1. Finance 298 Analysis of Fixed Income Securities
Bond Mathematics
Finance 298
Analysis of Fixed Income Securities

2. A General Valuation Model
The basic components of valuing any asset are:
An estimate of the future cash flow stream from owning the asset
The required rate of return for each period based upon the riskiness of the asset
The value is then found by discounting each cash flow by its respective discount rate and then summing the PV’s (Basically the PV of an Uneven Cash Flow Stream)

3. The formal model
The value of any asset should then be equal to:

4. A Basic Bond
A bond is basically a debt contract issued by a corporation or government entity.
The buyer is lending the issuer an amount of money (the par value). The issue agrees to pay interest at specified intervals (coupon payments) to the buyer, and return the par value at the end of the contract (the maturity date).

5. Components of a bond: Par Value: Initial issue amount Coupon Payment:
Interest payments on the par value.
Coupon Rate:
The rate that determines the coupon payments.
Maturity Date:
The point in time when the par value and final coupon payment are made.
Embedded Options (Call and Put Provisions):
The issuer may be able to “call” the bond prior to its maturity.
Market Price:
The current price the bond is selling for in the market.

6. Applying the general valuation formula to a bond
What component of a bond represents the future cash flows?
Coupon Payment: The amount the holder of the bond receives in interest at the end of each specified period.
The Par Value: The amount that will be repaid to the purchaser at the end of the debt agreement.

7. Basic Bond Mathematics
Given
r: The interest rate per period or return paid on assets of similar risk
CP: The coupon payment
MV: The Par Value (or Maturity Value)
n: the number of periods until maturity
The value of the bond is represented as:

8. The coupon payment is the same every period and can be factored out, this shows that the first part is just an annuity. Therefore the formula can be rewritten as:

9. The interest factors can be calculated by:

10. A Simple Example
A Bond with 20 years left until maturity, with a 9% coupon rate and a par value of $1,000. Assume for now that the discount rate is also 9% and that it makes annual coupon payments.
What are the relevant PVIFA and PVIF?
PVIFA9%,20 = PVIF9%,20=.1748
How much is the coupon payment?
1,000(.09) = $90

11. Applying the formula On a Financial Calculator:
20 N 9 I 90 PMT 1,000 FV PV = 1,000

12. The Discount Rate
So far we assumed that the interest rate is the same as the coupon rate. When this is true the value of the bond equals the par value.
Are the two usually the same?
No, the discount rate should represent the current required return on assets of similar risk. This changes as the level of interest rates in the economy changes

13. Continuing our example
Assume that we bought our 9% yearly coupon bond with 20 years left to maturity and one year later the required return decreased to 7%.
What is the value of the bond?
19 N 7 I 90 PMT 1,000 FV PV=1,206.71

14. Why did the price increase?
New bonds of similar risk are only paying a 7% return. This implies a coupon rate of 7% and a coupon payment of $70.
The old bond has a coupon payment of $90, everyone will want to buy the old bond, (the increased demand increases the price)
Why does it stop at $1,206.71?
If you bought the bond for $ and received $90 coupon payments for the next 19 years you receive a 7% return.

15. Changes in Bond Value Over Time
Assuming that interest rate stay constant, what happens to the price of the bond as it gets closer to maturity?
N= # of Payments
I = k 7% 7% 7%
PMT = Coup Pay
FV = Par Value
PV = Bond Value

16. Calculating Return
Total Return (yearly) – The combined capital gains yield and interest (current) yield from holding the bond one year.
Yield to Maturity – The yearly return if you purchase the bond today and hold it until maturity
Yield to Call – The yearly return if you purchase the bond today and hold it until it is called.
Yield to Put – the yearly return if you purchase the bond today and hold it until the put option is exercised.

17. Total Return Example
If you bought the bond in the previous example for 1,000 and then sold it after the rate change for what is your total return from owning the bond?

18. Yield to Maturity
Before we were looking for the “value” of the bond given a required rate of return.
Now given the current market price we want to find the interest rate that makes the cash flows from the bond equal to its market price - this rate is known as the Yield to Maturity.
The YTM is the return you earn IF you buy the bond today and hold it until maturity.

