1. Economics 4340 Theory of Financial Markets
Professor Edwin T Burton
Economics Department
The University of Virginia

2. Fixed Income Securities
What is fixed income?
A method of borrowing money where the return to the lender is fixed.
Return is made up of two components:
Principal = Original amount of loan
Interest = Additional compensation to lender
This exchange is a liability to the borrower and an asset to the lender.
Risks can exist along many dimensions – one main one is “default risk”

3. Fixed Income Securities
Default-Free Securities
May have other risk but default risk is absent… lender 100% assured of receiving payment as specified in the contract
No real world examples of pure default-free securities
Closest example – US government debt
Never defaulted on any debt in 220 years of borrowing
Lenders operate assuming repayment is guaranteed

4. Fixed Income Securities
Main types of U.S. Treasury Securities
U.S. Treasury Bills – orig. maturity of under one year
U.S. Treasury Notes – orig. maturity of 1 to 10 years
U.S. Treasury Bonds – orig. maturity of over 10 years

5. Fixed Income Securities
Characteristics
Treasury Bills = Discount Securities – one payment to owner
Treasury Notes & Bonds = Coupon Securities – interest payments made every 6 months until maturity, on which interest + principal is paid
How are these debt instruments sold?
Most U.S. Treasuries are sold via “single-price Dutch auctions.”

6. U.S Treasury Bill Auctions
Let’s say the Treasury wants to sell $10 billion (the “offering”) of 6-month Treasury bills. After announcing the auction, it accepts bids from potential buyers. Each bid lists the amount the bidder is willing to purchase at various prices – essentially, it represents the bidder’s entire demand curve for that security.

7. U.S Treasury Bill Auctions
A bid may look something like (note that a low quote is a high price): Non-Com is short for non-competitive – this is like bidding a very low quote or an infinite price.
Quote
Amount
5.1
$100 million
5.05
$200 million
5.0
$300 million
Non-Com
$50 million

8. U.S Treasury Bill Auctions
The Fed fills bids via Dutch auction
Bids filled from the highest price down
Non-Coms filled first
Lowest quote (highest price) filled second
Process continued until entire offering accounted for
Quote at which market clears becomes the quote for everyone – “single-price” auction

9. U.S Treasury Bill Auctions
Let’s say the total of the bids across all bidders was:
$3 billion in Non-Coms
$2 billion at 4.90
$2 billion at 5.00
$1 billion at 5.02
Then the auction results would be:
Market clears at 5.08
Our firm’s tenders through 5.05 are filled (at 5.08)
Tenders at 5.10 are left unfilled
$1 billion at 5.05
$1 billion at 5.08
$1 billion at 5.10

10. U.S Treasury Bill Quotes
What does this quote mean?
It is the bill’s discount, annualized to a 360-day year
We want to calculate the bill’s yield, annualized to a 365-day year. To do so:
Calculate the adjusted discount: 180/360 * 5.08 = 2.54
This means the price today is 2.54% less than the bill’s face value. E.g., face value = $1,000,000, today’s price is $1,000,000 * ( ) = $974,600
(1+r) =
Lastly, we annualize this yield: 2.606% * 365/180 ¼ 5.28% $x PV
$1,000,000
$974,600 = ¼
! r ¼ 2.606%

11. U.S Treasury Bill Quotes
Let’s say 80 days pass, and the discount moves to To calculate the new yield:
Calculate the adjusted discount: 100/360 * 4.5 = 1.25
Today’s price = $1,000,000 * ( ) = $987,500
= PV !
So the 100-day yield is:
Lastly, we annualize this yield: 1.266% * 365/100 ¼ 4.62% $x
(1+r) $x PV
$x - PV PV
= (1+r)
! r =
$12,500
$987,500
¼ 1.266%

12. U.S Treasury Bill Quotes
More generally, let:
d = quote,
Tsm = Time from settlement to maturity,
F = T-bill’s principal amount, or face value
Then:
Bill’s absolute discount D = d ¢ F ¢
Bill’s price P = F – D
Bill’s annualized yield y = ¢
Note that, in the Treasury bill market, the actual yield will generally exceed the quote, with some rare exceptions.
Tsm
360 D P
365
Tsm

13. U.S Treasury Bills Five types of Treasury Bills: Four-week bills
Three-month bills
Six-month bills
The year bill (none issued since Feb. 27, 2001)
Cash Management Bill
Used to provide cash when Treasury accounts are low
Rarely issued – only for short-term cash needs

14. Treasury Notes & Bonds
As opposed to Treasury Bills, Treasury Notes and Bonds have an orig. maturity over 1 year
Examples of Notes and Bonds:
2-Year Notes
5-Year Notes
20-Year Bonds (last auctioned January 1986)
30-Year Bonds
Only difference between notes and bonds is the original maturity – other than that, we will treat them as interchangeable (including the names)

15. U.S Treasury Notes and Bonds
As opposed to Treasury Bills, Treasury Notes and Bonds are coupon securities – pay two interest payments annually until maturity How do we know when these interest payments take place? All the information we need to determine interest payment amounts and timing exists in the bond’s name.

