a one basis point (0.01%) change in yield • It tells you how much money your positions will gain/lose for a 0.01% parallel movement in the yield curve
•
DV01 = duration*0.0001*bond value • DV01 = (Bond Value*modified duration)/10,000 • To compute DV01, the bond is re-priced after the yield is shocked up/down by one basis point
Pros and Cons of DV01
•
Pros
? Relatively simple to calculate ? Easy to understand and has gained universal acceptance ? Use the same approach to financial instruments that have known cash flows
•
Cons
? Don?t know how much the yield curve can move on a day to day basis ?Yield curve don?t necessarily move up or down in a parallel manner
Modified Duration
• •
•
It can be interpreted as the approximate percentage change in the price of bond from any given change in yield. Modified duration = (1/P)*(1/1+periodic yield)*(sum of time weighted present values of bonds cash flows) It is an extension of Macaulay duration
• •
It expresses the measurable change in the value of a security in response to a change in interest rates. It follows the concept that interest rates and bond prices move in opposite directions.
Macaulay Duration
It is the weighted average term to maturity of a bond?s cash flows. It is measured in number of periods (years). ? Macaulay duration= (sum of time-weighted
?
present value of the bonds cash flows)/ bond price
Macaulay Duration, Coupon Rate and Maturity
?Duration is equal to maturity for ZCBs. Long-term ZCBs have greatest duration ?As maturity increases, duration of bonds increases ?As coupon rate increases, the duration of bonds decreases
45 40 35
Maculay duration
30 25 20 15 10 5
0
0 5 10 15 20 25 30 35
Maturity Zero Coupon bond par bond perpetuity deep discount bond premium bond
Example on Duration
semiannual period
0.5 1 1.5 2 2.5 3 Total PV of Bond Price (P) Modified Duration Macaulay duration Present value of Cash flows Discount Cash flows (C+P) Factor (D ) [D*(C+F)] n*PVCF 2 2 2 2 2 102 112 0.9828 0.9659 0.9493 0.9330 0.9169 0.9011 1.9656 1.9318 1.8986 1.8659 1.8338 91.9165 101.4122 0.9828 1.9318 2.8479 3.7318 4.5846 275.7496 289.8285
par value years to maturity coupon, % yield Semi-annual equivalents:coupon, % Coupon, $ periods semi-annual yield
$ 100 3 4% 3.50%
2% $2.0 6 1.75%
101.412 2.809 3.019
Limitations of Duration
Duration is a good approximation of price changes for an option-free bond, but it is only good for relatively small changes in interest rates. ? Like DV01, duration is a linear estimate since it assumes that price change will be same regardless of whether interest rate goes up or down. ? However, as rate change grows larger, the curvature of the bond price/yield relationship becomes more important.
?
Presence of Embedded Options
?
? ? ?
The derivative of bond price volatility relative to yield changes that results in the „modified duration? formula has embedded in it the assumption that the cash flows of the bond do not change as the rates change It cannot be used to value bonds that include call or put options A decrease in interest rate can lead to utilization of a call option and thus decrease duration An increase in interest rate can lead to the exercise of a put option and thus decrease duration
Option Adjusted Spread (OAS)
? ? ? ?
?
Project a series of interest rate paths into the future Most simple assumption model (for interest rates into the future) can be the binomial model Find the PV of cash flows for each path and take the average of PV?s The spread over the representative rate (Treasury bill rate) that will result in the average of PVs to be equal to the current price of the bond is OAS The rest of the spread over the treasury bill rate is the value of the embedded option
Effective Duration
? ?
This duration figure takes into account the presence of embedded options Once the OAS has been found, price changes for corresponding changes in yields are found out by discounting at treasury bill rate +OAS
Portfolio Duration
?
?
Weighted Average of the durations of the individual securities in the portfolio Weights assigned are equal to the relative market value of the security as compared to the market value of the portfolio
Limitations of Portfolio Duration
? ?
?
