Description
This is a presentation highlights the bonds basics, concepts like straight bond prices, yield to maturity, calculating yields, coupon rate, current yield, yield to maturity, duration concept.
BONDS-Fixed Income Securities
?Typically investors are fascinated by potential rewards associated with stocks ?Investors lack interest in Fixed Income Securities as an investment vehicle ?Bonds pay Fixed Income and their prices are not subject to wide fluctuations ?Topics ?Bond Basics ?Terminology ?Type Of Bonds ?Straight Bond Prices & Yield to Maturity ?Calculating Yields ?Coupon Rate, Current Yield, and YTM 1 ?Duration
Bond Characteristics ?Bond is a borrowing arrangement ?Borrower(of money) issues /sells an obligation “ I owe you” (IOU) to the investor ?Issuer makes specified payments on specified dates (Coupon payments) for the life of the bond ?Pay on maturity date Bonds par value(face value) ? A Bond is a security that offers a series of fixed interest payments, and a fixed principal payment at maturity
2
Terminology ? Par or Face value - Amount repaid at maturity; usually Rs1,000 ? Maturity – Date/Period on/after which principal (par) is repaid ? Coupon Rate - Determines the amount of periodic Cash Flows; interest is typically paid semi-annually or annually
?Coupon Rate =Coupon Payment/Bonds Par Value
3
Bond Example ?Bond with par value Rs.1000 ?Coupon Rate =9% per annum paid semi - annually ?Maturity Period = 20 years ?Sold to Buyer at Rs.1000
?Bondholder gets (9/2)% x Rs.1000 ?Rs.45 / 6 months for 40 periods ?After 20 years, Principal Amount of Rs.1000
4
Types of Bonds ? Coupon (Fixed Interest) as discussed previously ? Zero Coupon - no periodic coupon payment
? Sold at discount; matures at par ? Also called „Pure Discount Bond?; like T-Bill ? Original Issue Discount (OID)
? Floating Rate - coupon rate is reset periodically depending on market
? Ex: T-Bill + 2%
5
? Callable Bond allows issuer to repurchase bond at a specified call price prior to maturity ?Call Date: date after which bond may be bought back ?Typically bonds are available with call protection initial time during which bonds are not callable ?Call Price: pre specified price; usually a premium is paid ?Usually after falling rates ? Puttable Bond allows an option to Buyer/Holder to sell bond back to the issuer ?Usually after rising rates
6
? Convertible Bonds gives an option to Buyer/Holder to exchange bonds for pre specified # of shares after a pre specified period of time ?Lower coupons & YTM ? Investment Grade Bonds rated BBB (S&P) or Baa (Moody’s) and above ? Junk Bonds: speculative grade; rated below investment grade, highly risky in nature ? Debenture - unsecured bonds, could be convertible or non convertible
7
? International - Bonds issued in a country different from borrower (eg: Resurgent India Bonds in the US) ? Eurobond – Bonds issued in currency of one country but sold in another ?eg: Eurodollar bonds - dollar denominated sold outside US
8
BOND PRICING
? The present value of a claim on future Cash Flows is the market price of that claim ? Since a bond is a claim on future Cash Flows, the present value of the claim is the price ? The coupon is an annuity and principal is a single cash flow ?Some Basic Facts!! ?There is an inverse relationship between market interest rates and bond prices ? If rates go up, bond prices MUST go down; if rates go down, bond prices MUST rise
9
Financial Asset Values
0 k Value CF1 CF2 1 2 n
...
CFn
PV =
?1 + k?
CF1
1
+
?1+ k?
CF2
2
+ ... +
?1 + k?
CFn
n
.
10
Basics of Bond Pricing
• Symbolically: Price =
C P ? ? i M i ?1 ( ? R ) 1 (1? R)
M
Where:
M = maturity R = required return on bond C = coupon payment ( rate of interest as specified) P = Principal
11
What?s the value of a 10-year, 10% coupon bond if kd = 10%?
0 10% V=? 100 100 1 2 10
...
100 + 1,000
VB ?
Rs100
?1 + k d ?
1
+ . . . +
Rs100
?1 + k d ?
10
+
Rs1 ,1000
?1+ k d ?
10
= Rs90.91 + = Rs1,000.
. . . + Rs38.55 + Rs385.54
12
Or…conceptually….
• The present value of an annuity (the stream of interest payments) plus • The present value of a single future value (the maturity value).
13
Bond Yields
? Coupon rate or coupon Yield ? Current Yield ? Yield to Maturity
14
BOND YIELDS
Annual coupon Coupon rate ? Par value
Annual coupon Current yield ? Bond price
Example Assume a bond has 12 years to maturity, a 8% coupon (paid annually), and the price is Rs1039.11. Coupon rate = Rs80/Rs1000= 8% Current yield = Rs80/Rs1039.11=7.7%
15
Yield to Maturity(YTM) ? The discount rate that equates a bond?s price with the present value of all future cash flows ? Average annual rate earned by investor if bond is held to maturity (and if coupons reinvested at same rate) ? Also referred as investor?s required or expected return on bond YTM is calculated assuming
1. 2. 3. Investors purchased the bond at current price The bond is held until its maturity Coupons received intermittently will be reinvested at the same rate
16
Consider the following example: Investors A & B are considering the purchase of a 5year Rs. 1000 par value bond, with the Coupon Rate of 7 %. Investor A?s required rate of return is 8 %, Investors B?s rate of return is 6%
What will they be willing to pay to purchase the bond if it matures at par?
