1. Dr. Himanshu Joshi FORE School of Management New Delhi
Bond Price Volatility
Dr. Himanshu Joshi
FORE School of Management
New Delhi

2. Bond Price Volatility
To employ the effective bond portfolio strategies, it is necessary to understand the price volatility of bonds resulting from changes in interest rate.
In this session we will discuss the price volatility characteristics of a bond and to present some measures to quantify price volatility.

3. A 9% coupon bond with 5 years to maturity.
Review of the Price-Yield Relationship for Option Free Bonds C:\Documents and Settings\himanshu\Desktop\SRPM June-Sept 12\price-yield-relationship.xlsx
Six hypothetical bonds, where the bond prices are shown assuming a par value of $100 and interest paid semiannually:
A 9% coupon bond with 5 years to maturity.
A 9% coupon bond with 25 years to maturity.
A 6% coupon bond with 5 years to maturity.
A 6% coupon bond with 25 years to maturity.
A zero coupon bond with 5 years to maturity.
A zero coupon bond with 25 years to maturity.

4. Price-Yield Relationship

5. The Inverse Relationship Between Bond Prices and Yields

6. Price-Yield Relationship
Notice that as the required yield rises, the price of the option-free bond declines. This relationship is not linear, however. (i.e., it not a straight line). The shape of the price-yield relationship for any option free bond is referred to as convex.
The price-yield relationship that we have discussed refer to an instantaneous change in the required yield.
The price of bond will change over time as a result of: (1) a change in the perceived credit risk of the issuer.
(2) a discount or premium bond approaching the maturity date, and
(3) a change in the market interest rates.

7. Price Volatility Characteristics of Option Free Bond

8. Price Volatility Characteristics of Option Free Bond
The Exhibit above shows, for six hypothetical bonds, the percentage change in the bond’s price for various changes in the required yield, assuming that the initial yield for all six bonds is 9%. (for varying coupon rates of 9%, 6% and 0%.)
A close examination of exhibit above reveals several properties concerning the price volatility of an option free bond.

9. Price Volatility Characteristics of Option Free Bond: Some Properties:
Property 1. although the prices of all option free bonds move in the opposite direction from the change in the yield required, the percentage change is not the same for all the bonds.
Property 2. for very small changes in the yield required, the percentage price change for a given bond is roughly the same, whether required yield increases or decreases.

10. Price Volatility Characteristics of Option Free Bond: Some Properties:
Property 3. for large changes in required yield, the percentage price change is not the same for an increase in the required yield as it is for a decrease in the required yield.
Property 4. for a given large change in basis points, the percentage price increase (on decrease on req. yield) is greater than the percentage price decrease (on increase in req. yield).

11. Price Volatility Characteristics of Option Free Bond: Some Properties:
Implication of Property 4: (Convexity)
The implication of property 4 is that if an investor owns a bond (i.e., long a bond), the price appreciation that will be realized if the required yield decreases is greater than the capital loss that will be realized if required yield rises by the same number of basis points.
For an investor who is “short” a bond, the reverse is true: the potential capital loss is greater than the potential capital gain if the required yield changes by a given number of basis points.

12. Characteristics of a bond that affect its price volatility
Characteristic 1: for a given term to maturity and initial yield, the price volatility of a bond is greater, lower the coupon rate.
(compare 9%, 6% and zero coupon bond with same maturity.)
Characteristic 2: For a given coupon rate and initial yield, the longer the term to maturity, the greater the price volatility.
(compare five year bond with 25 year bond with the same coupon say 9%, 6% and 0% bonds)

13. Characteristics of a bond that affect its price volatility: An Implication
An implication of the second characteristic is that investors who want to increase a portfolio’s price volatility (they expect interest rate to fall), all other factor being constant, they should hold bonds with longer maturities in the portfolio.
To reduce a portfolio’s price volatility in anticipation of a rise in interest rates, bond with shorter-term maturities should be held in portfolio.

14. Effects of Yield to Maturity
Holding other factors constant, does the YTM affect a bond’s price volatility?
Higher the yield to maturity at which bond trades, the lower the price volatility.