Yields might not change on all portfolio securities equally With different maturities, credit risks and embedded options, yields on individual securities will not change in equal amounts Portfolio duration is useful only when there are parallel changes in the yield curve (assuming everything else constant)
Immunizing Bond Portfolios using Interest Rate Futures
?
By trading Interest Rate Futures in conjunction of holding the Bond Portfolio, one can effectively adjust the duration of the portfolio Holding of the Bond Portfolio itself need not be altered The Portfolio manager may maintain the Bond Portfolio itself without disturbing favored maturities or issues Problem of lack of marketability of the bonds Transaction costs associated with adjusting the duration are much lower if one uses Interest Rate Futures, and the task can be accomplished with little or no capital, since one must make only a margin deposit to trade the futures contract
? ? ? ?
Duration and Immunization
?
As discussed earlier, Macaulay introduced the concept of Duration (D) which he defined as:
Where: Cit: Cash flow from the ith financial Instrument at time t. Ki: the Instrument?s yield to maturity. t: An element of a time vector ranging over the time to maturity. Pi: The Instrument?s price.
?
For Infinitesimal changes in (1+Ki):
?
Although the previous equation applies strictly to a single instrument, the duration of the Portfolio (Dp) with N assets each having a weight Wi in the Portfolio is given by:
Uses of the Concept of Duration
?
The concept of Duration has two distinct uses: Use 1: “The Bank Immunization Case” assumes that one agent holds both an asset and liability portfolio of equal value. By setting the duration of both the asset and liability portfolios equal, any change in the yield to Maturity Ki affects both portfolios equally. Consequently, the portfolio holder incurs no wealth change due to a shift in Interest Rates. Use 2: “The Planning Period Case” is directed towards a portfolio holder who has some planning period in mind, after which he plans to liquidate the portfolio.
?
?
How to Immunize using Interest Rate Futures
The Planning Period Case
Coupon Bond A Bond B Bond C T-Bond Futures T-Bill Futures 8.00% 10.00% 4.00% 8.00% Maturity 4 yrs 10 yrs 15 yrs 20 yrs 0.25 yrs Yield 12.00% 12.00% 12.00% 12.00% 12.00% Price 885.59 903.47 463.05 718.75 972.07 Duration 3.475 6.265 9.285 8.674 0.250
Given: $100million Bond Portfolio of Bond C with a duration of 9.285 years. Objective: Shorten the portfolio duration to 6 years to match the planning period.
?
Strategy 1: Sell Bond C and Buy Bond A until the following condition is met: WADA + WCDC = 6 YEARS
WA + WC = 1
Where Wi = percentage of portfolio funds committed to Asset I Result: 56.54% of the $100 million must be put in Bond A, the funds coming from the sale of Bond C.
?
Strategy 2: Continue to hold $100 million in Bond C and occupy an appropriate position in T-Bill futures satisfying the following condition:
?PP = ?PCNC + ?FPTBILLNTBILL
Where:
PP = value of the Portfolio Pc = Price of Bond C FPTBILL = T-Bill Futures Price NC = Number of C Bonds NTBILL = number of T-bills
?
Applying the same price change formula discussed earlier, we get:
Which reduces to DpPp = DcPcNc + DTBILLFPTBILLNTBILL
? The goal is to mimic portfolio 1 which has a total value of
$100 million and a duration of 6 years, it must be that: Pp = $ 100 million Dp = 6 years Dc = 9.285 Pc = $463.05 Nc = 215,959 DTBILL= 0.25
FPTBILL = 972.07
?
Solving for NTBILL = -1,351,747 indicates that this many T-Bills (assuming $1000 par value) must be sold short in the futures market. Since T-Bill futures are denominated in $1,000,000 face value, this strategy requires that 1,352 contracts must be sold.
?
Strategy 3: Continue to hold $100 million in Bond C and occupy an appropriate position in T-Bond futures satisfying the following condition:
DpPp = DcPcNc + DTBondFPTBondNTBond
?
Result: On Solving, we get NTBond = -52,691.Since T-Bond Futures contracts have a face value denomination of $100,000, 527 T-Bond futures contracts must be sold.
