1. Treasury Bonds and Duration
Fin 525 Week 4
Treasury Bonds and Duration

2. An Overview of the Bond Markets
A bond is a promise to make periodic coupon payments and to repay principal at maturity— breech of this promise is an event of default
Bonds carry original maturities greater than one year, so bonds are instruments of the capital markets
Bonds issued with maturities of 10 years or less are sometimes called notes
Issuers are corporations and government units
Professor Ross Miller • Fall 2007

3. Bonds are Bundles of Cash Flows
Bonds can be viewed as a “bundle” of zero- coupon securities—Treasury bonds are literally bundles of STRIPS
New cash flow scheme
Cash flows out now to buy the security
Cash flows in several times until the security matures
Later, we will incorporate options that can accelerate the payments at the borrower’s discretion
Professor Ross Miller • Fall 2007

4. Treasury Notes and Bonds
T-notes and T-bonds issued by the U.S. Treasury to finance the national debt and other federal government expenditures
Backed by the full faith and credit of the U.S. government and are essentially default risk free
Have significant interest-rate risk due to their longer maturities
Pay interest twice a year (interest payments are at half the “coupon rate”)
Professor Ross Miller • Fall 2007

5. Anatomy of the 5-year Treasury Note due 8/15/09 with a 3
Anatomy of the 5-year Treasury Note due 8/15/09 with a 3.50% Coupon Rate
$1,000 Principal Payable 8/15/09
$17.50 Coupon Payable 2/15/05
$17.50 Coupon Payable 8/15/05
$17.50 Coupon Payable 2/15/06
$17.50 Coupon Payable 8/15/06
$17.50 Coupon Payable 2/15/07
$17.50 Coupon Payable 8/15/07
$17.50 Coupon Payable 2/15/08
$17.50 Coupon Payable 8/15/08
$17.50 Coupon Payable 2/15/09
$17.50 Coupon Payable 8/15/09
Professor Ross Miller • Fall 2007

6. “On-the-Run” (Most Recently Issued) Treasury Notes and Bonds from 9/14/2007
Professor Ross Miller • Fall 2007

7. Bond Complications Bond prices are given using a par value of 100
Bonds are traded in multiples of $1,000, so “one bond” typically represents a principal amount of $1,000
While Treasury securities routinely trade in multiples of $1,000, many corporate bonds only trade in multiples of $1,000,000 or $5,000,000
Like STRIPS, bonds can be quoted in 32nds (and fractions thereof)
Professor Ross Miller • Fall 2007

8. Two Important Points
The on-the-run bonds (those most recently issued) are more popular than “seasoned” or “off-the-run” bonds
Their popularity slightly depresses their yields
It has makes them more liquid—the spread between the bid and asked yield tends to be lower and it is easier to buy or sell large quantities of them quickly
The longer the maturity of bond, the more sensitive it is to changes in interest rates
Professor Ross Miller • Fall 2007

9. Two Special Treasury Securities
10-year Treasury Note
Is used by mortgage lenders to hedge the interest rate risk of the typical fixed-rate mortgage
“30-year” Treasury Bond
It (and the STRIPS derived for it) are the basis of many insurance products
First auction since 2002 was held on February 9, 2006
Professor Ross Miller • Fall 2007

10. What Is Easy and What Is Difficult about Bonds
Finding a present value (PV) is easy
The present value of the bond is just the sum of the present values of every cash flow it generates
With regular coupon payments, annuity factors make this a snap to do
Finding the future value (FV) is difficult
The obvious point in time to seek an FV for the bond is at its maturity
The problem is what happens to all the cash flows received before then
This problem is known as “reinvestment risk” and certain assumptions (often unrealistic) are used to attempt to deal with it
Professor Ross Miller • Fall 2007

11. Computing a Bond’s Price from Its Yield Step 1: Listing the Cash Flows
An easy example: 4% coupon rate that matures in 1 year and that just paid its semiannual coupon interest payment
It has only two cash flows remaining:
$2 in ½ year (all interest)
$102 in 1 year ($2 interest + $100 principal)
The value of the bond (and its price in a competitive market) should be the PV of these cash flows
Professor Ross Miller • Fall 2007

