1. Yield Measures, Spot Rates, and Forward Rates by Frank J. Fabozzi
PowerPoint Slides by
David S. Krause, Ph.D., Marquette University
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2. Chapter 6 Yield Measures, Spot Rates, and Forward Rates
Major learning outcomes:
Yield and yield spread measures, which help investors gauge relative bond value.
Computing the spot rates from on-the-run Treasury issues.
Understanding the limitations of the nominal spread measure.
Explaining the two measures that overcome the limitation:
Zero-volatility spread
Option-adjusted spread
Computing forward rates.

3. Key Learning Outcomes
Explain the sources of return from investing in a bond (coupon interest payments. Capital gain/loss, and reinvestment income).
Compute the traditional yield measures for fixed-rate bonds (current yield, yield to maturity, yield to first call, yield to first par call date, yield to refunding, yield to put, yield to worst, and cash flow yield).
Explain the assumptions underlying traditional yield measures and the limitations of the traditional yield measures.
Calculate the reinvestment income required to generate the yield computed at the time of purchase.
Explain the factors affecting reinvestment risk.
Calculate the spread for life and discount margin measure for a floating-rate security and explain the limitations of both.

4. Key Learning Outcomes
Calculate the yield on a discount basis for a Treasury bill and explain its limitations.
Compute using the method of bootstrapping the theoretical Treasury spot rate curve given the Treasury yield curve derived from the on-the-run Treasury issues.
Explain the limitations of the nominal spread.
Describe and compute a zero-volatility spread given a spot rate curve.
Explain why the zero-volatility spread will diverge from, and is superior to, the nominal spread.
Explain an option-adjusted spread for a bond with an embedded option and explain what is meant by the option cost.

5. Key Learning Outcomes
Explain why the nominal spread hides the option risk for bonds with embedded options.
Define a forward rate and compute forward rates from spot rates.
Demonstrate the relationship between short-term forward rates and spot rates.
Explain why valuing a bond using spot rates and forward rates produces the same value.
Calculate the forward discount factor from forward rates.
Calculate the value of a bond given forward rates.

6. Sources of Bond Return
A bond investor receives a return from one or more of the following:
The coupon interest payments made by the issuer;
Any capital gain or loss when the bond matures or is sold;
Income from reinvestment of the interim cash flows (interest/and or principal payments)
Yield calculations should consider all three.

7. Sources of Bond Return
Coupon interest – periodic income, usually semi-annually payments.
Capital gain or loss – the profit or loss of market value (selling / maturity price minus purchase price).
Reinvestment income is the interest income generated by reinvesting coupon interest payments and any principal payments from the time of receipt to the bond’s maturity.

8. Bond Yields
The current yield relates the annual dollar coupon interest to the market price and fails to recognize any capital gain or loss and reinvestment income.
The yield to maturity is the interest rate that will make the present value of the cash flows from a bond equal to the price plus accrued interest.
The market convention to annualize a semiannual yield is to double it and the resulting annual yield is referred to as a bond-equivalent yield.
When market participants refer to a yield or return measure as computed on a bond-equivalent basis it means that a semiannual yield or return is doubled.

9. Current Yield
Current yield is greater (less) than the coupon rate, when the bond sells at a discount (premium)
The current yield is a weak measure because it does not measure any cash flows other than the coupon payments:
No consideration is given to the capital gain or loss
Reinvestment income is also not considered
Time value of money concepts are not followed

10. Yield to Maturity YTM is the most popular measure
It is the interest rate that will make the present value of the bond’s cash flow equal to its market price plus accrued income.
It is the same as the Internal Rate of Return (IRR)
Calculations:
Trial and error / approximations
Calculator
Spreadsheet (Excel)

11. YTM Computation It’s simply a present value problem (solve for y):
P is the bond price
C is the periodic coupon payment
N is the number of years to maturity
M is the (face value) payment at maturity
y is the “risk-adjusted discount rate” (or yield to maturity, or IRR)

12. Yield to Maturity
For bonds that pay interest semi-annually, the market convention adopted is to annualize the semi-annual yield to maturity by doubling it (multiplying by 2). This is the annual YTM.
This is also referred to as the: bond-equivalent yield (BEY)
For a bond selling at a discount (premium), the coupon rate is < (>) the current yield, and < (>) the yield to maturity.