Portfolio Characteristics for Planning Period Case:
Portfolio 1 (Bonds only) WA Portfolio Weights WC WCASH NA Number of Instruments NC NTBILL NTBOND NAPA NCPC Value of each Instrument NTBILLFPTBILL NTBONDFPTBOND CASH PORTFOLIO VALUE 56.54% 43.46% 0.00% 63,844 93,856 56,539,608 43,460,021 371 100,000,000 Portfolio 2 (Short T-Bill Futures) 100.00% 0.00% 0 215,959 -1,351,747 99,999,815 1,313,992,706 185 100,000,000 Portfolio 3 (Short T-Bond Futures) 100.00% 0.00% 0 215,959 -52,691 99,999,815 37,871,656 185 100,000,000
How Immunization Works?
?
Now we assume an instantaneous drop in rates for all maturities from 12% to 11%. We also assume that all coupon receipts during the six-year planning period can be re-invested at 11% until the end of the planning period.
Coupon Maturity 4 yrs 15 yrs 20 yrs 0.25 yrs Old Yield 12.00% 12.00% 12.00% 12.00% New Yield 11.00% 11.00% 11.00% 11.00% Old Price 885.59 463.05 718.75 972.07 New Price 913.57 504.33 778.13 974.25
Bond A Bond C T-Bond Futures T-Bill Futures
8.00% 4.00% 8.00% -
Effect of Interest Rate Shift
?
Finally, we examine the effect of Interest Rate shift on portfolio values, terminal wealth at the horizon( year 6) and on the total wealth position of the portfolio holder.
Portfolio 1 100,000,000 105,660,731 0 5,660,731 197,629,369 Portfolio 2 100,000,000 108,914,787 -2,946,808 5,967,979 198,204,050 Portfolio 3 100,000,000 108,914,787 -3,128,792 5,785,995 197,863,664
Original Portfolio Value New Portfolio Value Gain/Loss on futures Total Wealth change Terminal Value of all funds at t=6 Annualized Holding Period Return over 6 years
1.120234
1.120776
1.120455
Transaction Costs for Planning Period Case
?
One important concern in the implementation of Immunization strategies is the transaction costs involved.
Portfolio 1 Number of Instruments Traded Bond A Bond C T-Bill Futures Contracts T-Bond Futures Contracts One-way Transaction Cost Bond A @ $2 Bond C @ $2 T-Bill Futures @ $15 T-Bond Futures @ $15 Total Cost of becoming Immunized 63,844 -122,103 -
Portfolio 2 1,352 -
Portfolio 3 527
127,688 244,206 $371,894
20,280 $20,280
7,905 $7,905
?
Clearly, there is a tremendous difference between trading the cash and futures instruments. In an extreme example of this type, the transaction cost for “Bonds Only” case is prohibitive, amounting to almost 4% of the total portfolio value. The volume of Bonds to be traded is enormous, exceeding any reasonable volume for bonds of even the largest issue.
?
References
? ? ? ? ?
http://www.cunacfocouncil.org/news/435.htmlhttp://www.treasurer.ca.gov/Cdiac/publications/duration.pdfhttp://en.wikipedia.org/wiki/Bond_durationhttp://www.investopedia.com/university/advancedbond/advan cedbond5.asp Fixed Income Securities, Level 1&2, CFA Book 4
doc_178617508.ppt
It explains Duration is a measurement of how long in years it takes for the price of a bond to be repaid by internal cash flows
Concept of Duration
Introduction
• Duration is a measurement of how long, in years, it
takes for the price of a bond to be repaid by its internal cash flows • Source of interest rate risk is measured by duration. Interest rate risk refers to the effect of changes in the prevailing market rate of interest on bond values
Market Value
YTM
•Duration
= - % change in bond price/ %change in yield
For each of the two basic types of bonds the duration is the following
?
1. Zero-Coupon Bond: Duration is equal to its time to maturity
?
2. Vanilla Bond : Duration will always be less than its time to maturity
Determinants of duration
•
Coupon rate • Yield • Time to maturity
•
Holding coupon rate and maturity constant –
Increases in market yield rates cause a decrease in the present value factors of each cash flow. Since Duration is a product of the present value of each cash flow and time, higher yield rates also lower Duration. Therefore Duration varies inversely with yield rates.
