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### Fin 525 Week 4

Treasury Bonds and Duration

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 2006

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 2006

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 2006

Anatomy of the 5-year Treasury Note due 8/15/09 with a 3.50% Coupon Rate

$1,000 Principal Payable 8/15/09

$17.50CouponPayable2/15/07

$17.50CouponPayable2/15/05

$17.50CouponPayable8/15/05

$17.50CouponPayable2/15/06

$17.50CouponPayable8/15/06

$17.50CouponPayable8/15/08

$17.50CouponPayable2/15/09

$17.50CouponPayable8/15/09

$17.50CouponPayable8/15/07

$17.50CouponPayable2/15/08

Professor Ross Miller • Fall 2006

“On-the-Run” (Most Recently Issued)Treasury Notes and Bonds on 9/15/2006

Professor Ross Miller • Fall 2006

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, quotes are normally in 32nds

Professor Ross Miller • Fall 2006

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 2006

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 2006

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 2006

Computing a Bond’s Price from Its YieldStep 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 2006

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 2006

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 = $97.0851
- Add them together to get the bond’s value:$1.9512 + $97.0851 = $99.0363 or about $99.04
- Notice that because the yield is above the coupon rate, the bond trades at a discount

Professor Ross Miller • Fall 2006

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 2006

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 2006

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 2006

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 2006

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 2006

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 2006

Putting it all Together:The General Bond Valuation Formula

Annuity Factor

Professor Ross Miller • Fall 2006

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 2006

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 2006

Current Yield vs. Yield to Maturity

- Current Yield = Annual Coupon Payment/Price
- An example from RWJ, page 110
- 10% coupon, 1 year to maturity, $1,035.67 price
- Current yield = $100/$1,035.67 = 9.66%
- The problem is that in 1 year the bond matures at $1,000, so $35.67 is being forfeited over that year and should be reflected in the yield (and reduce it substantially in this case)

Professor Ross Miller • Fall 2006

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 2006

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 2006

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 2006

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 2006

Computing Accrued Interest

Accrued Interest = INT x Actual days since last coupon payment 2 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) = 1.29314%

Professor Ross Miller • Fall 2006

Getting the Dirty Price

Dirty Price = Clean Price + Accrued Interest

- Suppose the bond with the 5.875% coupon rate trades at 101-11, which is 101.34375% of face value, so Dirty price = 101.34375% + 1.29314% = 102.63689%

Professor Ross Miller • Fall 2006

The Treasury Yield Curve from 9/15/2006

Professor Ross Miller • Fall 2006

Four Yield Curve Shapes

- Upward-sloping
- The situation that occurs most of the now
- 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 2006

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 2006

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 2006

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 2006

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 2006

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 = $38.10 + $943.31 = $981.41

Professor Ross Miller • Fall 2006

Duration Example Timelines

Professor Ross Miller • Fall 2006

The Idea Behind Computing Duration

- Duration is the cash-flow weighted time to maturity

Duration = (38.10)(0.5) + (943.31)(1.0) 981.41 = 0.9806 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 2006

The General Formula for Duration

Here is the general formula for duration:

N N

CFt tPVt t

t = 1(1 + R)tt = 1

D = N = N

CFt PVt

t = 1 (1 + R)t t = 1

Professor Ross Miller • Fall 2006

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 2006

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 2006

A Big 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 2006

The Error in this Picture is the Consequence of Convexity

Professor Ross Miller • Fall 2006

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 e-mail, or at office hours
- Do the problems on the 9 slides that follow this one
- Attempt the Spring 2006 version of the mid-term (available on WebCT and bear in mind that financial conditions have changed since it was given several months ago)

Professor Ross Miller • Fall 2006

True-False Statements (Page 1 of 3)

- 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.
- As the Fed has raised the fed funds target rate from 1% to 5¼%, the yield curve has become more steeply sloped.

Professor Ross Miller • Fall 2006

True-False Statements (Page 2 of 3)

- The duration of a bond can never increase.
- The shape of the yield curve does not depend on the expected future actions of the Federal Reserve Board.
- 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 2006

True-False Statements (Page 3 of 3)

- 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 2006

“On-the-Run” (Most Recently Issued)Treasury Notes and Bonds as of 9/21/2005

Professor Ross Miller • Fall 2006

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 2006

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 99-13 (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 2006

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.

SecurityCoupon RateMaturity DatePrice

Treasury STRIP 0.000% 15-Aug-2008 89:07

Treasury Note 4.125% 15-Aug-2008 100:13

Professor Ross Miller • Fall 2006

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 2006

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 2006

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