13. Yield to Maturity
The yield to maturity takes into account all three sources of return but assumes that the coupon payments and any principal repayments can be reinvested at an interest rate equal to the yield to maturity.
The yield to maturity will only be realized if the interim cash flows can be reinvested at the yield to maturity and the bond is held to maturity.
Reinvestment risk is the risk an investor faces that future reinvestment rates will be less than the yield to maturity at the time a bond is purchased.

14. Yield to Maturity Relationship
The following relationships between the price of a bond, coupon rate, current yield, and yield to maturity hold:

15. Bond-Equivalent Yield Convention
The market convention to annualize a semiannual yield is to double it and the resulting annual yield is referred to as a bond-equivalent yield.
When market participants refer to a yield or return measure as computed on a bond-equivalent basis it means that a semiannual yield or return is doubled.

16. Bond-Equivalent Yield Convention
The idea of simply doubling the semi-annual yield to get the YTM should be somewhat troubling to those who understand the concept of the time value of money. You would think that computing the effective rate would be proper.
However, it is still done this way by many because “that’s the way it always has been done.” This is the market convention, much the same as the Dow Jones Industrial Average is used to express how the overall stock market is performing.
This is okay as long as all bond yields are calculated this way and will not lead to arbitrage opportunities.
Even though the BEY is not the true effective annual yield, we’ll soon see that the use of yields has many problems – the least of which involves the doubling of the semi-annual rate to obtain a YTM estimate.

17. Limitations of YTM
YTM is better than current yield because it considers the coupon, capital gain/loss, and reinvestment income. It is also based on the time value of money concepts.
However, it assumes that the coupon payments from a bond can be reinvested at an interest rate equal to the yield to maturity.
This assumption can be highly misleading because of reinvestment risk
We have seen that a term structure of interest rates exists, which can result in different yields at different points in time.

18. Example of the Limitation of YTM
Suppose an investor has a 15-year 8% semi-annual coupon bond purchased at par ($100).
The YTM is 8%
Translated into total future dollars this is:
$100 x (1.04)30 = $324.34
Decomposed this is $100 of principal return and $ of total dollar return
Without reinvestment income, the dollar return would be:
$120 of coupon income and $0 capital gain (because the bond is purchased at par)

19. Example of the Limitation of YTM (continued)
The dollar return shortfall is $ $120 = $104.34
This shortfall is made up if the coupon payments are reinvested at a yield of 8% (the interest rate at the time the bond was purchased)
For this bond, the reinvestment income is 46.5% ($104.34/224.34) of the total dollar return needed to produce a yield of 8%
The investor will only realize the YTM of 8% if:
The coupon payments can be reinvested at the YTM of 8% (this is reinvestment risk)
The bond is held to maturity (if the bond is not held to maturity, the investor faces the risk of selling for less than the purchase price which is known as interest rate risk)
These are large and questionable assumptions

20. Factors Affecting Reinvestment Risk
Reinvestment risk is the risk an investor faces that future reinvestment rates will be less than the yield to maturity at the time a bond is purchased.
Interest rate risk is the risk that if a bond is not held to maturity, an investor may have to sell it for less than the purchase price.
The longer the maturity and the higher the coupon rate, the more a bond’s return is dependent on reinvestment income to realize the yield to maturity at the time of purchase.

21. Reinvestment Risk The two factors affecting reinvestment risk are:
For a given YTM and a given non-zero coupon rate, the longer the maturity, the more the bond’s total dollar return depends on reinvestment income to realize the YTM at the time of purchase. That is, the greater the reinvestment risk.
The implication is that the YTM for a long-term, high coupon bond may have a large amount of the total dollar return as reinvestment income (which is more risky).

22. Reinvestment Risk (continued)
The two factors affecting reinvestment risk are:
2. For a coupon bond, for a given YTM and maturity, the higher the coupon rate, the more dependent the bond’s total dollar return will be on the reinvestment of the coupon payments in order to produce the YTM at the time of the purchase.
The implication is that bonds selling at a premium will be more dependent on reinvestment income than bonds selling at par. This is because the reinvestment income has to make up the capital loss due to amortizing the price premium when holding the bond to maturity.
Conversely, a bond selling at a discount will be less dependent on reinvestment income.