15. Price Change for a 100 basis point change in yield for 9%, 25 year Bond trading at different Yield Levels

16. Measures of Bond Price Volatility
Money managers, arbitrageurs, and traders need to have a way to measure a bond’s price volatility to implement hedging and trading strategies.
Three measures are commonly employed are:
Price value of a basis point
Yield value of a basis point
Duration.

17. 1. Price Value of a Basis Point
Bond
Initial Price (9% yield)
Price at 9.01% yield
Price Value of a Basis Point
Price Value f a Basis Point/Initial Price
9% Coupon/5 Year
100
0.0396
9% Coupon/25 Year
0.0987
6% Coupon/5 Year
0.0364
6% Coupon/25 Year
0.0746
0% Coupon/5 Year
0.0308
0% Coupon/25 Year
0.0265
Because this measure of price volatility is in terms of dollar price change, dividing
The price value of a basis point by the initial price gives the percentage price
change for a 1 basis point change in yield.

18. 3. Duration
P =C/1+y + C/(1+y) C/(1+y)n + M/(1+y)n First differentiation is duration.

19. Duration
To deal with the ambiguity of the maturity of a bond making many payments, we need a measure of the average maturity of the bond’s promised cash flows to serve as useful summary statistics of the effective maturity of a bond.
We would like use the measure as a guide to sensitivity of a bond to interest rate changes.
Since we have noted that price sensitivity tends to increase with time to maturity.

20. Macaulay’s Duration A measure of the effective maturity of a bond
The weighted average of the times until each payment is received, with the weights proportional to the present value of the payment
Duration is shorter than maturity for all bonds except zero coupon bonds
Duration is equal to maturity for zero coupon bonds

21. Duration: Calculation

22. Duration: Calculation
Numerator: present value of the cash flow occurring at time t,
while the denominator is the value of all the payments forthcoming from the bond.
These weights sums to 1.0, because of the cash flows discounted at the Yield to maturity equals the bond price
Excel…..

23. Spreadsheet 16.1 Calculating the Duration of Two Bonds

24. Duration/Price Relationship
Price change is proportional to duration and not to maturity
D* = modified duration

25. Rule 5. Duration for level perpetuity
Duration of perpetuity = 1+y/y
At 10% yield, the duration of a perpetuity that pays $100 once a year forever is
1.10/.10 = 11 years.
But at an 8% yield
1.08/0.08 = 13.5 years.

26. Duration Calculation….

27. Macaulay Duration and Modified Duration for Six Hypothetical Bonds
9%, 5 Year
4.13
3.96
9%, 25 Year
10.33
9.88
6%, 5 Year
4.35
4.16
6%, 25 Year
11.10
10.62
0%, 5 Year
5.00
4.78
0%, 25 Year
25.00
23.92

28. Properties of Duration
As can be seen from the various durations computed for the six hypothetical bonds, the Modified duration and Macaulay Duration of a coupon bond are less than the maturity.
It is obvious from the formula that Macaulay Duration of the zero coupon bond is equal to its maturity.
A zero coupon bond’s modified duration is however, less than its maturity.
Also, lower the coupon, generally higher the modified and Macaulay duration of the bond.

29. Properties of Duration..
There is a consistency between the properties of bond price volatility we discussed earlier and the properties of modified duration:
We showed earlier that when all other factors remain unchanged, the longer the maturity, the greater the price volatility.
A property of Modified Duration is that when all other factors are constant, the longer the maturity, the greater the Modified Duration.

30. Properties of Duration..
2. We also showed that the lower the coupon rate, all other factors being constant, the greater the bond price volatility. In case of duration, the lower the coupon rate, the greater the Modified Duration, the greater the Price Volatility.

31. Properties of Duration..
3. Finally, as we noted earlier, another factor that will influence price volatility is YTM. All other factors being constant, the higher the yield level, the lower the price volatility. The same property holds for modified duration, as can be seen in the following table, which shows the modified duration of a 25 year 9% coupon bond at various yield levels:

32. Properties of Duration..
Yield (%)
Modified Duration 7
11.21 8
10.53 9
9.88 10
9.27 11
8.70 12
8.16 13
7.66

33. Approximating the Percentage Price Change..
dP/dy* 1/P = - Modified Duration
Multiplying both the side with dy (change in the required yield)
dP/P = - Modified Duration * dy
This equation can be used to approximate the percentage change for a given change in required yield.

