3. U.S. Treasuries

from fixedincome2025 import table

Reasons To Consider U.S. Treasuries

• High credit quality

  • As of Sept 2025, 58% of disclosed global official foreign exchange reserves is in USD-denominated assets, mostly in the Treasuries

• Tax advantages

• Highly liquid

Types of Fixed Coupon U.S. Treasuries

\[\]

Type	Term Lengths	Payment Freq
T-Bills	4, 8, 13, 17, 26, 52 weeks	None (sold at discount and redeemed at par)
T-Notes	2, 3, 5, 7, 10 years	6M
T-Bonds	20, 30 years	6M

\[\]

Example: 2-Year Treasury Note Cash Flows

• Assuming:

  • Today is Sep 2025

  • Face Value: \(\$1,000\)

  • Coupon Rate: \(4.5\%\) annually

  • Semiannaual Coupon Payment:

    \[1000\times 0.045 /2 = \$22.5\]

Date	Payment
Mar 2026	$22.50
Sep 2026	$22.50
Mar 2027	$22.50
Sep 2027	$1,022.50

How To Buy

• You can buy on treasurydirect.gov or in your brokerage account

• You can buy newly issued in an auction

• You can buy or sell in a liquid secondary market

• Auction frequency:

\[\]

Type	Term Lengths	Payment Freq	Auction Freq
T-Bills	4, 8, 13, 17, 26, 52 weeks	None	Weekly
T-Bills	52 weeks	None	Every 4 weeks
T-Notes	2, 3, 5, 7 years	6M	Monthly
T-Notes	10 years	6M	Feb, May, Aug, Nov
T-Bonds	20, 30 years	6M	Feb, May, Aug, Nov

\[\]

Bond Pricing

• Given the Treasury yield curve, the bond price is sum of all cash flows discounted by \(P(t, T) = e^{-(T-t)R(t, T)}\)

• Assume today is Sep 2025 and the cash flow table

Date	Payment
Mar 2026	$22.50
Sep 2026	$22.50
Mar 2027	$22.50
Sep 2027	$1,022.50

• Bond price is

  \[22.5 e^{-0.5\times 0.0385} + 22.5 e^{-1\times 0.0365} + 22.5 e^{-1.5\times 0.0358} + 1022.5 e^{-2\times 0.0351} = \$1018.27\]

Bond Pricing: Exercise I

• Recall that the discount factor \(P(t, T) = e^{-(T-t)R(t, T)}\)

• Consider a 2-year T-Note with face value \(\$100\) and annual coupon rate \(5\%\)

• What’s the coupon payment?

• What’s the cash flow table?

• What’s the bond price?

Bond Pricing: Exercise I (Cont.)

• Semiannual coupon payment

  \[100\times 0.05 /2 = \$2.5\]

• Cash flow table

Date	Payment
Mar 2026	$2.50
Sep 2026	$2.50
Mar 2027	$2.50
Sep 2027	$102.50

• Bond price

  \[2.5 e^{-0.5\times 0.0385} + 2.5 e^{-1\times 0.0365} + 2.5 e^{-1.5\times 0.0358} + 102.5 e^{-2\times 0.0351} = \$102.78\]

Bond Pricing: Exercise II

• Recall that the discount factor \(P(t, T) = e^{-(T-t)R(t, T)}\)

• Consider a 1.5-year T-Note with face value \(\$100\) and annual coupon rate \(4\%\)

• What’s the coupon payment?

• What’s the cash flow table?

• What’s the bond price?

Bond Pricing: Exercise II (Cont.)

• Semiannual coupon payment

  \[100\times 0.04 /2 = \$2\]

• Cash flow table

Date	Payment
Mar 2026	$2.00
Sep 2026	$2.00
Mar 2027	$102.00

• Bond price

  \[2 e^{-0.5\times 0.0385} + 2 e^{-1\times 0.0365} + 102 e^{-1.5\times 0.0358} = \$100.56\]

U.S. Treasury Yield Curve Construction

• The yield curve can be backed out from the bond market data

• Bond market data example for yield curve construction from Section 4.7 of Options, Futures, and Other Derivatives 11th Edition:

\[\]

Principal ($)	Term (years)	Annual Coupon ($)	Bond Price ($)
100	0.25	0	99.6
100	0.50	0	99.0
100	1.00	0	97.8
100	1.50	4	102.5
100	2.00	5	105.0

\[\]

• Our goal is to back out zero rates for each term on the table

• Be careful of the “bond yield” in the book, which is defined differently as the zero rate

U.S. Treasury Yield Curve Construction (Cont.)

\[\]

Principal ($)	Term (years)	Annual Coupon ($)	Bond Price ($)
100	0.25	0	99.6
100	0.50	0	99.0
100	1.00	0	97.8
100	1.50	4	102.5
100	2.00	5	105.0

\[\]

• Assuming \(t=0\), we can already determine the zero rate at \(T=0.25\), \(T=0.5\) and \(T=1\), as we know the bonds have zero coupon

• Recall that \(P(t, T) = e^{-(T-t)R(t, T)}\)

• Solve

  \[P(0, 0.25) = 99.6/100 = 0.996 = e^{-0.25 \times R(0, 0.25)}\]

• \(R(0, 0.25) = 1.603\%\)

Exercise: Zero Rate for \(T=0.5\) and \(T=1\)?

\[\]