23. Reinvestment Risk (continued)
Exhibit 1 shows an example of three bonds with different coupons (and selling prices)
Based on the years to maturity and the coupon rate (selling price) it shows:
Reinvestment risk is greatest for longer maturity bonds
Reinvestment risk is greatest for higher coupon bonds (those selling at a premium)

24. Reinvestment Risk

25. Semi-Annual versus Annual Coupon Paying Bonds
Some bonds (especially those issued outside the U.S.) pay interest annually rather than semi-annually
The bond-equivalent yield for an annual paying bond is computed as follows:
2 x [(1+ yield on annual-pay bond)0.5 -1]
The term in the square brackets involves determining what semi-annual yield, when compounded, produces the yield on an annual-paying bond.
Doubling this semi-annual yield gives the bond-equivalent yield.

26. Semi-Annual versus Annual Coupon Paying Bonds Example
Suppose the YTM on an annual-pay bond is 6%. Then the bond-equivalent yield is:
2 x [(1+ .06)0.5 -1] = 5.91%
The bond-equivalent yield will always be less than the annual-paying bond’s YTM

27. Semi-Annual versus Annual Coupon Paying Bonds Example
It is also possible to convert the bond-equivalent yield to an annual-pay basis using the following:
[(1+ (yield on bond-equivalent basis)/2)2 -1]
The yield on an annual-pay basis is always greater than the yield on a bond-equivalent basis because of compounding.

28. Yield to Call
The yield to call is the interest rate that will make the present value of the expected cash flows to the assumed call date equal to the price plus accrued interest.
Yield measures for callable bonds include yield to first call, yield to next call, yield to first par call, and yield to refunding.
The yield to call considers all three sources of potential return but assumes that all cash flows can be reinvested at the yield to call until the assumed call date, the investor will hold the bond to the assumed call date, and the issuer will call the bond on the assumed call date.

29. Yield to Call
For callable bonds, the practice has been to calculate a YTM and a yield to call (YTC)
The YTC assumes the issuer will call the bond on the call date at the specified call price
Investors calculate the yield to first call, yield to next call, yield to first par call, and a yield to refunding.
Yield to refunding is used when bonds are currently callable but have some restrictions on the source of the funds used to buy back the debt when a call is exercised.

30. Yield to Call
Calculating the YTC is the same as that for any YTM calculation.
In the case of yield to first call, the expected cash flows are the coupon payments to the first call date and the call price.
For the yield to first par call, the expected cash flows are the coupon payments to the first par call date and the par value.
Exhibit 2 contains an example of a YTC under varying interest rate levels.

31. Yield to Call

32. Yield to Call
The YTC considers all three sources of potential return from owning a bond; however, like YTM it is assumed that all cash flows can be reinvested at the YTC until the assumed call date.
As we have learned previously, this may be an inappropriate assumption.
Other problems with YTC occur when comparisons are made with YTM because of the different maturity/call dates.

33. Yield to Put
When a bond is putable, the yield to the put date is computed.
The yield to put is computed assuming that the issue will be put on the first put date.
The yield to put is the interest rate that will make the present value of the cash flows to the first put date equal to the price plus accrued interest.
Like other yield measures, the yield to put assumes that any interim coupon payments can be reinvested at the yield calculated.

34. Yield to Worst
The yield to worst is the lowest yield from among all possible yield to calls, yield to puts, and the yield to maturity.

35. Yield to Worst
A yield can be computed for every possible call and put date, as well as the YTM.
The lowest of all the yields is called the yield to worst.
The yield to worst has little meaning as a measure of potential return because
It does not identify the potential return over some investment horizon
It does not recognize that each yield calculation has different exposures to reinvestment risk.

36. Cash Flow Yield
Mortgage- and asset-backed securities (which are secured with pools of loans or receivables) have periodic cash flows that include interest and principal. These are typically monthly.
Because the borrowers can prepay, the timing of the principal payments are not known with certainty.
It is necessary to make an assumption about the rate at which principal prepayments occur. This is referred to as:
Prepayment rate, or
Prepayment speed (this is measured in units of PSA).
Given an assumed payment schedule, a cash flow yield can be computed.