Determinants of duration
•Holding yield rate and maturity constant –
Increases in coupon rates raise internal cash flows. Thus it takes less time for the price of a bond to be repaid Therefore Duration varies inversely to coupon rate. • Holding
yield rate and coupon rate constant
An increase in maturity increases Duration and cause the bond to be more sensitive to changes in market yields. Decreases in maturity decrease Duration and render the bond less sensitive to changes in market yield. Therefore Duration varies directly with time-to-maturity (t) • Presence of Embedded presence of embedded options.
options: Duration varies inversely to
DV01 (Dollar value of a basis point)
•
It is the absolute change in bond price from a one basis point (0.01%) change in yield • It tells you how much money your positions will gain/lose for a 0.01% parallel movement in the yield curve
•
DV01 = duration*0.0001*bond value • DV01 = (Bond Value*modified duration)/10,000 • To compute DV01, the bond is re-priced after the yield is shocked up/down by one basis point
Pros and Cons of DV01
•
Pros
? Relatively simple to calculate ? Easy to understand and has gained universal acceptance ? Use the same approach to financial instruments that have known cash flows
•
Cons
? Don?t know how much the yield curve can move on a day to day basis ?Yield curve don?t necessarily move up or down in a parallel manner
Modified Duration
• •
•
It can be interpreted as the approximate percentage change in the price of bond from any given change in yield. Modified duration = (1/P)*(1/1+periodic yield)*(sum of time weighted present values of bonds cash flows) It is an extension of Macaulay duration
• •
It expresses the measurable change in the value of a security in response to a change in interest rates. It follows the concept that interest rates and bond prices move in opposite directions.
Macaulay Duration
It is the weighted average term to maturity of a bond?s cash flows. It is measured in number of periods (years). ? Macaulay duration= (sum of time-weighted
?
present value of the bonds cash flows)/ bond price
Macaulay Duration, Coupon Rate and Maturity
?Duration is equal to maturity for ZCBs. Long-term ZCBs have greatest duration ?As maturity increases, duration of bonds increases ?As coupon rate increases, the duration of bonds decreases
45 40 35
Maculay duration
30 25 20 15 10 5
0
0 5 10 15 20 25 30 35
Maturity Zero Coupon bond par bond perpetuity deep discount bond premium bond
Example on Duration
semiannual period
0.5 1 1.5 2 2.5 3 Total PV of Bond Price (P) Modified Duration Macaulay duration Present value of Cash flows Discount Cash flows (C+P) Factor (D ) [D*(C+F)] n*PVCF 2 2 2 2 2 102 112 0.9828 0.9659 0.9493 0.9330 0.9169 0.9011 1.9656 1.9318 1.8986 1.8659 1.8338 91.9165 101.4122 0.9828 1.9318 2.8479 3.7318 4.5846 275.7496 289.8285par value years to maturity coupon, % yield Semi-annual equivalents:coupon, % Coupon, $ periods semi-annual yield
$ 100 3 4% 3.50%
2% $2.0 6 1.75%
101.412 2.809 3.019
Limitations of Duration
Duration is a good approximation of price changes for an option-free bond, but it is only good for relatively small changes in interest rates. ? Like DV01, duration is a linear estimate since it assumes that price change will be same regardless of whether interest rate goes up or down. ? However, as rate change grows larger, the curvature of the bond price/yield relationship becomes more important.
?
Presence of Embedded Options
?
? ? ?
The derivative of bond price volatility relative to yield changes that results in the „modified duration? formula has embedded in it the assumption that the cash flows of the bond do not change as the rates change It cannot be used to value bonds that include call or put options A decrease in interest rate can lead to utilization of a call option and thus decrease duration An increase in interest rate can lead to the exercise of a put option and thus decrease duration
Option Adjusted Spread (OAS)
? ? ? ?
?