12. Computing a Bond’s Price from Its Yield: Step 2: Get An Interest Rate
Bond payment periods are at ½-year intervals, so use a ½-year (or semiannual) interest rate (but it is usually stated as an annual rate, that is then halved)
Suppose the rate is quoted as 5.00% annually
This means it is really 2.50% semiannually
Recalling compounding from a few weeks ago, the effective annual interest rate is:
Professor Ross Miller • Fall 2007

13. Computing a Bond’s Price from Its Yield: Step 3: Discount the Cash Flow and Add Up
The first cash flow is $2, so discounted at 2.5%, it is worth $2/(1.025) = $1.9512
The second cash flow is $102, so discounted at 2.5% (twice), it is worth $102/(1.025)2 = $
Add them together to get the bond’s value: $ $ = $ or about $99.04
Notice that because the yield is above the coupon rate, the bond trades at a discount
Professor Ross Miller • Fall 2007

14. Extending the Bond Valuation Process
Note that in order to find the FV of the bond one year from now, we need to specify what happens to the $2 interest paid in ½ year between the time it is paid and the time the bond matures
Also, it would be nice to deal with more than one cash flow prior to maturity
The regularity of the cash flows makes this easy to do
It is best to think to the bond’s value as the sum of the PV of its coupon interest payments and the single return of principal payment at maturity
Professor Ross Miller • Fall 2007

15. Bond Value = PV(Coupon Payments) + PV(Principal Payment)
Since bonds pay semiannual interest, they have two “periods” per year, so T = 2 x years to maturity
The principal payment is easy, since it can be viewed as a pure discount bond that matures in T periods, so PV(Principal Payment) = F/(1+r)T
More difficult is the adding together of a potentially large number of cash flows; however it helps greatly that each cash flow (coupon interest payment) is for the same amount of cash
Professor Ross Miller • Fall 2007

16. The Perpetuity Trick
A perpetuity is a series of cash flows that starts one period from now and continues without end (in perpetuity)
Perpetuities never “mature,” so there is no payment that corresponds to the return of principal
$100 of a 4% coupon rate perpetuity would pay $2 in ½ year, $2 in 1 year, $2 in 1½ years, and so on, forever
Professor Ross Miller • Fall 2007

17. The Perpetuity Trick (continued)
At an quoted bond interest rate of 5% (which, as before, is 2.5% on a semiannual basis), The present value of this perpetuity would then be:
$2/(1.025) +$2/(1.025)2 + $2/(1.025)3 +$2/(1.025)4….
This infinite series converges to a value of $2/(.025) = $80
In general, if C is the coupon interest payment and r is the periodic interest rate, the value of the perpetuity is:
Professor Ross Miller • Fall 2007

18. The Perpetuity Trick (continued)
But, with the exception of some British bonds that are now essentially extinct, bonds do not pay interest forever
That is where the trick comes in
Suppose the bond makes 60 semiannual interest payments
Consider a perpetuity that begins when that bonds matures, so its first payment is in the 61st period
The difference in cash flows between a perpetuity that starts now and one that starts 60 periods from now is exactly the bond’s coupon interest cash flows
Professor Ross Miller • Fall 2007

19. The Perpetuity Trick (finished)
The PV of a perpetuity that starts in T periods is just:
Therefore, using the algebra described in the textbook, the PV of equal cash flow of C for T periods (known as an annuity) is the different between the value of a perpetuity now and one that starts in T periods:
Professor Ross Miller • Fall 2007

20. Putting it all Together: The General Bond Valuation Formula
Annuity Factor
Professor Ross Miller • Fall 2007

21. Some Comments on the Bond Valuation Formula
The PV functions in both Excel and standard financial calculators do all of this automatically
The annuity factor comes in handy for doing amortization
A fully-amortizing loan (like the typical fixed-rate mortgage), has a face value (F) of 0—it is all “paid off” at maturity
If we know the loan’s periodic interest rate (r) and the loan amount (PV), then each loan payment is just the PV divided by the annuity factor (Excel and calculators also do this)
Professor Ross Miller • Fall 2007