37. Cash Flow Yield
For mortgage-backed and asset-backed securities, the cash flow yield based on some prepayment rate is the interest rate that equates the present value of the projected principal and interest payments to the price plus accrued interest.
The cash flow yield assumes that all cash flows (principal and interest payments) can be reinvested at the calculated yield and that the assumed prepayment rate will be realized over the security’s life.
For amortizing securities, reinvestment risk is greater than for standard coupon nonamortizing securities because payments are typically made monthly and include principal as well as interest payments.

38. Bond-Equivalent Cash Flow Yield
The interest rate that will make the present value of the projected principal and interest payments equal to the market price plus accrued interest is a monthly yield that gets annualized as follows:
The semi-annual effective yield is computed from monthly yield by compounding it for six months: (1 + monthly yield)6 – 1
Effective semi-annual yield is doubled to get the annual cash flow yield on a bond-equivalent basis: 2 x effective semi-annual yield
Again, it seems odd to be doubling up the semi-annual rate by multiplying by 2, but this is the market convention (“that’s the way it has always been done”).
Remember, too, that this is a minor issue compared to the other problems with using yield as a relative measure.

39. Limitations of Cash Flow Yield
The shortcomings of cash flow yield are:
The projected cash flows are assumed to be reinvested at the cash flow yield
The mortgage- and asset-backed securities are assumed to be held until the final payoff of all loans (or based on some assumed prepayment rate)
The reinvestment risk is especially significant since the payments are monthly and include principal payments.
If the actual prepayment differs significantly from the actual principal payment rate, the cash flow yield will not be close to the actual yield.

40. Spread/Margin Measures for Floating Rate Bonds
For floating-rate securities, instead of a yield measure, margin measures (i.e., spread above the reference rate) are computed.
Two margin measures commonly used are spread for life and discount margin.

41. Spread/Margin Measures for Floating Rate Bonds
The coupon rate for a floating (variable) rate bond (floater) changes periodically according to a reference rate (i.e. LIBOR).
Since the future reference rate is not known, it is not possible to determine the cash flows – meaning that YTM cannot be calculated.
Margin measures are used instead of YTM for floaters.
Margin is simply some spread above the floater’s reference rate.
There are several spread or margin measures:
Spread for life
Discount margin

42. Spread for Life
When a floater is selling at a premium (discount), investors consider the premium (discount) as another source (reduction) of the return.
Spread for life (or simple margin) is a measure of potential return that accounts for the amortization (accretion) of the premium (discount) as well as the constant quoted margin over the bond’s remaining life.

43. Spread for Life Spread for life is calculated as follows: Where
[(100(100 – Price))/Maturity + Quoted Margin] x (100/Price)
Where
Price = market price per $100 of par value
Maturity = number of years to maturity
Quoted Margin = quoted margin in the coupon reset formula measured in basis points
Example: a floater with a quoted margin of 80 basis points is selling for and matures in 6 years:
[(100(100 – ))/ ] x (100/ ) = bps
This is a simple calculation, but has the limitations that it considers only the accretion / amortization of the discount/premium over the floater’s remaining term to maturity and does not consider the level of the coupon rate or the time value of money.

44. Discount Margin
The discount margin assumes that the reference rate will not change over the life of the security and that there is no cap or floor restriction on the coupon rate.
The discount margin estimates the average margin over the reference rate that the investor can expect to earn over the life of the security.

45. Discount Margin The discount margin is calculated as follows:
Determine the cash flows assuming the reference rate does not change over the life of the security
Select a margin
Discount the cash flows by the current value of the reference rate plus the margin
Compare the present value of the cash flows to the price plus accrued interest.
If the present value is equal to the price plus accrued interest, the discount margin is the same as the selected margin
If the present value is not equal, then use a different margin to equate the two.

46. Discount Margin (continued)
Exhibit 3 shows the ‘trail and error’ approach to calculating the discount margin for a floating rate bond
The drawbacks of the discount margin as a measure of potential return from investing in a floating rate bond are as follows:
The measure assumes that the reference rate will not change over the life of the security
If the floating-rate security has a cap or floor, is not taken into consideration

47. Discount Margin

48. Yield on Treasury Bills
Because T-bills have a maturity of one year or less, the standard convention is to compute a yield on a discount basis. There are two variables:
The settlement price per $1 of maturity value (p)
The number of days to maturity which is calculated as the number of days between the settlement date and the maturity date (NSM)
The yield on a discount basis is compute as:
Discount = (1 – p) x (360/NSM)