34. Approximating the Percentage Price Change..Excel..
Example: Consider the 25 year 6% bond selling at to yield 9%. The modified duration for this bond is if yield increase instantaneously from 9% to 9.10%, a yield change of ( ) = (10 Basis Points)

35. Portfolio Duration
The duration of the portfolio is simply the weighted average of the bonds in the portfolios.
Bond
Market Value
Portfolio Weight
Duration A
$10 million
0.10 4 B
$40 million
0.40 7 C
$30 million
0.30 6 D
$20 million
0.2 2

36. Portfolio Duration 0.1*4 + 0.4*7 +0.3*6+ 0.2*2 = 5.4
The portfolio duration is 5.4 and interpretation is as follows:
If all the yields affecting the four bonds in the portfolio change by 100 basis points, the portfolio’s value will change by approximately 5.4%.
Portfolio managers look at their interest rate exposure to a particular issue in terms of its contribution to portfolio duration. This measure is found by multiplying the weight of the issue in the portfolio by duration of the individual issue:

37. Portfolio Duration
Contribution to portfolio Duration = weight of issue in portfolio * duration of issue.
A = 0.10* 4 = 0.40
B= 0.40* 7 = 2.8
-----

38. Portfolio Duration…
Moreover, portfolio managers look at portfolio duration for sectors of the bond market.
Sector
Portfolio Weight
Sector Duration
Contribution to Portfolio Duration
Treasury
0.00
4.95
Agency
0.121
3.44
0.42
Mortgages
0.449
3.58
1.61
Commercial Mortgage backed securities
0.139
5.04
0.70
Asset Backed Securities
0.017
3.16
0.05
Credit
0.274
6.35
1.74
Total
1.00
4.52

39. Convexity
The relationship between bond prices and yields is not linear
Duration rule is a good approximation for only small changes in bond yields

40. Bond Price Convexity: 30-Year Maturity, 8% Coupon; Initial Yield to Maturity = 8%

41. Correction for Convexity

42. Calculation of Convexity measure and dollar convexity measure for five year 9% bond selling to yield 9%.
Coupon Rate- 9.0%
Term (years)- 5 Years
Initial Yield- 9.0%
Price-$100

44. Example. Convexity
Bond in the figure has a 30 year maturity, an 8% coupon, and sells at an initial yield to maturity of 8%.
Because coupon rate equals YTM, the bond sells at par value $1000.
Modified duration for the bond is years, and convexity is
If the bond’s yield increases from 8% to 10%, bond price will fall to $

45. Example… Change of 18. 85% The duration rule ∆P/P = -D* ∆y
= * .02 = %.
Duration with Convexity:
∆P/P = -D* ∆y +1/2 * Convexity* (∆y)2
=11.26*.02 +1/2 * * (.02)2
= %.

47. Approximating A Bond’s Duration and Convexity Measure
Step 1. increase the yield on the bond by a small number of basis points and determine the new price at this higher yield level. Denote this price by P+.
Step 2. decrease the yield on the bond by same number of points and calculate the new price. We will denote this new price by P-.
Step 3. Letting P0 be the initial price, duration can be approximated using the following formula:

48. Approximating A Bond’s Duration and Convexity Measure
Approximate Duration = [P- - P+]/[2x(P0)(∆y)]
Approximate Convexity Measure
= [P+ + P- - 2P0]/[P0x(∆y)2]

50. PASSIVE BOND PORTFOLIO MANAGEMENT STRATEGIES: Immunization
In contrast to indexing strategies many institutions try to insulate their portfolios from interest rate risk altogether.
The net worth of the firm or ability to meet future obligations fluctuate with interest rates.
Immunization refers to strategies used by such investors to shield their overall financial status from interest rate fluctuations.
Example: Pension Funds (Fixed Future obligation) and Banks (asset Liability maturity mismatch).
The lesson is that funds should match the interest rate exposure of assets and liabilities so that the value of assets will track the value of liabilities whether rate rises or falls.