22. Another Difficulty: Bond Yield
For a zero-coupon bond, one can find an APY (annual percentage yield) by solving FV=PV(1+r)T for r, which gives:
r = (FV/PV)(1/T) – 1
With multiple cash flows, there is no general formula
One can use Excel’s (or a financial calculator’s) IRR or YIELD function to compute a yield to maturity (and this is standard practice), but yield to maturity assumes that all cash flows can be reinvested at that yield
Professor Ross Miller • Fall 2007

23. Current Yield vs. Yield to Maturity
Current Yield = Annual Coupon Payment/Price
An example from RWJ, pages 133–134
10% coupon, 2 years to maturity, $1, price
Current yield = $100/$1, = 9.66%
The problem is that in 2 years when the bond matures at $1,000, $35.67 is being forfeited over that 2-year period and should be reflected in the yield (and reduce it substantially in this case)
Professor Ross Miller • Fall 2007

24. Current Yield vs. Yield to Maturity (continued)
Yield to maturity (sometimes simply referred to as the bond’s yield) is a way of adjusting yield to account for the expected gain or loss in the bond over its lifetime
In the case of the previous textbook example, the yield to maturity is the value of y (which turns out to be 8%) that solves:
Professor Ross Miller • Fall 2007

25. The NPV of a Bond Whose Cash Flows are Discounted at its Yield to Maturity is 0
In other words, the price of the bond (your current cash outflow) is equal to the PV of its interest and principal payments discounted at a rate equal to the yield to maturity
This only works for liquid bonds, where the market price is presumed to be an accurate reflection of what it costs to purchase the bond
This is a sneak preview of the concept of an efficient market that is coming up in the equity part of the course
Professor Ross Miller • Fall 2007

26. But What About Bonds that Pay More than U. S. Treasuries
But What About Bonds that Pay More than U.S. Treasuries? They Must Have NPV>0.
No.
Their higher interest rate covers the risk that the issuer will default on the bond. (More on this next week.)
Last semester, the risk premiums for bonds were near an all-time low (Bloomberg article about European bonds)
Currently, risk premiums for corporate bonds are close to their historical average
Professor Ross Miller • Fall 2007

27. Par, Premium, and Discount Bonds
Bonds that trade at 100 are trading at par
Their yield to maturity is the same as their coupon rate
Bonds that trade above 100 are trading at a premium to par
Their yield to maturity is less than their coupon rate because receiving less than the bond’s price at maturity diminishes the interest payments
Bonds that trade below 100 are trading at a discount to par
Their yield to maturity is more than their coupon rate because receiving more than the bond’s price at maturity supplements the interest payments
Professor Ross Miller • Fall 2007

28. How a Treasury Bond’s Yield Changes
All of a Treasury bond’s cash flows are determined at the time of issuance (that is why it is called a fixed-income security)
The one variable is the price that it trades for in the market
In order for the yield on a bond to increase, that price must decrease
In order for the yield on a bond to decrease, that price must increase
Professor Ross Miller • Fall 2007

29. One Last Complication
Quoted bond prices (such as those on Bloomberg on in the Wall Street Journal) do not include interest accrued since the last coupon payment
This is known as the “clean price”
Adding in the accrued interest gives the “dirty price” (also known as the sale or invoice price), which is what is really paid for the bond
Professor Ross Miller • Fall 2007

30. Computing Accrued Interest
Accrued Interest = INT x Actual days since last coupon payment Actual days in coupon period
An example: 5.875% coupon rate, 81 days since last coupon payment and 184 days in coupon period: Accrued interest = (5.875%/2) x (81/184) = %
Professor Ross Miller • Fall 2007

31. Getting the Dirty Price
Dirty Price = Clean Price + Accrued Interest
Suppose the bond with the 5.875% coupon rate trades at , which is % of face value, so Dirty price = % % = %
Professor Ross Miller • Fall 2007