49. Example of Yield on Treasury Bills
Settlement date of 8/6/06 with maturity of 1/8/07 and a price of The number of days from settlement to maturity is 155.
The yield on a discount basis is:
5.18% = (1 – ) x (360/155)
For a given yield on a discount basis, the price of a bill (per $1 maturity value) is computed as:
p = 1 – (d x (NSM/360))
p = 1 – (.0518 x (155/360)) =

50. Yield on Treasury Bills
The quoted yield on a discount basis is not a meaningful measure of return because:
The measure is based on a maturity investment value rather than on the actual dollar amount
The yield is annualized according to a 360-day year rather than a 365-day year, making it difficult to compare yield on T-bills with T-notes and bonds which pay interest based on the actual number of days in a year
Market participants recognize these limitations of yield on discount basis and make adjustments to make the yield on a T-bill comparable to other Treasury investments.

51. Theoretical Spot Rates
The theoretical spot rate is the interest rate that should be used to discount a default-free cash flow.
Because there are a limited number of on-the-run Treasury securities traded in the market, interpolation is required to obtain the yield for interim maturities; hence, the yield for most maturities used to construct the Treasury yield curve are interpolated yields rather than observed yields.
Default-free spot rates can be derived from the Treasury yield curve by a method called bootstrapping.
The basic principle underlying the bootstrapping method is that the value of a Treasury coupon security is equal to the value of the package of zero-coupon Treasury securities that duplicates the coupon bond’s cash flows.

52. Theoretical Spot Rates “Bootstrapping”
In order to value default-free cash flows, the theoretical spot rate for Treasury securities must be determined. (This was given in Chapter 5 – now we calculate it).
The default-free theoretical spot rate curve is constructed from the observed Treasury yield curve.
Several techniques are used to create the yield curve; however, the most commonly employed method is called “bootstrapping”

53. Bootstrapping Spot Rates
Bootstrapping uses the yield for the on-the-run Treasury issues (since there are no credit or liquidity risks).
A problem exists because there may be an insufficient number of data points for on-the-run issues to construct a yield curve.
Issuance of Treasury securities
3-month, 6-month, 2-year, 3-year, 5-year, and 10-year notes (the 30-year has recently been reissued)
This leaves gaps in the yield curve which can be filled in with simple linear interpolation

54. Bootstrapping Spot Rates
To fill in the gap for each missing one year maturity, it is possible to start with the lowest maturity and work up to the highest maturity with the following formula:
(yield at higher maturity – yield at lower maturity)
Number of years between two observed maturity points
The estimated on-the-run yield for all intermediate whole-year maturities is found by adding the amount computed to the yield at the lower maturity.

55. Bootstrapping Spot Rates
Example: 2-year 4.52%, 5-year 4.66%, 10-year 4.80%, 30-year 5.03%
Using the above information, to bootstrap the 3- and 4-year Treasury rates, the following interpolation of .0466% was computed as follows:
(4.66% – 4.52%)
3 years
Then the interpolated 3-year rate would be:
4.52% % = 4.567%
The interpolated 4-year rate would be:
4.567% % = 4.614%
Therefore, when a yield curve is shown, many of the points are only approximations. Exhibits 4 and 5 show an interpolated “bootstrapped” Treasury yield curve.
This method produces only a ‘crude approximation’

56. Bootstrapping Spot Rates

57. Bootstrapping Spot Rates

58. Theoretical Spot Rates
The basic principle is that the value of a Treasury coupon series should be equal to the value of a package of zero-coupon Treasuries that duplicates the coupon bond’s cash flows.
Using the arbitrage-free method, it is possible to compute the approximate yield of bonds (spot rates) over any maturity range (including months) and going forward in time.
These will be more precise that the linear interpolated results from bootstrapping.
Exhibit 6 shows the plot of the theoretical spot rates and the par value Treasury yield curve.

59. Theoretical Spot Rates

60. Method of Bootstrapping Spot Rates from the Par Yield Curve
Begin with the 6-month spot rate.
Set the value of the 1-year bond equal to the present value of the cash flows with the 1-year spot rate divided by 2 as the only unknown.
Solve for the 1-year spot rate.
Use the 6-month and 1-yar spot rates and equate the present value of the cash flows of the 1.5 year bond equal to its price, with the 1.5 year spot rate as the only unknown.
Solve for the 1.5 year spot rate.