19. Calculating YTM
To solve for YTM we are solving for the interest rate (r) in the bond valuation formula:
We cannot solve for r algebraically, only by trial and error

20. Calculating YTM Unfortunately calculating YTM is difficult:
You can approximate it by using the PVIFA and PVIF tables
Solve for I on the Financial Calculator (make sure to enter both (-) and (+) CF’s
on excel use the Yield command
=Yield(settlement, maturity, rate, price, redemption value, frequency, basis)

21. YTM Vs. Total Return
Yearly total return equals the YTM only if the required return does not change over the year.
In the previous example assume you bought the 7% YTM bond with 19 periods left and held it one year.
The price at the end of the year is 1,201.18

22. YTM and Risk
The YTM will change as the level of interest rates in the economy change and as the risk associated with the firm and its projects change.
The YTM is a representation of the probability of default and the current level of interest rates in the economy.

23. Promised or Expected Return
You will earn the YTM if the bond does not default and you hold it to maturity.
The expected return should encompass the chance of default, probability the bond is called or a put option is exercised, and the possibility of interest rate fluctuations.
The YTM is only the expected return if the prob. of default is zero, the prob. of call or put is zero, and interest rates remain unchanged.

24. Yield to Call
The yield to call is the yield paid on the bond assuming that a call option is exercised, given the current market price.
It represents the yield you would earn if you bought the bond today and held it until the call option was exercised.

25. YTC Example
Assume you bought the bond in the previous example and that it had a call option that could be exercised in 9 years. If exercised, the firm is required to pay a $90 premium.

26. Yield to Worst
After calculating all the possible Yields (yield to call, yield to put) the one with the lowest return is the termed the yield to worst.

27. Quick Facts
If the level of interest rates in the economy increases the bond price decreases and vice versa.
If r>Coupon rate the price of the bond is below the par value - it is selling at a discount.
If r<Coupon rate the price of the bond is above the par value - it is selling at a premium.
Keeping everything constant the value of the bond will move toward par value as it gets closer to maturity.

28. Complications
Most bonds make payments every six months instead of each year.
We have assumed that the next coupon payment is exactly 6 months away, often that is not the case. When the time frame is less than 6 months you need to account for interest over the shortened period.
We have assumed that the interest rate is constant, some bonds pay a floating rate of interest.

29. Semiannual Compounding
Most bonds make coupon payments twice a year, to account for this:
Divide the annual coupon interest payment by 2.
Multiply the number of periods by 2.
Divide the annual interest rate by 2

30. Example: Semiannual Compounding
What is the most you would be willing to pay for a 10% coupon bond that makes semiannual coupon payments, 30 years left to maturity and an annual required return of 12%
PMT = 1,000(.10) / 2 = 50 (each 6 months)
I = 12%/2 = 6% each six months
N = 30 (2) = 60 FV = 1,000
PV = ? =

31. Less than 6 months until next coupon payment

32. Floating Interest Rates
A useful measure is the effective margin (Or spread compared to a base rate).
Use the spread to calculate a value and compare that to the current price. If the two are different then there should be a change to the margin. You then develop an estimate of the margin the market is currently using to price the bond.

33. Bond Price Volatility
Assuming an option free bond, we have shown that the price and yield move in an opposite direction, however there are some important details:
Given similar bonds that differ only in maturity or coupon rate, The % price change associated with the same size change in yield will differ.
For a given bond the % price change associated with a small change in yield is the same regardless of whether the yield increases or decreases.

34. Bond Price Volatility continued
For a given bond
the % price change associated with a large increase in yield will not be the same as the % price change associated with the same size decrease in yield
For a large change in yield the % price increase is greater than the % change decrease associated with the same size yield change.

35. Different Coupons
Compare two 30 year semiannual coupon bonds, both with a current yield of 12%. Let Bond A have a coupon rate of 10% and Bond B have a coupon rate of 8%. What is the associated price change if the yield changes to 11%? 13%?
Bond A 10% coupon Bond B 8% coupon
Yield Price Change Price Change
11%
(8.87%) (9.08%)
12%
(7.62%) (7.77%)
13%

36. Impact of Maturity
Compare two 10% coupon semiannual bonds, both with a current yield of 12%. Let Bond A have 30 years to maturity and Bond B have 15 years. What is the associated price change if the yield changes to 11%? 13%?
Bond A 30 Years Bond B 15 Years
Yield Price Change Price Change
11%
(8.87%) (7.54%)
12%
(7.62%) (6.75%)
13%

37. Impact of Change in Rates (30 year bond on earlier slide)

38. Measuring Bond Price Volatility
Price value of a basis point
Measures the price change for a one basis point (.0001 or.01%) change in the yield of the bond.
Yield value of a basis point
Measures the change in the yield of the bond for a given price change.
Duration
Measures the price elasticity of the bond.