The Interest received annually by the Investor = Rs. 70 & Investor shall receive Rs.1000 on maturity
17
Net Present Value (NPV) of the Bond can be found :B 0 = Int 1 + Int 2 + Int 3 .................. INtn + B n (1 + Kd) (1 + Kd)2 (1 + Kd)3 (1+Kd)n (1+Kd)n Where B0 =Present Value of Bond / debenture. INT1 =Amount of interest in period Kd =Required rate of return on bond ( % ) Bn =Terminal or maturity, value in period N = number of years to maturity.
18
A
= 70 +
70
+ 70 +
70 +
70 +
1000
(8%) (1.08 )1 ( 1.08 )2 (1.08 )3 (1.08 ) 4 (1.08 )5 (1.08 )5 A = Rs. 70 x 3.993 + 1000 x .681 (8%) = Rs. 279.51 + Rs. 681 = Rs. 960.51 B = 70 + 70 + 70 + 70 + 70 + 1000 (6%) (1.06 )1 ( 1.06 )2 (1.06 )3 (1.06)4 (1.06)5 (1.06)5
B (6%)
= Rs. 70 x 4.212 + 1000 x .747 = Rs. 294.84 + Rs. 747 = Rs. 1041.84
19
Bond Value Properties
• The following factors have an effect on Bond Value
1. Relationship between required rate of return & coupon rate 2. Number of years to maturity 3. Yield to Maturity
1. Relationship between required rate of return & coupon rate
? Inverse relationship between Required Rate of Return (market interest rates) and Bond Prices
When Required Rate of Return = Coupon Rate Value of Bond=Par Value When Required Rate of Return >Coupon Rate Value of Bond<Par Value When Required Rate of Return <Coupon Rate Value of Bond>Par Value
20
Bond Value Properties (continued)
2. Effect of Number of years to maturity on Bond Values
?
When required rate of return is greater than the coupon rate the discount on the bond declines as maturity approaches
When required rate of return is less than the coupon rate the premium on the bond declines as maturity approaches
?
Premium Bond
Price Time Maturity Date Discount Bond
21
Bond Value Properties (continued)
3.
3.1
Effect of YTM on Bond’s Price
For a given change in YTM, Longer maturity Bonds post larger price changes
X Face Value Y
Rs.1000 Rs.1000
Coupon Rate
Maturity Period YTM Effect on Value if YTM changes to 10% Price of Bond at YTM 10% % change in Bonds Price
8%
4years 8%
8%
6 years 8%
936.6 6.34%
912.4 8.76%
22
Bond Value Properties (continued)
?
3.2
Effect of YTM on Bond’s Price
Bond Price movements resulting from equal absolute increases or decreases in yield are not symmetrical. ? A decrease in yield raises Bond Prices by more than an increase in yield of the same amount lowers price This property is called as Convexity because of the shape of the Bond Price Curve
YTM =8% YTM =10% Face Value Coupon Rate Rs.1000 8% 4years Rs.1000 4years Rs. 936.60 6.34% Rs.1000 8% 4 years Rs.1069.20 6.92% YTM =6% Rs.1000 8%
Maturity Period 4years
Price of Bond for given YTM % change in Bond's Price
23
Bond Value Properties (continued)
?
3.3
Effect of YTM on Bond’s Price
For a given change in YTM, Percentage Price change in case of Bonds with High Coupon Rate will be smaller than in case of Bonds with low coupon rate
X Face Value Coupon Rate Maturity Period YTM Rs.1000 8% 4years 8% Y Rs.1000 9% 4years 8%
Effect on Value if YTM changes to 10%
Price of Bond at YTM 10% % change in Bonds Price 936.6 6.34% 968.3 3.17%
24
Bond Value Properties (continued)
?
3.4
Effect of YTM on Bond’s Price
A change in YTM affects the Bond with higher YTM more than Bond with lower YTM
X Face Value Coupon Rate Maturity Period YTM Price of Bond Price of Bond if YTM increases by 20% Rs.1000 Rs. 948.84 YTM=9.6% 4years 8% Rs.1000 8%
Y Rs.1000 9% 4years 10% Rs.936.60 Rs. 878.96 YTM=12%
% change in Bonds Price
5.12%
6.15%
25
Bond Maturity & Interest rate risk
The value of Bond depends upon the interest rate. There is an inverse relationship between value of bond & interest rate. ?The intensity of interest rate risk will be higher for bonds with long term maturities than bonds for short term periods ?Price of Bonds with smaller Coupon Rate tend to be more volatile
?However, Bonds may have different coupons and different maturity dates
?YTM may be considered, but does not indicate which bond prices tend to be more volatile ?YTM on Bonds with different maturities and different coupons cannot be compared ?HOW DO YOU COMPARE THEM
26
Example: Compare Coupon Bond to Zero Coupon Bond
(Both Bonds are of 10 year maturity) Coupon Bond has lower sensitivity. Why?