61. Example
Consider the yields on coupon Treasury bonds trading at par (given in the table).
YTM for the bonds is expressed as a bond equivalent yield (semi-annual YTM).

62. 1. Begin with the 6-month spot rate.
The bond with six months left to maturity has a semi-annual discount rate of 5%/2 = 2.5% or 5% on a bond equivalent yield basis.
Since this bond will only make one payment of $ in six months, the YTM is the spot rate for cash flows to be received six months from now.
The bootstrapping process proceeds from this point using the fact that the 6-month annualized spot rate is 5%.

63. where z2 is the annualized 1-year spot rate
2. Set the value of the 1-year bond equal to the present value of the cash flows with the 1-year spot rate divided by 2 as the only unknown. Solve for the 1-year spot rate.
The 1-year bond will make two payments, one in six months of $3.0 and one in one year of $103.0, and that the appropriate spot rate to discount the coupon payment (which comes 6 months from now), is written as:
$3.0/(1.025)1 + $103.0/(1+z2/2)2 = $100
where z2 is the annualized 1-year spot rate

64. 3. Solve for the 1-year spot rate.
$3.0/(1.025)1 + $103.0/(1+z2/2)2 = $100
where z2 is the annualized 1-year spot rate.
Solve for z2/2 as: $103.0/(1+z2/2)2 = $100 - $3/ = $100 - $2.927 = $97.073
or:
$103.0/$ = (1+z2/2)2
So: sq. root of ($103.0/$97.073) -1 = z2/2 = %
1-yr Spot rate (z2) = % times 2 = %

65. where z3 is the annualized 1.5-year spot rate.
4. Use the 6-month and 1-year spot rates and equate the present value of the cash flows of the 1.5 year bond equal to its price, with the 1.5 year spot rate as the unknown.
Now that we have the 6-month and 1-year spot rates, this information can be used to price the 18-month bond.
Set the bond price equal to the value of the bond’s cash flows as:
$3.5/(1.025)1 + $3.5/( )2 + $103.5/(1+z3/2)3 = $100
where z3 is the annualized 1.5-year spot rate.

66. 5. Solve for the 1.5 year spot rate.
$3.0/(1.025)1 + $3.5/( )2 + $103.5/(1+z3/2)3 = $100
where z3 is the annualized 1.5-year spot rate.
Solve for z3/2 as: $103.5/(1+z3/2)2 = $100 - $3.5/ $3.5/( )2 = $100 - $ $3.30 =$93.285
or:
$103.5/$ = (1+z3/2)2
So: cube root of ($103.5/$93.285) -1 = z3/2 = %
1.5-yr Spot rate (z3) = % times 2 = %

67. Yield Spreads Relative to the Spot Rate Curve
The nominal spread is the difference between a non-Treasury bond’s yield and the YTM for a benchmark Treasury coupon security.
The nominal yield spread measures the compensation for the additional credit risk, option risk, and liquidity risk an investor is exposed to by investing in a non-Treasury security with the same maturity.
The problems with the nominal spread measure are:
For both bonds, the yield fails to take into consideration the term structure of spot rates
In the case of a call or put bond, expected interest rate volatility may alter the cash flows of the non-Treasury bond

68. Nominal Spread
The nominal spread is the difference between the yield for a non-Treasury bond and a comparable-maturity Treasury coupon security.
The nominal spread fails to consider the term structure of the spot rates and the fact that, for bonds with embedded options, future interest rate volatility may alter its cash flows.

69. Nominal Spread

70. Z-Spread
The zero-volatility spread or Z-spread is a measure of the spread that the investor will realize over the entire Treasury spot rate curve if the bond is held to maturity, thereby recognizing the term structure of interest rates.
Unlike the nominal spread, the Z-spread is not a spread off one point on the Treasury yield curve but is a spread over the entire spot rate curve.

71. Zero-Volatility Spread
The zero-volatility or Z- spread is a measure of the spread the investor would realize over the entire Treasury spot rate curve if the bond is held to maturity.
It is not the spread off of one point on the Treasury yield curve (nominal spread), it is an average over all spot rates.
The Z-spread is also called a static spread – and is calculated as the spread which will make the present value of the cash flows from the non-Treasury bond, when discounted at the Treasury spot rate plus the spread, equal to the non-Treasury bond’s price.
Trial and error is used to determine the Z-spread.