39. Price Value of a Basis Point (PVBP)
For small changes in yield the price change will be very close regardless of the whether the yield change is an increase or a decrease.
Use the semi annual bond above (30 years, 10% coupon, 12% YTM).
YTM Price Change % change
12.01% %
12.00%
11.99% %

40. PVBP
In the previous example whether the yield increased or decreased the price changed by approximately 82 cents.
To find a larger price change you can scale the price change. For example if you had a 1% (100Bp) change you cold estimate the price change to be 100(.82) or $82.

41. Yield Value of a Price Change
The change in yield for a given change in the price of a bond.
Example using the semi annual bond above, what is the yield change if the price changes by 1/32 of 1% of par value?
Price Yield Change
% % %
% %

42. Duration: The Big Picture
Duration: Measures the sensitivity of the PV of a cash flow stream to a change in the discount rate.
Keeping everything else constant the change in PV is greater:
The longer the time prior to receiving the cash flow
The larger the cash flow
(we just showed both of these)

43. Duration: The Big Picture
Calculation: Given the PV relationships, we need to weight the Cash Flows based on the time until they are received. In other words we are looking for a weighted maturity of the cash flows where the weight is a combination of timing and magnitude of the cash flows

44. Calculating Duration
One way to measure the sensitivity of the price to a change in discount rate would be finding the price elasticity of the bond (the % change in price for a % change in the discount rate)

45. Duration Mathematics
Macaulay Duration is the price elasticity of the bond (the % change in price for a percentage change in yield).
Formally this would be:

46. Estimating Duration
There are multiple methods for estimating the duration of a bond we will look at three different approaches.
Weighted Discounted Cash Flows (Macaulay)
Modified Duration
Averaging the price change

47. Duration Mathematics
Taking the first derivative of the bond value equation with respect to the yield will produce the approximate price change for a small change in yield.

48. Duration Mathematics
The approximate price change for a small change in r

49. Duration Mathematics Macaulay Duration
substitute

50. Macaulay Duration of a bond

51. Duration Mathematics (2nd derivation)
Taking the first derivative of the bond value equation with respect to the yield will produce the approximate price change for a small change in yield.

52. Duration Mathematics
The approximate price change for a small change in r

53. Duration Mathematics
Divide both sides by the original price to get the
% change in price associated with a given change in r

54. Duration Mathematics
Multiply both sides by (1+r)

55. Macaulay Duration of a bond

56. Duration Example
10% 30 year coupon bond, current rates =12%, semi annual payments

57. Example continued
Since the bond makes semi annual coupon payments, the duration of periods must be divided by 2 to find the number of years.
/ 2 = years
Another interpretation of duration is shown here: Duration indicates the average time taken by the bond, on a discounted basis, to pay back the original investment.

58. Using Duration to estimate price changes
Rearrange
% Change in Price
Estimate the % price change for a 1 basis point
increase in the yearly yield
Multiply by original price for the price change
( )=

59. Using Duration Continued
Using our 10% semiannual coupon bond, with 30 years to maturity and YTM = 12%
Original Price of the bond =
If YTM = 12.01% the price is
This implies a price change of
Our duration estimate was a difference of .0010

60. The estimated price change is then the same as before:
Note:
Previously yield increased from 12% a year to 12.01%.
We used the Duration represented in years,
We could have also used duration represented in semiannual periods, The change in yield needs to be adjusted to .0001/2 = however, the original yield (1+r) stays at 1.06.
The estimated price change is then the same as before:
( )=

61. Modified Duration Substitute DMOD The % Change in price
was given above as:
Substitute DMOD

62. Modified vs Macaulay Duration

63. Approximating Duration
Substitute

64. Approximating Modified Duration
For a given bond: We know the original price, given a change in rates it is easy to calculate the change in price. However, the price change will differ if yield increases compared to a yield decrease. In the PVBP example above, we showed for a small change in yield the price changes were close.