32. The Treasury Yield Curve from 9/18/2007: The Day the Fed Funds Target Rate was Cut
Professor Ross Miller • Fall 2007

33. The Yield Curve Has Been Steepening Lately
Professor Ross Miller • Fall 2007

34. Four Yield Curve Shapes
Upward-sloping
The situation that occurs most of the time
Provides a higher interest rate to lenders willing to assume the risk and reduced liquidity that comes from lending for a longer period of time
Downward-sloping (also known as inverted)
Usually the result of prolonged Fed tightening
Often the forerunner of a recession
Flat (a fleeting moment between the above two)
Humped (rare, but possibly the current shape)
Professor Ross Miller • Fall 2007

35. Why Sweat the Yield Curve?
It contains within it a forecast of future interest rates and provides the prices of forward rate agreements (FRAs) and interest rate swaps
The bond yield curve that is typically given in newspapers is not the one used by professionals because it is complicated by the coupon payments of the bonds
Professionals use the pure yield curve based on zero-coupon securities
It is a reminder that despite the fact that most of financial analysis assumes all cash flows from an investment can be discounted using the same interest rate, doing so it sometimes unwise
Professor Ross Miller • Fall 2007

36. For Those Who Cannot Get Enough of the Yield Curve
Animated yield curves abound on the Web
SmartMoney.com provides an animated yield curve with lots of additional explanation
StockCharts.com shows how the yield curve relates to the stock market
Professor Ross Miller • Fall 2007

37. Duration (also known as Macaulay Duration)
Duration is a measure of interest rate sensitivity that is expressed in units of years
Higher duration means a greater sensitivity to changes in interest rates
Doubling duration means doubling the interest rate sensitivity
A bond with a duration of 6 years will drop twice as much as a bond with a duration of 3 years when interest rates increase by 1 basis point (0.01%)
Professor Ross Miller • Fall 2007

38. Duration Example
Consider a Treasury bond (or note) with the following properties
One year left to maturity
$1,000 face value
8% coupon rate (paid semiannually)
Its yield (to maturity) is 10%
Cash flow from this bond
$40 (4% of $1,000) paid in ½ year
$1,040 (4% of $1,000 plus the $1,000 face value) paid in 1 year
Professor Ross Miller • Fall 2007

39. Computing the Present Value of the Bond
Value of $40 in ½ year = $40/(1.05) = $38.10
Value of $1,040 in 1 year = $1,040/(1.05)2 = $943.31
Total value = $ $ = $981.41
Professor Ross Miller • Fall 2007

40. Duration Example Timelines
Professor Ross Miller • Fall 2007

41. The Idea Behind Computing Duration
Duration is the cash-flow weighted time to maturity
Duration = (38.10)(0.5) + (943.31)(1.0) = years
Notice that if the first cash flow were eliminated, making the total value of the bond $943.31, the duration of the bond would simply be 1 year
Professor Ross Miller • Fall 2007

42. The General Formula for Duration
Here is the general formula for duration:
N N
 CFt  t  PVt  t
t = 1 (1 + R)t t = 1
D = N = N
 CFt  PVt
t = (1 + R)t t = 1
Professor Ross Miller • Fall 2007

43. Features of Duration Duration and Coupon Interest
The higher the coupon payment, the lower is a bond’s duration
Duration and Maturity
Duration increases with the maturity of a bond but at a decreasing rate
Professor Ross Miller • Fall 2007

44. Duration, Zero-Coupon Bonds, and Bond Portfolios
For a zero-coupon bond, duration equals maturity
The duration of a portfolio of bonds is the value- weighted average of the durations of the individual bonds
A portfolio with $1,000 in a bond with a duration of 2 years and $2,000 in a bond with a duration of 4 years has a duration of [2(1,000)+4(2,000)]/(3,000) = 3.33 years
Professor Ross Miller • Fall 2007

45. A Problem with Duration
It only applies to very small changes in the discount factor (as the price of the bond changes, so does duration)
The duration of zero-coupon bonds; however, only changes with time, not price
The measure of how much duration changes is known as convexity
Professor Ross Miller • Fall 2007