It is not just a 10 year maturity instrument, but A contract for a series of 20 payments of different maturities. ? Most of these payments occur well before the maturity of the bond ? The effective maturity is sort of an average of all these maturities
This Effective Maturity is DURATION Duration determines each Bonds sensitivity to changes in interest rates
27
Duration is defined As the average time it takes the Bondholder to receive the interest and principal It is the weighted average that encompasses the total amount of Bonds payments and their timings and then standardizes for Bonds prices
Weight= [ CFt / (1+y)t ] Price of Bond
y=YTM CFt= Cash Flow at any given time t t=time until payment
T
PV of that payment Price of Bond
Duration = ? t*Weight
t=1
28
? Weight of each payment is measured by the importance of that payment to the Value of the Bond ? Weight of each payment is the proportion of the total value of the bond accounted for by that payment ? This proportion is the present Value of the payment divided by the Bonds Price
29
Calculation of Duration Bond A 8% Semi Annual Coupon Bond 2 years Time to Maturity. Prevailing Discount Rate 10%
Time until Payment (years) 0.5 1 1.5 2
Payment (Rs.) 40 40 40 1040
Discounted Duration=Weightx Value of Time until Payment Weight Payment 38.10 36.28 34.55 855.61 0.0395 0.0376 0.0358 0.8871 0.0197 0.0376 0.0537 1.7741
Sum
964.54
1.000
1.8852
30
Calculation of Duration Bond B Zero Coupon Bond 2 years Time to Maturity. Prevailing Discount Rate 10%
Time until Payment (years) 0.5 1 1.5 2
Payment (Rs.) 0 0 0 1000
Discounted Duration=Weightx Value of Time until Payment Weight Payment 0 0 0 822.70 0 0 0 1 0 0 0 2
Sum
822.70
1
2
31
Duration Properties
? Sensitivity of a Bond Price to changes in interest rates is influenced by 3 key factors ?Time to Maturity ?Coupon Rates ?Yield to Maturity
? Rule 1 –Duration of a Zero Coupon Bond equals Time to Maturity ? See previous example 2 year coupon Bond has lower Duration than Zero Coupon ? Early coupon reduces the bonds weighted average time until payments
32
DURATION PROPERTIES (continued) ? Rule 2 – Holding Maturity constant, a Bonds Duration is higher when the coupon rate is lower. ? The higher the coupons the more they reduce the weighted average maturity of the payments ? Rule 3 – Holding the Coupon Rate constant, a Bonds Duration increases with its Time to Maturity ? Rule 4- Holding other factors constant, Duration of a Coupon Bond is higher when the Bonds YTM is lower ? At lower yields the more distant payments have a relatively greater Present Value (PV) ? In weighted average calculation distant payments receive greater weights
33
FIXED INCOME PORTFOLIO MANAGEMENT ? Bonds pay a Fixed Income and mature at a specified date ? Intuitively - Leads to a Buy & Hold Strategy –Passive Management ? However, investors may follow Active Management, actively buying and selling ? Passive Management Takes Bond Prices as fairly set and seeks to control only the risk of the Fixed Income Portfolio 2 strategies are followed: 1. Indexing Strategy ? Attempts to replicate the performance of a Bond Index ? Has the same Risk Reward profile as Bond Market Index ? Index includes vast number of securities, difficult to purchase each security in the index in proportion to its 34 Market Value. Some Bonds are thinly traded
Bond Index Portfolio – similar to Stock Market Indexing ? More rebalancing problems as compared Stock Index . Bonds are continually dropped from the Index as their Maturities fall below 1 year ? As new Bonds are issued they are added to the Index Bonds used to compute the index are constantly changing ? Bond Index Portfolio limits the interest rate exposure to that of the whole market – as captured by the index
?Not Good enough for Financial Institutions (FI) ?FI want to protect the value of their portfolios ?Banks – maintain Net Worth for Regulatory reasons Pension funds must guarantee a certain future value when liabilities mature.
35
Bond Index Portfolio – Passive Management 2. Immunization Techniques ? Used by Pension Funds & Insurance Companies ? Protects the Financial Status from interest rate fluctuations ? Attempt a Zero Risk Profile. Interest rate fluctuations have no effect on value of the firm Net Worth Immunization Techniques Typically Banks have a mismatch between asset & Liability maturity structures ? Liabilities – Deposits by Customers; ST; Short Duration ?Assets – Commercial & Consumer Loans LT in nature; Longer Duration Assets are of Longer Duration than Deposits Therefore Asset Value are more sensitive to Interest Rate fluctuations Interest Rate rise Asset Value drops much more than Liabilities
36
Net Worth Immunization (cont’d) ? Gap Management developed to reduce the gap between the Asset and Liability Durations ? Floating Rate Loans reduce the Duration of Bank Asset Portfolios ? Such loans do not fall in value when market interest rate rise ? On the Liability side of B/S, Fixed Deposits lengthen the Duration reducing the Gap ? Gap Management ? Banks attempt to equate the Duration of Assets and Liabilities to immunize itself from interest rate risk ? If Duration of Assets & Liabilities is kept equal and since Assets & Liabilities are equal in size, changes in interest rates affect values of A &L equally ? No effect on Net Worth, if Gap (Portfolio Duration) = ZERO 37
Bond Index Portfolio – Passive Management Target Date Immunization
? Pension Funds, Insurance Companies use this Obligation to provide flow of income after retirement ? As interest rates fluctuate, Value of assets held by fund and the rate at which they generate income also fluctuates ? Therefore the need to protect or immunize the future value at some target date against interest rate movements ? With Duration matched Assets & Liabilities, Ability of the firm to meet its obligations should be unaffected by interest rate movements
38
Example: Pension fund has to pay Rs.14,693.28, 5years later, Current Market Interest Rate=8% PV of Obligation=Rs.10,000 Firm chooses to fund this obligation with Rs.10,000 of 8% Annual Coupon Bonds, selling at par with 6 years to Maturity. Duration of the Bond is 5 years