72. Zero-Volatility Spread Example
Exhibits 8 and 9 show how trial and error is used to compute the Z-spread
The Z-spread is measured relative to the Treasury spot rate curve and represents a spread to compensate for the non-Treasury bond’s credit risk, liquidity risk, and any embedded option risk.

73. Zero-Volatility Spread

74. Zero-Volatility Spread

75. Zero-Volatility and Nominal Spread
For bullet bonds, unless the yield curve is very steep, the nominal spread will not differ significantly from the Z-spread; for securities where principal is paid over time rather than just at maturity there can be a significant difference, particularly in a steep yield curve environment.

76. Z-Spread and the Nominal Spread
The Z- and nominal spreads will not differ much for standard coupon-paying bullet bonds. They will diverge when:
The slope of the term structure is steep
The principal is paid off before maturity (i.e. mortgage- and asset-back bonds)
The Z-spread can be calculated to any benchmark spot rate curve.
When used for the same issuer, it is possible to isolate liquidity risk

77. Option-Adjusted Spread
The option-adjusted spread (OAS) converts the cheapness or richness of a bond into a spread over the future possible spot rate curves.
An OAS is said to be option adjusted because it allows for future interest rate volatility to affect the cash flows.
The OAS is a product of a valuation model and, when comparing the OAS of dealer firms, it is critical to check on the volatility assumption (and other assumptions) employed in the valuation model.
The cost of the embedded option is measured as the difference between the Z-spread and the OAS.
Investors should not rely on the nominal spread for bonds with embedded options since it hides how the spread is split between the OAS and the option cost.
OAS is used as a relative value measure to assist in the selection of bonds with embedded options.

78. Option-Adjusted Spread
The Z-spread, which looks at measuring the spread over a spot rate curve, has a problem in that it fails to take future interest rate volatility into consideration – which could change the cash flows for bonds with embedded options.
The option-adjusted spread (OAS) was developed to take the dollar difference between the fair valuation and the market price and convert it to a yield spread measure.
The OAS is used to reconcile the fair price (value) and the market price by finding a return (spread) that will equate the two.
The spread is measured in basis points.

79. Option-Adjusted Spread
The OAS depends upon the valuation model employed.
OAS models primarily differ in how they forecast interest rate changes.
What are the key modeling differences?
Interest rate volatility is a crucial assumption. The higher the interest rate volatility, the lower the OAS.
The OAS is a spread over the Treasury spot rate curve or the issuer’s benchmark. In the model, the spot rate curve is the result of a series of assumptions that allow for changes in interest rates.
The spread is referred to as “option adjusted” because the bond’s embedded options can change the cash flows and therefore the value of the security. Note: the Z-spread ignores how interest rate changes can impact cash flows – which is why it is referred to as the zero-volatility OAS.

80. Option Cost
The implied cost of the option embedded in a bond can be obtained by calculating the difference between the OAS at the assumed interest rate or yield volatility and the Z-spread.
The Z-spread is the OAS plus the option cost. Therefore, the option cost equals the Z-spread minus the OAS.
Thus, the option cost is the difference between the spread that would be earned in a static interest rate environment (Z-spread) and the spread after adjusting for the option (OAS).

81. Option Cost
For callable bonds and mortgage- and asset-backed securities, the option cost is positive.
This is because the issuer’s ability to alter the cash flows will result in an OAS that is less than the Z-spread.
In the case of a putable bond, the OAS is larger than the Z-spread because of the investor’s ability to alter the cash flows.
In general, when the option cost is positive (negative), the investor has sold (bought) an option to the issuer or borrower.
An investor that relies only on the nominal spread may not be adequately compensated for taking on option risk – which is one of the strengths of the OAS approach.

82. Summary of Spread Measures
Benchmark
Reflects compensation for:
Nominal
Treasury yield curve
Credit risk, option risk, liquidity risk
Zero-volatility
Treasury spot rate curve
Option-Adjusted
Credit risk, liquidity risk

83. Forward Rates
Besides default-free theoretical spot rate curves extrapolated from the Treasury yield curve, it is possible to compute forward rates.
Since forward rates are extrapolated from the default-free theoretical spot rate curve, these rates are referred to as implied forward rates.
Besides using the Treasury yield curve, it is possible to compute forward rates from other interest rate curves (i.e. LIBOR).