Project a series of interest rate paths into the future Most simple assumption model (for interest rates into the future) can be the binomial model Find the PV of cash flows for each path and take the average of PV?s The spread over the representative rate (Treasury bill rate) that will result in the average of PVs to be equal to the current price of the bond is OAS The rest of the spread over the treasury bill rate is the value of the embedded option
Effective Duration
? ?
This duration figure takes into account the presence of embedded options Once the OAS has been found, price changes for corresponding changes in yields are found out by discounting at treasury bill rate +OAS
Portfolio Duration
?
?
Weighted Average of the durations of the individual securities in the portfolio Weights assigned are equal to the relative market value of the security as compared to the market value of the portfolio
Limitations of Portfolio Duration
? ?
?
Yields might not change on all portfolio securities equally With different maturities, credit risks and embedded options, yields on individual securities will not change in equal amounts Portfolio duration is useful only when there are parallel changes in the yield curve (assuming everything else constant)
Immunizing Bond Portfolios using Interest Rate Futures
?
By trading Interest Rate Futures in conjunction of holding the Bond Portfolio, one can effectively adjust the duration of the portfolio Holding of the Bond Portfolio itself need not be altered The Portfolio manager may maintain the Bond Portfolio itself without disturbing favored maturities or issues Problem of lack of marketability of the bonds Transaction costs associated with adjusting the duration are much lower if one uses Interest Rate Futures, and the task can be accomplished with little or no capital, since one must make only a margin deposit to trade the futures contract
? ? ? ?
Duration and Immunization
?
As discussed earlier, Macaulay introduced the concept of Duration (D) which he defined as:
Where: Cit: Cash flow from the ith financial Instrument at time t. Ki: the Instrument?s yield to maturity. t: An element of a time vector ranging over the time to maturity. Pi: The Instrument?s price.
?
For Infinitesimal changes in (1+Ki):
?
Although the previous equation applies strictly to a single instrument, the duration of the Portfolio (Dp) with N assets each having a weight Wi in the Portfolio is given by:
Uses of the Concept of Duration
?
The concept of Duration has two distinct uses: Use 1: “The Bank Immunization Case” assumes that one agent holds both an asset and liability portfolio of equal value. By setting the duration of both the asset and liability portfolios equal, any change in the yield to Maturity Ki affects both portfolios equally. Consequently, the portfolio holder incurs no wealth change due to a shift in Interest Rates. Use 2: “The Planning Period Case” is directed towards a portfolio holder who has some planning period in mind, after which he plans to liquidate the portfolio.
?
?
How to Immunize using Interest Rate Futures
The Planning Period Case
Coupon Bond A Bond B Bond C T-Bond Futures T-Bill Futures 8.00% 10.00% 4.00% 8.00% Maturity 4 yrs 10 yrs 15 yrs 20 yrs 0.25 yrs Yield 12.00% 12.00% 12.00% 12.00% 12.00% Price 885.59 903.47 463.05 718.75 972.07 Duration 3.475 6.265 9.285 8.674 0.250
Given: $100million Bond Portfolio of Bond C with a duration of 9.285 years. Objective: Shorten the portfolio duration to 6 years to match the planning period.
?
Strategy 1: Sell Bond C and Buy Bond A until the following condition is met: WADA + WCDC = 6 YEARS
WA + WC = 1
Where Wi = percentage of portfolio funds committed to Asset I Result: 56.54% of the $100 million must be put in Bond A, the funds coming from the sale of Bond C.
?
Strategy 2: Continue to hold $100 million in Bond C and occupy an appropriate position in T-Bill futures satisfying the following condition:
?PP = ?PCNC + ?FPTBILLNTBILL
Where:
PP = value of the Portfolio Pc = Price of Bond C FPTBILL = T-Bill Futures Price NC = Number of C Bonds NTBILL = number of T-bills
?
Applying the same price change formula discussed earlier, we get:
Which reduces to DpPp = DcPcNc + DTBILLFPTBILLNTBILL
? The goal is to mimic portfolio 1 which has a total value of
$100 million and a duration of 6 years, it must be that: Pp = $ 100 million Dp = 6 years Dc = 9.285 Pc = $463.05 Nc = 215,959 DTBILL= 0.25
FPTBILL = 972.07
?