51. Immunization
The notion of immunization was introduced by F. M. Redington, an actuary for a life insurance company.
The idea behind immunization is that duration matched assets and liabilities let the assets portfolio meet the firm’s obligations despite interest rate movements.

52. Immunization..
The procedure is termed immunization because it “immunizes” the portfolio value against interest changes.
The procedure and its refinements, is in fact one of the most widely used analytical techniques of investment science, shaping portfolios consisting of billions of dollars of fixed income securities held by pension funds, insurance companies, and other financial institutions.

53. Immunization..
Let us more fully consider its purpose first. A portfolio can not be structured meaningfully without a statement of its purpose.
Suppose you wish to invest in money now that will be used next year for a major household expense. If you invest in 1 year T-bills you know exactly how much money these bills will be worth in a year, hence there is no relative risk to your purpose.
Conversely if you invest your money in 10 year T- Bill, value of this T-Bill after one year will be quite variable. (Price Risk)
The situation will be reversed if you were saving to pay off an obligation that was due in 10 years, then a 10 year zero coupon bond would provide completely predictable returns, but 1 year T-Bill would impose (Reinvestment Risk).

54. Immunization..
Now suppose that you face a series of cash obligations and you wish to acquire a portfolio that you will use to pay these obligations as they arise. (LICs).
One way to do this is to purchase a set of zero-coupon bonds that have maturities and face values exactly matching the separate obligations.
Not applicable with Corporate Bonds.
If perfect matching is not possible, you may instead acquire a portfolio having a value equal to the present value of the stream of obligations.
You can sell some of your portfolio whenever cash is needed to meet particular obligation; or if portfolio delivers more cash than needed at a given time, you can buy more bonds.
Hence you will meet the obligations exactly.
What is the limitation of this method?

55. Immunization..
A problem with this value matching technique arises if the yield change. The value of your portfolio and Present Value of Stream of cash flows both will change, but differently.
Your portfolio will no longer be matched.
Immunization solves this problem- at least approximately-by matching duration as well as present values.

56. Immunization..
If the duration of the portfolio matches that of the obligation stream, then the cash value of the portfolio and the present value of the obligation stream will respond identically to a change in yield.
If yield increases present value of the asset portfolio will decrease, but the present value of the obligations will decrease by approximately the same amount, so the value of portfolio will still be adequate to cover the obligation.

57. Pension Funds Lost Ground Despite Broad Market Gains
With the S&P 500 providing a rate of return in excess of 25%, 2003 was a banner year for the stock market. Not surprisingly, this performance showed up in the balance sheets of US pension funds: assets in these funds rose by more than $100 billion.
Despite this boost, the pension funds actually lost the ground in 2003, the gap between assets and liabilities growing by about $45 billion..
How could this happen?

58. Immunization Example
Suppose you are trying to immunize a year 10 obligation whose present value is $1000; that is, at the current interest rate of 6%, its future value is $1000*(1.06)10 = $
You intend to immunize the obligation by purchasing $1000 worth of a bond or a combination of bonds. You have three options:
Bond 1 has 10 years remaining until maturity, a coupon rate of 6.7 percent and face value of $1000.
Bond 2 has a 15 years maturity, a coupon rate of percent, and a face value of $1000.
Bond 3 has 30 years until maturity, a coupon rate of 5.9 percent and a face value of $1000.

59. Immunization Example for 30 year bond..
Year 10: Future Obligation of $ due
Excel.. 30
Buy $1040 face value of 30 year bond.
Sell bond for PV of remaining coupons and redemption in 30 years
Reinvest Coupon from bond during 1-10 years
When the Interest rate increases
Value of Reinvested coupon increases
Value of bond in year 10 decreases
When the interest rate decreases:

60. Immunization.. Limitations..
The method assumes that all yield are equal, whereas in fact they usually are not. Indeed it is quite unrealistic to assume that both long term and short term bonds can be found with identical yields.
Also in practice more than two bonds will be used, partly to diversify default risk if the bonds included are not Treasury bonds.

61. Figure Immunization

62. Table 16.5 Market Value Balance Sheet