46. The Error in this Picture is the Consequence of Convexity
Professor Ross Miller • Fall 2007

47. For Week 5 Follow the links on the slides
Review everything that has been covered in class from RWJ Chapters 4 and 5
Figure out any questions you have up to this point in the course and ask them either in class, via , or at office hours
Do the problems on the 10 slides that follow this one
Attempt the Fall 2006 Monday edition of the mid-term (available on WebCT and bear in mind that financial conditions have changed since it was given).
Professor Ross Miller • Fall 2007

48. True-False Statements (Page 1 of 4)
After the U.S. Treasury issues a Treasury bond, its price never changes.
A premium bond will always have a lower yield to maturity than its coupon rate.
The actions of the Fed do not affect the duration of the 30-year Treasury Bond.
Professor Ross Miller • Fall 2007

49. True-False Statements (Page 2 of 4)
The duration of a bond can never increase.
The number of Treasury notes available for purchase changes over time.
Perpetuities are totally insensitive to changes in interest rates.
A 2-year Treasury note with a coupon rate of 6% will always have twice the yield to maturity of a 2-year Treasury note with a coupon rate of 3%.
Professor Ross Miller • Fall 2007

50. True-False Statements (Page 3 of 4)
The duration of a Treasury bond is always greater than its time to maturity.
If during the course of a trading day the price of a bond with several coupon payments left increases, its duration will also increase.
Everything else being equal (including yield to maturity), a bond with a higher coupon rate will be more sensitive to changes in interest rates than one with a lower coupon rate.
Professor Ross Miller • Fall 2007

51. True-False Statements (Page 4 of 4)
Over the past year many speculators have funded their operations by borrowing in yen because interest rates are the lowest in Japan.
Concerns about future U.S. inflation have increased the price of gold and decreased the price of the 30-year Treasury bond in recent weeks.
Professor Ross Miller • Fall 2007

52. “On-the-Run” (Most Recently Issued) Treasury Notes and Bonds as of 9/21/2005
Professor Ross Miller • Fall 2007

53. Treasury Note and Bond Questions Using The Previous Bond Price/Yield Slide
On 9/21/2005, what was the shape of the Treasury yield curve? (circle one)
Downward-sloping Flat Upward-sloping
Circle the Treasury Notes/Bond(s) that were trading at a premium.
2-year 3-year 5-year 10-year 30-year
If the 10-year Treasury note decreased in price by 15/32, what would its yield to maturity be?
Professor Ross Miller • Fall 2007

54. Coupon rate Yield to Maturity Duration
Treasury Note and Bond Questions Using The Previous Bond Price/Yield Slide (continued)
What is the effective annual interest rate of the 10-year Treasury note at the new price you found in the previous question? (Round to the nearest basis point.)
Suppose that you look at the price of the 5-year Treasury note today and see that it is unchanged at (99 and 13/32). Circle every value that will be lower then than it is now.
Coupon rate Yield to Maturity Duration
Professor Ross Miller • Fall 2007

55. Comparison of Treasury STRIPS and Note
Consider the following Treasury STRIPS and Treasury Note. The Treasury Note pays semiannual interest and both return $100 in principle at maturity. These quotes are from September 23, 2004, which is 39 days after the last coupon payment.
Security Coupon Rate Maturity Date Price
Treasury STRIP % Aug :07
Treasury Note % Aug :13
Professor Ross Miller • Fall 2007

56. Comparison of Treasury STRIPS and Note (continued)
What was the current yield of the Treasury note?
Was the yield to maturity of the Treasury note higher or lower than its current yield?
How much interest (as a percent of face value) did the Treasury note accrued in the 39 days since the last interest payment if there were 182 days in the interest period.
How much cash flow will $10,000 face value of the Treasury note in part 3 generate on August 15, 2008?
Professor Ross Miller • Fall 2007

57. Comparison of Treasury STRIPS and Note (continued)
An announcement of slower housing starts causes yields on all Treasury securities to decline by 0.01%. Which security, the Note or the STRIPS, will increase by the most as a percentage of its price?
Professor Ross Miller • Fall 2007