Solving for NTBILL = -1,351,747 indicates that this many T-Bills (assuming $1000 par value) must be sold short in the futures market. Since T-Bill futures are denominated in $1,000,000 face value, this strategy requires that 1,352 contracts must be sold.
?
Strategy 3: Continue to hold $100 million in Bond C and occupy an appropriate position in T-Bond futures satisfying the following condition:
DpPp = DcPcNc + DTBondFPTBondNTBond
?
Result: On Solving, we get NTBond = -52,691.Since T-Bond Futures contracts have a face value denomination of $100,000, 527 T-Bond futures contracts must be sold.
Portfolio Characteristics for Planning Period Case:
Portfolio 1 (Bonds only) WA Portfolio Weights WC WCASH NA Number of Instruments NC NTBILL NTBOND NAPA NCPC Value of each Instrument NTBILLFPTBILL NTBONDFPTBOND CASH PORTFOLIO VALUE 56.54% 43.46% 0.00% 63,844 93,856 56,539,608 43,460,021 371 100,000,000 Portfolio 2 (Short T-Bill Futures) 100.00% 0.00% 0 215,959 -1,351,747 99,999,815 1,313,992,706 185 100,000,000 Portfolio 3 (Short T-Bond Futures) 100.00% 0.00% 0 215,959 -52,691 99,999,815 37,871,656 185 100,000,000
How Immunization Works?
?
Now we assume an instantaneous drop in rates for all maturities from 12% to 11%. We also assume that all coupon receipts during the six-year planning period can be re-invested at 11% until the end of the planning period.
Coupon Maturity 4 yrs 15 yrs 20 yrs 0.25 yrs Old Yield 12.00% 12.00% 12.00% 12.00% New Yield 11.00% 11.00% 11.00% 11.00% Old Price 885.59 463.05 718.75 972.07 New Price 913.57 504.33 778.13 974.25
Bond A Bond C T-Bond Futures T-Bill Futures
8.00% 4.00% 8.00% -
Effect of Interest Rate Shift
?
Finally, we examine the effect of Interest Rate shift on portfolio values, terminal wealth at the horizon( year 6) and on the total wealth position of the portfolio holder.
Portfolio 1 100,000,000 105,660,731 0 5,660,731 197,629,369 Portfolio 2 100,000,000 108,914,787 -2,946,808 5,967,979 198,204,050 Portfolio 3 100,000,000 108,914,787 -3,128,792 5,785,995 197,863,664
Original Portfolio Value New Portfolio Value Gain/Loss on futures Total Wealth change Terminal Value of all funds at t=6 Annualized Holding Period Return over 6 years
1.120234
1.120776
1.120455
Transaction Costs for Planning Period Case
?
One important concern in the implementation of Immunization strategies is the transaction costs involved.
Portfolio 1 Number of Instruments Traded Bond A Bond C T-Bill Futures Contracts T-Bond Futures Contracts One-way Transaction Cost Bond A @ $2 Bond C @ $2 T-Bill Futures @ $15 T-Bond Futures @ $15 Total Cost of becoming Immunized 63,844 -122,103 -
Portfolio 2 1,352 -
Portfolio 3 527
127,688 244,206 $371,894
20,280 $20,280
7,905 $7,905
?
Clearly, there is a tremendous difference between trading the cash and futures instruments. In an extreme example of this type, the transaction cost for “Bonds Only” case is prohibitive, amounting to almost 4% of the total portfolio value. The volume of Bonds to be traded is enormous, exceeding any reasonable volume for bonds of even the largest issue.
?
References
? ? ? ? ?
http://www.cunacfocouncil.org/news/435.htmlhttp://www.treasurer.ca.gov/Cdiac/publications/duration.pdfhttp://en.wikipedia.org/wiki/Bond_durationhttp://www.investopedia.com/university/advancedbond/advan cedbond5.asp Fixed Income Securities, Level 1&2, CFA Book 4
doc_178617508.ppt