16. U.S Treasury Notes and Bonds
Example: “ 5¼’s of Feb/29 ” “Feb/29” tells us the security matures on February 15th, 2029 (all new/recent securities use the 15th). This also tells us interest payments will occur on February 15th and August 15th every year until (and including) the maturity date.
Security’s Coupon
Security’s Maturity Date

17. U.S Treasury Notes and Bonds
Example: “ 5¼’s of Feb/29 ” “5¼’s” tells us the security pays a total of 5¼% of the principal in interest every year This payment is made in two equal installments semiannually (in this case, on February 15th and August 15th of each year)
Security’s Coupon
Security’s Maturity Date

18. U.S Treasury Notes and Bonds
Example: 5¼’s of Feb/29, principal = $100,000
From this, we know all future payments from this bond.
Principal on February 15th, 2029 ! $100,000 payment
Total interest paid yearly = 5¼% * $100,000 = $5,250
Half of this interest ($2,625) paid every February 15th, other half ($2,625) paid every August 15th until 2/15/29
This represents all future cash flows related to this bond.

19. U.S Treasury Notes and Bonds
Example: 4½’s of Aug/33, principal = $1,000,000
Bond’s future payments:
Principal on August 15th, 2033 ! $1,000,000 payment
Total interest paid yearly = 4½% * $1,000,000 = $45,000
Until February 15th, 2033, half of this interest ($22,500) paid every August 15th, other half ($22,500) paid every February 15th
Again, this represents all future cash flows related to this bond.

20. U.S Treasury Notes and Bonds
So the name of the bond and its principal tells us everything we get from the bond - but how do we know what we have to pay to buy it?
The price is listed in the market, as a percentage of the principal amount.
Example: Bond with $1,000,000 principal value
Listed Price = 90 ! Actual Price = $900,000
Listed Price = 110 ! Actual Price = $1,100,000
Listed Price = 100 ! Actual Price = $1,000,000

21. U.S Treasury Notes and Bonds
Prices can be listed in 1/32nd increments.
Example: Bond with $1,000,000 principal
Listed Price = 98 5/32 ! Actual Price = $981,562.50
Listed Price = 100 3/16 ! Actual Price = $1,001,875.00
Listed Price = 103 1/4 ! Actual Price = $1,032,500.00
Listed Price = 97 5/8 ! Actual Price = $976,250.00
Now let’s say the next coupon payment is on Feb. 15, 2018, and we buy the bond on Feb. 14, 2018 – do we essentially get that coupon payment for free?

22. U.S Treasury Notes and Bonds
No – we need to account for accrued interest.
Example: Let’s say we buy the 4½’s of Aug/33 with $1,000,000 principal on Oct 31, 2017 at 102 ¼.
Price = 102 ¼ * $1,000,000 / 100 = $1,022,500.00
Coupon Payments = $22,500 each
Most recent coupon payment = August 15th, 2017
Next payment on February 15th, 2018, 184 days later; 77 of those days have passed
Accrued interest = $22,500 * 77 / 184 = $9,415.76

23. U.S Treasury Notes and Bonds
We will receive the full $22,500 on February 15th, 2018, even though we only held the bond for 107 of the 184 days between coupon payments. Therefore, we need to compensate the current bondholder for losing this accrued interest. Total price = $1,022,500 + $9, = $1,031, This is called the “price plus accrued” – U.S. Treasury Securities always trade with accrued.

24. U.S Treasury Notes and Bonds
Example: Same bond, but we bought it on Sept. 2nd, 2017 at a price of 98 11/32.
Price = 98 11/32 * $1,000,000 / 100 = $983,437.50
Accrued interest = $22,500 * 18 / 184 = $2,201.09
Price with Accrued = $983, $2, = $985,638.59
Let’s say we buy this bond (4½’s of Aug/33) on February 15th, 2018, right after the coupon payment (so no accrued interest) at a price of 100… what return would we be getting on our investment?

25. U.S Treasury Notes and Bonds
First, what exactly are we getting?
$22,500 coupon payments every 6 months from August 15th, 2018 until August 15th, 2033 (31 total)
$1,000,000 maturity payment on August 15th, 2033
And what exactly are we paying?
Listed Price = 100 ! Actual Price = $1,000,000.00
No accrued, so total cost = $1,000,000.00
So, we need to find the interest rate that equates the current price to the PV of future cash flows.