84. Forward Rates
Using arbitrage arguments, forward rates can be extrapolated from the Treasury yield curve or the Treasury spot rate curve.
The spot rate for a given period is related to the forward rates; specifically, the spot rate is a geometric average of the current 6-month spot rate and the subsequent 6-month forward rates.

85. Forward Rates Notation: 1f1
Definition of forward rate: The implied rate of return on a security to be issued at some future date.
Definition of spot rate: The rate of return on securities already issued.
when issued
time to maturity

86. Spot and Forward Rates for Fixed Income Securities
A spot rate is a rate agreed upon today, for a loan that is to be made today. (e.g. r1 = 5% indicates that the current rate for a one-year loan is 5%).
A forward rate is a rate agreed upon today, for a loan that is to be made in the future. (e.g. 2f1 = 7% indicates that we could contract today to borrow money at 7% for one year, starting two years from today). 35 35

87. Forward Rates
Forward rates of interest are implicit in the term structure of interest rates
t = …
Note the notation: 3f4 means “the forward rate from period 3 to period 4.”
When the beginning subscript is omitted, it is understood that the forward rate is for one period only: 3f4 = f4 . r1
1f2 r2
2f3 r3
3f4

88. General Formula for Forward Rates
One-period forward rates:
n-period forward rates:

89. Example: Forward Rates
What one-year forward rates are implied by the following spot rates?
Maturity Year
Spot Rate (rt)
Forward Rate (ft) 1
4.0% – 2
5.0%
6.01% 3
5.5%
6.507%

90. Implied Forward Rate Example
Suppose the spot term structure of zero-coupon yields is: {r1=0.08, r2=0.10, r3=0.13, r4=0.14,…}
If investors wish to invest $1,000,000 for two years. They can choose between:
buying a 2-yr. discount bond, and
buying a sequence of two 1-yr. bonds, i.e., one now and one in one year from now.

91. What Will the Investor Choose?
The alternative that pays the higher cumulative return over the 2-yr time horizon.
Caveat: The rate of return on the bond issued one year from now is uncertain.
How do we estimate it?
With the implied forward rate

92. Estimating the Implied Forward Rate
Underlying assumption: These must be equal cumulative returns, with no arbitrage possible.

93. Estimation of Implied Forward Rates (using the spot term structure from a previous slide)

94. Estimation of Implied Forward Rates (continued)

95. General Formula for Implied Forward Rates
Note that implied fwd rates are internally consistent, e.g.,

96. Deriving a 6-Month Forward Rate
To compute a 6-month forward rate, it is necessary to utilize a yield curve and the corresponding spot rate curve.
The following 2 investments should have the same value:
1-year Treasury bill and
2 six-month Treasury bills (one purchased now and the other in six months)
An investor should be indifferent since they should produce the same investment income over the same investment horizon.

97. Deriving a 6-Month Forward Rate
Although an investor does not know the interest rate of the second 6-month T-bill, it is possible to compute it because the “forward” rate must such that it equalizes the dollar return between the two alternatives.
Exhibit 11 shows the timeline for the two investment alternatives:
The value of first six-month T-bill is: X(1 + z1)
The value of the total investment following the second six-month T-bill is: X(1 + z1)(1 + f)
Where z1 is one-half the bond-equivalent yield of the 6-month spot rate and f is one-half the forward rate on a 6-month Treasury bill available 6 months from now. X is the amount of the investment.

98. Deriving a 6-Month Forward Rate

99. Relationship Between Spot Rates and Short-Term Forward Rates
The value of alternative investment (a 1-year T-bill) is computed as:
X(1 + z2)2
Because the two alternatives should generate identical returns:
X(1 + z1)(1 + f) = X(1 + z2)2
Solving for f = [(1 + z2)2 / (1 + z1)] -1
Multiplying f by 2 to get the forward rate on a bond-equivalent yield basis.
Forward rates can be computed on various combinations of short- and longer-term interest rates. Exhibit 12 provides the six-month forward rates for the entire yield curve. Exhibit 13 is a graph of the forward rate curve.