65. Approximating Duration
We can use the fact that the price change associated with a small yield change will be very close regardless of whether the yield increases or decreases.
To get an approximation of the price change you can average the two price changes associated with a change in yield.

66. The change in price
Let P0 be the original price and P- be the price following a small decrease in yield. The associated price change be equal to (P-- P0).
Let P0 be the original price and P+ be the price following a small increase in yield. The associated price change be equal to (P0- P+).
The average price change is then given by

67. Approximating Modified Duration
Let the change in price be approximated by the
average price change calculated on the last slide

68. Example
Using the 30 year 10% coupon bond above that makes semiannual coupon payments and has a current YTM of 12%, consider a 20 Bp change in yield.
Yield Price
11.8%
12.0%
12.2%

69. Comparing results
Previously we calculated modified duration to be the shortcut approximation result was
The short cut assumed a 20 Bp change in yield. Our original estimate assumed a 1 Bp change when we calculated the price change.
Using a 1 Bp change in the duration equation results in
( )/2(.0001)( ) =

70. How much does the size of Bp change matter?
What is the duration if we had assumed a 50 Bp change? Or a 100 Bp Change?
Generally with a shorter maturity bond, the duration estimates will be very close, but for longer maturity bonds the duration estimates may differ by a small amount.

71. Bond Comparison

72. Duration Characteristics
Keeping other factors constant the duration of a bond will:
Increase with the maturity of the bond
Decrease with the coupon rate of the bond
Will decrease if the interest rate is floating making the bond less sensitive to interest rate changes

73. Effective Duration vs. Other Definitions
Macaulay Duration and the most frequently used definition of modified duration assume that the cash flows do not change as the discount rate (yield) changes.
Effective Duration accounts for an associated change in the cash flow, for example if a bond is called, or if mortgages are prepaid early.
The linear approximation of Duration also implicitly assumes that cash flows can change. The value of the security should include any changes in the cash flows.

74. Using Modified Duration
The approximate price percentage change will equal: – Modified duration (change in yield)
In the 1 BP change example above we found duration to be Given the 1 Bp change in yield this implies a (.0001)= change in price.
Given the initial price, this implies a price change of ( ) =

75. Using duration continued
What if interest rates change by 100 Bp or 1%?
Given the 1 Bp change in yield this implies a (.01)= or % change in price.
In this case the price change would be ( ) =
The actual change depended on whether yield increased of decreased.
13%
12%
11%

76. Duration Intuition
The previous slide provides a good explanation of the intuition underlying duration. The duration of was shown to create an approximate price change of % for a 100 Bp change in yield. However, the actual price change was not equal to the approximation in either case due to the shape of the price yield relationship.

77. PVBP and Duration We defined PVBP as the price change in a bond
associated with a one basis point change in yield.
Using a change in yield of one basis point
rearrange

78. Duration and Convexity
Using duration to estimate the price change implies that the change in price is the same size regardless of whether the price increased or decreased.
The price yield relationship shows that this is not true.

79. Impact of Change in Rates

80. Duration
Duration provides a linear approximation of the price change, the actual relationship is convex

81. Duration and Convexity

82. Using Duration Continued
Using our 10% semiannual coupon bond, with 30 years to maturity and YTM = 12%
Rate Actual Price Duration Est.
Price Change Est Change Diff
10% 1,
12%
13%

83. Duration Estimate of Change
Actual Price Change
Duration Estimate of Change
Duration
Under
Est. Price
Increase
Duration
Over
Est. Price
Decrease

84. Convexity
The amount of curvature in the yield price relationship is often referred to as the convexity.
The curvature is measuring the change in the duration for a given change in yield (return).

85. Positive Convexity
Generally: as the yield of the bond increases, the convexity of the bond decreases (positive convexity)
This implies
As Yield increases, each successive price decline is less (there is a decline in the duration of the bond)
As Yield decreases, each successive price increase is greater (there is an increase in the duration of the bond)

86. Convexity Generally the following can also be said:
For a given return and maturity, the lower the coupon the greater the convexity
For a given return and modified duration, the lower the coupon the lower the convexity