26. U.S Treasury Notes and Bonds
In other words, we’re looking for the r that solves this equation (note that here, r is the 6-month rate): Could solve this via trial and error… However, note that this is the same stream of cash flows we’d get if we put $1M into an account bearing 2.25% every 6 months.
$22.5K
(1+r)1 +
$22.5K
(1+r)2 +
$22.5K
(1+r)3 +
$22.5K
(1+r)4
$1M = + … +
$22.5K
(1+r)29 +
$22.5K
(1+r)30 +
$22.5K
(1+r)31 +
$1M
(1+r)31

27. U.S Treasury Notes and Bonds
Since we can replicate the future cash flows with a 2.25% 6-month rate, that must be the value of r. We need to annualize this rate… normally, we would consider the compounding from two periods of 6-month interest. However, by convention, we simply double the 6-month rate… here, the annual rate is 4.5%.
This is called the bond’s “yield-to-maturity” – it is the most important metric on which we measure bonds.

28. U.S Treasury Notes and Bonds
Note that the bond had a 4.5% coupon rate and also ended up yielding a 4.5% interest rate… why is this? In general, when a bond’s listed price is 100, its coupon rate and yield will be the same. What if the price dropped to 98 – what would happen to the yield?

29. U.S Treasury Notes and Bonds
Now, we’re looking for the 6-month rate r that solves: The present value of the same future stream of cash flows has gone down. Therefore, the implied interest rate, or the yield, must have increased.
$22.5K
(1+r)1 +
$22.5K
(1+r)2 +
$22.5K
(1+r)3 +
$22.5K
(1+r)4 + …
$980K = +
$22.5K
(1+r)29 +
$22.5K
(1+r)30 +
$22.5K
(1+r)31 +
$1M
(1+r)31

30. U.S Treasury Notes and Bonds
What if the price had been 102? Here, the present value of the same future stream of cash flows has gone up. Therefore, the implied interest rate, or the yield, must have decreased.
$22.5K
(1+r)1 +
$22.5K
(1+r)2 +
$22.5K
(1+r)3 +
$22.5K
(1+r)4 + …
$1.02M = +
$22.5K
(1+r)29 +
$22.5K
(1+r)30 +
$22.5K
(1+r)31 +
$1M
(1+r)31

31. U.S Treasury Notes and Bonds
This demonstrates an important relationship between bond prices and yields… We’ve covered what happens to bonds in the market, but how are they issued in the first place?
Bond prices and yields vary inversely. That is, as prices go up, yields go down, and vice versa.

32. U.S Treasury Notes and Bonds
New T-note and T-bond issues are sold via Dutch auction, just like T-bills, except the bids submitted are yields-to-maturity, not discount quotes. Example: The Treasury is auctioning off $15 billion of 30-year bonds. One firm’s bid may look something like:
Yield-to-maturity
Amount
5.8
$100 million
5.7
$200 million
5.5
$300 million
Non-Com
$50 million

33. U.S Treasury Notes and Bonds
The Fed then fills bids from the lowest yield upwards, eventually reaching a market-clearing yield. The bond’s coupon is then set at that yield, rounding to the nearest 1/8th, so that the starting price is as close to 100 as possible. Examples:
Auction Result
Coupon Rate
4.55
4.5
4.97
5.0
5.11
5.125
5.25

34. U.S Treasury Notes and Bonds
If the coupon ends up above the yield, the purchase price will be just above 100. If the coupon ends up below the yield, the purchase price will be just below 100. After the bonds are issued, they are traded openly in the market, and market forces determine price movements. More buyers than sellers ! the price goes up (and hence the implied yield goes down), and vice versa.

35. U.S Treasury Securities
We’ve seen securities under one year (Bills) have no intermediate payments, and those over one year (Notes/Bonds) have semiannual payments. What if you want a long-term Treasury security with only one payment at maturity? The Treasury does not provide such an instrument… how could we create one? By buying a note or bond and selling off the pieces individually.

36. U.S Treasury Securities
This is a process known as “stripping” the note or bond.
To strip a security:
Buy a note or bond (say, $1M of the 5’s of Aug/19),
Give it to a custodian bank and split up the payments,
Sell each of the payments – in this case, the coupon payments of $25,000 on Feb. 15th 2018 and 2019 and Aug. 15th 2018 and 2019, plus the principal repayment of $1M on Aug. 15th, 2019.
Each strip is known as a “zero-coupon bond.”