Single Payment obligation is immunized by the Bond Duration matching balances the reinvestment rate Risk (accumulated value of coupon payments) & Price Risk (the Sale Value of the Bond)
39
Terminal Value of Bond Portfolio after 5 Years (All proceeds reinvested) Rs.10,000, 8% Annual Coupon Bonds, Selling at Par, Six years to maturity
Payment Number Rates at 8% 1 2 3
Time Remainig Accumulated until Value of Obligatio Invested n Payment 4 3 2 800x(1.08)^4 = 800x(1.08)^3 = 800x(1.08)^2 = 1088.39 1007.77 933.12
4 5 Sale of Bond
1 0 0
800x(1.08)^1 = 800x(1.08)^0 = 10800/1.08 =
864.00 800.00 10000.00 40 14693.28
Rates fall to 7% Time Payment remaini Number ng until Rates at obligati 7% on 1 4 2 3 3 2 4 1 5 0 Sale of Bond 0
Accumulated Value of Invested Payment 800x(1.07)^4 = 800x(1.07)^3 =
1048.64 980.03
800x(1.07)^2 = 800x(1.07)^1 = 800x(1.07)^0 = 10800/1.07 =
915.92 856.00 800.00 10093.46 41 14694.05
Rates rise to 9%
Payment Number Rates at 9% 1 2 3 4 5 Sale of Bond
Time Remaining Until obligation 4 3 2 1 0 0
Accumulated Value of Invested Payment 800x(1.09)^4 = 800x(1.09)^3 = 800x(1.09)^2 = 800x(1.09)^1 = 800x(1.09)^0 = 10800/1.09 =
1129.27 1036.02 950.48 872.00 800.00 9908.26 14696.03
Other solution is to have Zero Coupon Bonds matching the obligations 42
Problems with Conventional Immunization
? ? ? Uses YTM of Bond to calculate weight applied to each coupon ? Duration is valid for Flat Yield Curve (all payments are discounted at a common interest rate) Solution is in Weight calculation for Duration, PV of each Cash Flow should be discounted at appropriate rate from Yield Curve Duration Matching immunizes the Portfolio for parallel shifts in Yield Curve ? Change in YTM is same across all maturities Immunization makes sense only in case of nominal liabilities ? Does not help to immunize a projected obligation that will grow with price levels Duration changes by mere passage of time; Therefore immunized portfolios need to be continually rebalanced Other than Zero Coupon Bonds, Duration is not a linear function of time
?
?
?
43
Active Bond Management
? 2 sources of Potential Value in Active Bond Management 1. 2. Interest Rate Forecasting Anticipate interest rate changes, accordingly Buy or Sell Identification of mis-priced securities. The default premium on a bond may be large, so under priced
Interest Rate Forecasting One type of forecasting is Horizon Analysis i. Particular Holding period is selected ii. Yield Curve at the end of the period is selected iii. Bonds Time to Maturity at the end of the holding Period is known iv. Yield will be found out from the predicted Yield Curve v. End of Period Price is calculated vi. Add coupon income and capital gain on bond to obtain total return on the bond over the Holding Period
44
Horizon Analysis
Example ? Consider a 20 year maturity 10% Coupon Bond, YTM =9% Price of this Bond is Rs.1090.9 Investor Horizon is 5years After 5 years 1. Bond will have 15 years left for Maturity 2. Predict the Yield on 15 years to determine Bonds Expected Price 3. YTM expected for 15 year Bond after 5 years is 8% 4. Expected Price after 5 years is Rs.1170.9 5. Capital Gain is Rs.80 6. Intermediate Coupons paid every year also reinvested at say 8% 7. Value of Reinvested Coupons is Rs.586.7 8. Total Return provided by the Bond is Rs.80+Rs.586.7=Rs.666.7 9. Total Return is 61.11% 10. Repeat the procedure for many Bonds and select the one promising superior Holding Period Return 45
Horizon Analysis-Riding the Yield Curve
? Consider an upward sloping Yield Curve, Projection is that the Yield Curve will not shift during the Horizon Period ? As Bond maturities fall with passage of time, Yield will also fall as they “ride” the Yield Curve towards the Lower Yields of ST Bonds ? Decrease in Yield leads to Capital Gains on Bonds Example ? Consider 10 year maturity Bond, YTM =9% 9 year Maturity Bond , YTM = 8.8%
Consider a Zero Coupon Bond with Face Value Rs.1000 Price of the Bond is Rs.1000/1.09^10 =Rs.422.41
After 1 year if yield on 9 year Bonds are still 8.8% Price of 9 year Zero Coupon Bond will be 1000/1.088^9 = Rs.468.10 Return on Bond is 10.82% If Bonds YTM remained at 9% Price would be Rs.1000/1.09^9 = Rs.460.43 Return = 9% 46 PROBLEM What if Yield Curve Rises over Time
Contingent Immunization
Mixed Strategy – Active & Passive Conservative – T –Bills, but if little greedy, but with downward protection go for Contingent Immunization Example: Investor Horizon 2years Portfolio size Rs. 10million, YTM = 10% Using Conventional immunization, Value of Portfolio after 2 years V portfolio = Rs. 10million*1.10^2 = Rs.12.1million can be achieved ? ?
If little greedy & also ready to lose some value but with downward protection; Go For Contingent Immunization.
To achieve higher returns – Active Management required Investor ready to suffer losses only to the extent that Terminal Value of Portfolio does not drop below Rs. 11million ? Money required to achieve your goal = Rs.11million/1.1^2 = Rs. 9.09million This takes care of fear component that Portfolio Value does not drop below Rs.11 million after 2 years ? ? Now look at the GREED Part.
47
Contingent Immunization (continued)
Amount of Money Investor can afford to lose is Rs.10million - Rs.9.09million = Rs.0.91 million Investor can start with active management and can afford some losses at the outset. Calculate the funds required to lock in a future value of Rs.11million at the current interest rate T= Time left until Horizon R = market interest rate Value of funds required to reach desired Terminal Value = Rs.11million/(1+r)^T At this value if the Portfolio value is immunized it will grow risk free to Rs.11million by the Horizon date
Rs.11million/(1+r)^T is the Trigger Point, if actual Value dips to this point, STOP Active Management immunize the Portfolio and let the Value grow to Rs.11million
48
• Yield to Call • Components of Total Return on a Bond
49
doc_472006954.ppt
This is a presentation highlights the bonds basics, concepts like straight bond prices, yield to maturity, calculating yields, coupon rate, current yield, yield to maturity, duration concept.
BONDS-Fixed Income Securities
?Typically investors are fascinated by potential rewards associated with stocks ?Investors lack interest in Fixed Income Securities as an investment vehicle ?Bonds pay Fixed Income and their prices are not subject to wide fluctuations ?Topics ?Bond Basics ?Terminology ?Type Of Bonds ?Straight Bond Prices & Yield to Maturity ?Calculating Yields ?Coupon Rate, Current Yield, and YTM 1 ?Duration
Bond Characteristics ?Bond is a borrowing arrangement ?Borrower(of money) issues /sells an obligation “ I owe you” (IOU) to the investor ?Issuer makes specified payments on specified dates (Coupon payments) for the life of the bond ?Pay on maturity date Bonds par value(face value) ? A Bond is a security that offers a series of fixed interest payments, and a fixed principal payment at maturity
2
Terminology ? Par or Face value - Amount repaid at maturity; usually Rs1,000 ? Maturity – Date/Period on/after which principal (par) is repaid ? Coupon Rate - Determines the amount of periodic Cash Flows; interest is typically paid semi-annually or annually
?Coupon Rate =Coupon Payment/Bonds Par Value
3
Bond Example ?Bond with par value Rs.1000 ?Coupon Rate =9% per annum paid semi - annually ?Maturity Period = 20 years ?Sold to Buyer at Rs.1000
?Bondholder gets (9/2)% x Rs.1000 ?Rs.45 / 6 months for 40 periods ?After 20 years, Principal Amount of Rs.1000
4
Types of Bonds ? Coupon (Fixed Interest) as discussed previously ? Zero Coupon - no periodic coupon payment
? Sold at discount; matures at par ? Also called „Pure Discount Bond?; like T-Bill ? Original Issue Discount (OID)
? Floating Rate - coupon rate is reset periodically depending on market
? Ex: T-Bill + 2%
5
? Callable Bond allows issuer to repurchase bond at a specified call price prior to maturity ?Call Date: date after which bond may be bought back ?Typically bonds are available with call protection initial time during which bonds are not callable ?Call Price: pre specified price; usually a premium is paid ?Usually after falling rates ? Puttable Bond allows an option to Buyer/Holder to sell bond back to the issuer ?Usually after rising rates
6
? Convertible Bonds gives an option to Buyer/Holder to exchange bonds for pre specified # of shares after a pre specified period of time ?Lower coupons & YTM ? Investment Grade Bonds rated BBB (S&P) or Baa (Moody’s) and above ? Junk Bonds: speculative grade; rated below investment grade, highly risky in nature ? Debenture - unsecured bonds, could be convertible or non convertible
7
? International - Bonds issued in a country different from borrower (eg: Resurgent India Bonds in the US) ? Eurobond – Bonds issued in currency of one country but sold in another ?eg: Eurodollar bonds - dollar denominated sold outside US
8
BOND PRICING
? The present value of a claim on future Cash Flows is the market price of that claim ? Since a bond is a claim on future Cash Flows, the present value of the claim is the price ? The coupon is an annuity and principal is a single cash flow ?Some Basic Facts!! ?There is an inverse relationship between market interest rates and bond prices ? If rates go up, bond prices MUST go down; if rates go down, bond prices MUST rise
9
Financial Asset Values
0 k Value CF1 CF2 1 2 n
...
CFn
PV =
?1 + k?
CF1
1
+
?1+ k?
CF2
2
+ ... +
?1 + k?
CFn
n
.
10
Basics of Bond Pricing
• Symbolically: Price =
C P ? ? i M i ?1 ( ? R ) 1 (1? R)
M
Where:
M = maturity R = required return on bond C = coupon payment ( rate of interest as specified) P = Principal
11
What?s the value of a 10-year, 10% coupon bond if kd = 10%?
0 10% V=? 100 100 1 2 10
...
100 + 1,000
VB ?
Rs100
?1 + k d ?
1
+ . . . +
Rs100
?1 + k d ?
10
+
Rs1 ,1000
?1+ k d ?
10
= Rs90.91 + = Rs1,000.
. . . + Rs38.55 + Rs385.54
12
Or…conceptually….
• The present value of an annuity (the stream of interest payments) plus • The present value of a single future value (the maturity value).
13
Bond Yields
? Coupon rate or coupon Yield ? Current Yield ? Yield to Maturity
14
BOND YIELDS
Annual coupon Coupon rate ? Par value
Annual coupon Current yield ? Bond price
Example Assume a bond has 12 years to maturity, a 8% coupon (paid annually), and the price is Rs1039.11. Coupon rate = Rs80/Rs1000= 8% Current yield = Rs80/Rs1039.11=7.7%
15
Yield to Maturity(YTM) ? The discount rate that equates a bond?s price with the present value of all future cash flows ? Average annual rate earned by investor if bond is held to maturity (and if coupons reinvested at same rate) ? Also referred as investor?s required or expected return on bond YTM is calculated assuming
1. 2. 3. Investors purchased the bond at current price The bond is held until its maturity Coupons received intermittently will be reinvested at the same rate
16
Consider the following example: Investors A & B are considering the purchase of a 5year Rs. 1000 par value bond, with the Coupon Rate of 7 %. Investor A?s required rate of return is 8 %, Investors B?s rate of return is 6%
What will they be willing to pay to purchase the bond if it matures at par?
The Interest received annually by the Investor = Rs. 70 & Investor shall receive Rs.1000 on maturity
17
Net Present Value (NPV) of the Bond can be found :B 0 = Int 1 + Int 2 + Int 3 .................. INtn + B n (1 + Kd) (1 + Kd)2 (1 + Kd)3 (1+Kd)n (1+Kd)n Where B0 =Present Value of Bond / debenture. INT1 =Amount of interest in period Kd =Required rate of return on bond ( % ) Bn =Terminal or maturity, value in period N = number of years to maturity.
18
A
= 70 +
70
+ 70 +
70 +
70 +
1000
(8%) (1.08 )1 ( 1.08 )2 (1.08 )3 (1.08 ) 4 (1.08 )5 (1.08 )5 A = Rs. 70 x 3.993 + 1000 x .681 (8%) = Rs. 279.51 + Rs. 681 = Rs. 960.51 B = 70 + 70 + 70 + 70 + 70 + 1000 (6%) (1.06 )1 ( 1.06 )2 (1.06 )3 (1.06)4 (1.06)5 (1.06)5
B (6%)
= Rs. 70 x 4.212 + 1000 x .747 = Rs. 294.84 + Rs. 747 = Rs. 1041.84
19
Bond Value Properties
• The following factors have an effect on Bond Value
1. Relationship between required rate of return & coupon rate 2. Number of years to maturity 3. Yield to Maturity
1. Relationship between required rate of return & coupon rate
? Inverse relationship between Required Rate of Return (market interest rates) and Bond Prices
When Required Rate of Return = Coupon Rate Value of Bond=Par Value When Required Rate of Return >Coupon Rate Value of Bond<Par Value When Required Rate of Return <Coupon Rate Value of Bond>Par Value
20
Bond Value Properties (continued)
2. Effect of Number of years to maturity on Bond Values
?
When required rate of return is greater than the coupon rate the discount on the bond declines as maturity approaches
When required rate of return is less than the coupon rate the premium on the bond declines as maturity approaches
?
Premium Bond
Price Time Maturity Date Discount Bond
21
Bond Value Properties (continued)
3.
3.1
Effect of YTM on Bond’s Price
For a given change in YTM, Longer maturity Bonds post larger price changes
X Face Value Y
Rs.1000 Rs.1000
Coupon Rate
Maturity Period YTM Effect on Value if YTM changes to 10% Price of Bond at YTM 10% % change in Bonds Price
8%
4years 8%
8%
6 years 8%
936.6 6.34%
912.4 8.76%
22
Bond Value Properties (continued)
?
3.2
Effect of YTM on Bond’s Price
Bond Price movements resulting from equal absolute increases or decreases in yield are not symmetrical. ? A decrease in yield raises Bond Prices by more than an increase in yield of the same amount lowers price This property is called as Convexity because of the shape of the Bond Price Curve
YTM =8% YTM =10% Face Value Coupon Rate Rs.1000 8% 4years Rs.1000 4years Rs. 936.60 6.34% Rs.1000 8% 4 years Rs.1069.20 6.92% YTM =6% Rs.1000 8%
Maturity Period 4years
Price of Bond for given YTM % change in Bond's Price
23
Bond Value Properties (continued)
?
3.3
Effect of YTM on Bond’s Price
For a given change in YTM, Percentage Price change in case of Bonds with High Coupon Rate will be smaller than in case of Bonds with low coupon rate
X Face Value Coupon Rate Maturity Period YTM Rs.1000 8% 4years 8% Y Rs.1000 9% 4years 8%
Effect on Value if YTM changes to 10%
Price of Bond at YTM 10% % change in Bonds Price 936.6 6.34% 968.3 3.17%
24
Bond Value Properties (continued)
?
3.4
Effect of YTM on Bond’s Price
A change in YTM affects the Bond with higher YTM more than Bond with lower YTM
X Face Value Coupon Rate Maturity Period YTM Price of Bond Price of Bond if YTM increases by 20% Rs.1000 Rs. 948.84 YTM=9.6% 4years 8% Rs.1000 8%
Y Rs.1000 9% 4years 10% Rs.936.60 Rs. 878.96 YTM=12%
% change in Bonds Price
5.12%
6.15%
25
Bond Maturity & Interest rate risk
The value of Bond depends upon the interest rate. There is an inverse relationship between value of bond & interest rate. ?The intensity of interest rate risk will be higher for bonds with long term maturities than bonds for short term periods ?Price of Bonds with smaller Coupon Rate tend to be more volatile
?However, Bonds may have different coupons and different maturity dates
?YTM may be considered, but does not indicate which bond prices tend to be more volatile ?YTM on Bonds with different maturities and different coupons cannot be compared ?HOW DO YOU COMPARE THEM
26
Example: Compare Coupon Bond to Zero Coupon Bond
(Both Bonds are of 10 year maturity) Coupon Bond has lower sensitivity. Why?
It is not just a 10 year maturity instrument, but A contract for a series of 20 payments of different maturities. ? Most of these payments occur well before the maturity of the bond ? The effective maturity is sort of an average of all these maturities
This Effective Maturity is DURATION Duration determines each Bonds sensitivity to changes in interest rates
27
Duration is defined As the average time it takes the Bondholder to receive the interest and principal It is the weighted average that encompasses the total amount of Bonds payments and their timings and then standardizes for Bonds prices
Weight= [ CFt / (1+y)t ] Price of Bond
y=YTM CFt= Cash Flow at any given time t t=time until payment
T
PV of that payment Price of Bond
Duration = ? t*Weight
t=1
28
? Weight of each payment is measured by the importance of that payment to the Value of the Bond ? Weight of each payment is the proportion of the total value of the bond accounted for by that payment ? This proportion is the present Value of the payment divided by the Bonds Price
29
Calculation of Duration Bond A 8% Semi Annual Coupon Bond 2 years Time to Maturity. Prevailing Discount Rate 10%
Time until Payment (years) 0.5 1 1.5 2
Payment (Rs.) 40 40 40 1040
Discounted Duration=Weightx Value of Time until Payment Weight Payment 38.10 36.28 34.55 855.61 0.0395 0.0376 0.0358 0.8871 0.0197 0.0376 0.0537 1.7741
Sum
964.54
1.000
1.8852
30
Calculation of Duration Bond B Zero Coupon Bond 2 years Time to Maturity. Prevailing Discount Rate 10%
Time until Payment (years) 0.5 1 1.5 2
Payment (Rs.) 0 0 0 1000
Discounted Duration=Weightx Value of Time until Payment Weight Payment 0 0 0 822.70 0 0 0 1 0 0 0 2
Sum
822.70
1
2
31
Duration Properties
? Sensitivity of a Bond Price to changes in interest rates is influenced by 3 key factors ?Time to Maturity ?Coupon Rates ?Yield to Maturity
? Rule 1 –Duration of a Zero Coupon Bond equals Time to Maturity ? See previous example 2 year coupon Bond has lower Duration than Zero Coupon ? Early coupon reduces the bonds weighted average time until payments
32
DURATION PROPERTIES (continued) ? Rule 2 – Holding Maturity constant, a Bonds Duration is higher when the coupon rate is lower. ? The higher the coupons the more they reduce the weighted average maturity of the payments ? Rule 3 – Holding the Coupon Rate constant, a Bonds Duration increases with its Time to Maturity ? Rule 4- Holding other factors constant, Duration of a Coupon Bond is higher when the Bonds YTM is lower ? At lower yields the more distant payments have a relatively greater Present Value (PV) ? In weighted average calculation distant payments receive greater weights
33
FIXED INCOME PORTFOLIO MANAGEMENT ? Bonds pay a Fixed Income and mature at a specified date ? Intuitively - Leads to a Buy & Hold Strategy –Passive Management ? However, investors may follow Active Management, actively buying and selling ? Passive Management Takes Bond Prices as fairly set and seeks to control only the risk of the Fixed Income Portfolio 2 strategies are followed: 1. Indexing Strategy ? Attempts to replicate the performance of a Bond Index ? Has the same Risk Reward profile as Bond Market Index ? Index includes vast number of securities, difficult to purchase each security in the index in proportion to its 34 Market Value. Some Bonds are thinly traded
Bond Index Portfolio – similar to Stock Market Indexing ? More rebalancing problems as compared Stock Index . Bonds are continually dropped from the Index as their Maturities fall below 1 year ? As new Bonds are issued they are added to the Index Bonds used to compute the index are constantly changing ? Bond Index Portfolio limits the interest rate exposure to that of the whole market – as captured by the index
?Not Good enough for Financial Institutions (FI) ?FI want to protect the value of their portfolios ?Banks – maintain Net Worth for Regulatory reasons Pension funds must guarantee a certain future value when liabilities mature.
35
Bond Index Portfolio – Passive Management 2. Immunization Techniques ? Used by Pension Funds & Insurance Companies ? Protects the Financial Status from interest rate fluctuations ? Attempt a Zero Risk Profile. Interest rate fluctuations have no effect on value of the firm Net Worth Immunization Techniques Typically Banks have a mismatch between asset & Liability maturity structures ? Liabilities – Deposits by Customers; ST; Short Duration ?Assets – Commercial & Consumer Loans LT in nature; Longer Duration Assets are of Longer Duration than Deposits Therefore Asset Value are more sensitive to Interest Rate fluctuations Interest Rate rise Asset Value drops much more than Liabilities
36
Net Worth Immunization (cont’d) ? Gap Management developed to reduce the gap between the Asset and Liability Durations ? Floating Rate Loans reduce the Duration of Bank Asset Portfolios ? Such loans do not fall in value when market interest rate rise ? On the Liability side of B/S, Fixed Deposits lengthen the Duration reducing the Gap ? Gap Management ? Banks attempt to equate the Duration of Assets and Liabilities to immunize itself from interest rate risk ? If Duration of Assets & Liabilities is kept equal and since Assets & Liabilities are equal in size, changes in interest rates affect values of A &L equally ? No effect on Net Worth, if Gap (Portfolio Duration) = ZERO 37
Bond Index Portfolio – Passive Management Target Date Immunization
? Pension Funds, Insurance Companies use this Obligation to provide flow of income after retirement ? As interest rates fluctuate, Value of assets held by fund and the rate at which they generate income also fluctuates ? Therefore the need to protect or immunize the future value at some target date against interest rate movements ? With Duration matched Assets & Liabilities, Ability of the firm to meet its obligations should be unaffected by interest rate movements
38
Example: Pension fund has to pay Rs.14,693.28, 5years later, Current Market Interest Rate=8% PV of Obligation=Rs.10,000 Firm chooses to fund this obligation with Rs.10,000 of 8% Annual Coupon Bonds, selling at par with 6 years to Maturity. Duration of the Bond is 5 years
Single Payment obligation is immunized by the Bond Duration matching balances the reinvestment rate Risk (accumulated value of coupon payments) & Price Risk (the Sale Value of the Bond)
39
Terminal Value of Bond Portfolio after 5 Years (All proceeds reinvested) Rs.10,000, 8% Annual Coupon Bonds, Selling at Par, Six years to maturity
Payment Number Rates at 8% 1 2 3
Time Remainig Accumulated until Value of Obligatio Invested n Payment 4 3 2 800x(1.08)^4 = 800x(1.08)^3 = 800x(1.08)^2 = 1088.39 1007.77 933.12
4 5 Sale of Bond
1 0 0
800x(1.08)^1 = 800x(1.08)^0 = 10800/1.08 =
864.00 800.00 10000.00 40 14693.28
Rates fall to 7% Time Payment remaini Number ng until Rates at obligati 7% on 1 4 2 3 3 2 4 1 5 0 Sale of Bond 0
Accumulated Value of Invested Payment 800x(1.07)^4 = 800x(1.07)^3 =
1048.64 980.03
800x(1.07)^2 = 800x(1.07)^1 = 800x(1.07)^0 = 10800/1.07 =
915.92 856.00 800.00 10093.46 41 14694.05
Rates rise to 9%
Payment Number Rates at 9% 1 2 3 4 5 Sale of Bond
Time Remaining Until obligation 4 3 2 1 0 0
Accumulated Value of Invested Payment 800x(1.09)^4 = 800x(1.09)^3 = 800x(1.09)^2 = 800x(1.09)^1 = 800x(1.09)^0 = 10800/1.09 =
1129.27 1036.02 950.48 872.00 800.00 9908.26 14696.03
Other solution is to have Zero Coupon Bonds matching the obligations 42
Problems with Conventional Immunization
? ? ? Uses YTM of Bond to calculate weight applied to each coupon ? Duration is valid for Flat Yield Curve (all payments are discounted at a common interest rate) Solution is in Weight calculation for Duration, PV of each Cash Flow should be discounted at appropriate rate from Yield Curve Duration Matching immunizes the Portfolio for parallel shifts in Yield Curve ? Change in YTM is same across all maturities Immunization makes sense only in case of nominal liabilities ? Does not help to immunize a projected obligation that will grow with price levels Duration changes by mere passage of time; Therefore immunized portfolios need to be continually rebalanced Other than Zero Coupon Bonds, Duration is not a linear function of time
?
?
?
43
Active Bond Management
? 2 sources of Potential Value in Active Bond Management 1. 2. Interest Rate Forecasting Anticipate interest rate changes, accordingly Buy or Sell Identification of mis-priced securities. The default premium on a bond may be large, so under priced
Interest Rate Forecasting One type of forecasting is Horizon Analysis i. Particular Holding period is selected ii. Yield Curve at the end of the period is selected iii. Bonds Time to Maturity at the end of the holding Period is known iv. Yield will be found out from the predicted Yield Curve v. End of Period Price is calculated vi. Add coupon income and capital gain on bond to obtain total return on the bond over the Holding Period
44
Horizon Analysis
Example ? Consider a 20 year maturity 10% Coupon Bond, YTM =9% Price of this Bond is Rs.1090.9 Investor Horizon is 5years After 5 years 1. Bond will have 15 years left for Maturity 2. Predict the Yield on 15 years to determine Bonds Expected Price 3. YTM expected for 15 year Bond after 5 years is 8% 4. Expected Price after 5 years is Rs.1170.9 5. Capital Gain is Rs.80 6. Intermediate Coupons paid every year also reinvested at say 8% 7. Value of Reinvested Coupons is Rs.586.7 8. Total Return provided by the Bond is Rs.80+Rs.586.7=Rs.666.7 9. Total Return is 61.11% 10. Repeat the procedure for many Bonds and select the one promising superior Holding Period Return 45
Horizon Analysis-Riding the Yield Curve
? Consider an upward sloping Yield Curve, Projection is that the Yield Curve will not shift during the Horizon Period ? As Bond maturities fall with passage of time, Yield will also fall as they “ride” the Yield Curve towards the Lower Yields of ST Bonds ? Decrease in Yield leads to Capital Gains on Bonds Example ? Consider 10 year maturity Bond, YTM =9% 9 year Maturity Bond , YTM = 8.8%
Consider a Zero Coupon Bond with Face Value Rs.1000 Price of the Bond is Rs.1000/1.09^10 =Rs.422.41
After 1 year if yield on 9 year Bonds are still 8.8% Price of 9 year Zero Coupon Bond will be 1000/1.088^9 = Rs.468.10 Return on Bond is 10.82% If Bonds YTM remained at 9% Price would be Rs.1000/1.09^9 = Rs.460.43 Return = 9% 46 PROBLEM What if Yield Curve Rises over Time
Contingent Immunization
Mixed Strategy – Active & Passive Conservative – T –Bills, but if little greedy, but with downward protection go for Contingent Immunization Example: Investor Horizon 2years Portfolio size Rs. 10million, YTM = 10% Using Conventional immunization, Value of Portfolio after 2 years V portfolio = Rs. 10million*1.10^2 = Rs.12.1million can be achieved ? ?
If little greedy & also ready to lose some value but with downward protection; Go For Contingent Immunization.
To achieve higher returns – Active Management required Investor ready to suffer losses only to the extent that Terminal Value of Portfolio does not drop below Rs. 11million ? Money required to achieve your goal = Rs.11million/1.1^2 = Rs. 9.09million This takes care of fear component that Portfolio Value does not drop below Rs.11 million after 2 years ? ? Now look at the GREED Part.
47
Contingent Immunization (continued)
Amount of Money Investor can afford to lose is Rs.10million - Rs.9.09million = Rs.0.91 million Investor can start with active management and can afford some losses at the outset. Calculate the funds required to lock in a future value of Rs.11million at the current interest rate T= Time left until Horizon R = market interest rate Value of funds required to reach desired Terminal Value = Rs.11million/(1+r)^T At this value if the Portfolio value is immunized it will grow risk free to Rs.11million by the Horizon date
Rs.11million/(1+r)^T is the Trigger Point, if actual Value dips to this point, STOP Active Management immunize the Portfolio and let the Value grow to Rs.11million
48
• Yield to Call • Components of Total Return on a Bond
49
doc_472006954.ppt