Principal ($)	Term (years)	Annual Coupon ($)	Bond Price ($)
100	0.25	0	99.6
100	0.50	0	99.0
100	1.00	0	97.8
100	1.50	4	102.5
100	2.00	5	105.0

\[\]

• \(P(t, T) = e^{-(T-t)R(t, T)}\), \(t=0\)

• \(R(0, 0.5) = 2.01\%\)

• \(R(0, 1) = 2.225\%\)

What if Coupon Is Not Zero?

\[\]

Principal ($)	Term (years)	Annual Coupon ($)	Bond Price ($)
100	1.50	4	102.5
100	2.00	5	105.0

\[\]

• We’ve established that the cash flow schedule of a 1.5 year bond that pays \(\$4\) annual coupon is

table('cashflow_18m')

	Payment
$T$
0.5	$2
1.0	$2
1.5	$102

• And we know the zero rates \(R(0, 0.5) = 2.01\%\), \(R(0, 1) = 2.225\%\), and the bond price

  \[2 e^{-0.5 \times R(0, 0.5)} + 2 e^{-1\times R(0, 1)} + 102 e^{-1.5\times R(0, 1.5)} = \$102.5\]

• Thus we can back out \(R(0, 1.5) = 2.284\%\)

Some Math

• Given

  • \(R(0, 0.5) = 2.01\%\)

  • \(R(0, 1) = 2.225\%\) \begin{align*} &2 e^{-0.5 \times R(0, 0.5)} + 2 e^{-1\times R(0, 1)} + 102 e^{-1.5\times R(0, 1.5)} = 102.5 \end{align*}

• Thus \begin{align*} 102 e^{-1.5\times R(0, 1.5)} &= 102.5 - 2 e^{-0.5 \times R(0, 0.5)} + 2 e^{-1\times R(0, 1)}\\ &= 102.5 - 2 e^{-0.5 \times 0.0201} - 2 e^{-1\times 0.02225} = 98.564, \end{align*} and hence \begin{align*} R(0, 1.5) = -\frac{\log(98.564/102)}{1.5} = 0.02284 \end{align*}

Zero Rate From Non-Zero Coupon Bond: Exercise

\[\]

Principal ($)	Term (years)	Annual Coupon ($)	Bond Price ($)
100	2.00	5	105.0

• Cash flow schedule:

table('cashflow_2y')

	Payment
$T$
0.5	$2.5
1.0	$2.5
1.5	$2.5
2.0	$102.5

• And we know the zero rates: \(R(0, 0.5) = 2.01\%\), \(R(0, 1) = 2.225\%\), \(R(0, 1.5) = 2.284\%\)

• Bond price is

  \[2.5 e^{-0.5\times 0.0201} + 2.5 e^{-1\times 0.02225} + 2.5 e^{-1.5\times 0.02284} + 102.5 e^{-2\times R(0, 2)} = \$102.5\]

• Thus we can back out \(R(0, 2) = 2.416\%\)

Dirty Price and Clean Price

• Most of the time the bond price quoted is not the price you pay to buy it!

• If you are in the middle of a coupon period, you need to pay part of the coupon too

  • What happens if such mechanism doesn’t exist?

  • Why not just quote dirty price?

• The part of the coupon is called the accrued interest

• The quoted price called the clean price, and the cash price to pay is called the dirty price

\[\]

\[\text{Dirty price} = \text{Clean price} + \text{Accrued interest since last coupon date}\]

Dirty Price and Clean Price (Cont.)

• Picture from here

Day Count Conventions (DCCs)

• To understand how to compute the dirty price, we need to first introduce DCCs

• So far all our examples were (mistakenly) using the 30/360 DCC – assuming there are 360 days in a year and 30 days in each month

  • Only with the 30/360 DCC, 6M has year fraction 0.5, 9M is 0.75, and 18M is 1.5

• Common DCCs are

  • Act/Act: US Treasuries, UK gilts (UK government bonds)

  • 30/360: US corporate and municipal bonds

  • Act/360: US money market instruments

  • Act/365: AUD, CAD money market instruments

Accrued Interest – Act/Act

• Assuming a US Treasury bond has

  • Principal \(\$100\)

  • Coupon payment dates 3/1 and 9/1

  • Coupon rate \(8\%\)

• What’s the accrued interest from 3/1 to 7/3?

• The interest for the period is \(\$4\)

• The actual number of days between 3/1 and 7/3 is (counting the start but not the end date)

  \[31 + 30 + 31 + 30 + 2 = 124\]

• The first for terms are for March, April, May and June, respectively

• The actual number of days between 3/1 and 9/1 is

  \[31 + 30 + 31 + 30 + 31 + 31 = 184\]

• So the accrued interest is

  \[\frac{124}{184} \times 4 = 2.6957\]

Accrued Interest – 30/360

• Assuming a corporate bond in the US has the same principal, coupon payment dates and coupon rate, what’s the accrued interest from 3/1 to 7/3?

• The interest for the period is \(\$4\)

• Number of days between 3/1 and 7/3, under the 30/360 DCC, is

  \[4\times 30 + 2 = 122\]

• and between 3/1 and 9/1 is

  \[6\times 30 = 180\]

• So the accrued interest is

  \[\frac{122}{180} \times 4 = 2.7111\]

Other Treasuries

• Aside from fixed coupon Treasuries, there are also TIPS (Treasury inflation-protected securities), FRNs (Treasury floating rate notes), and Treasury STRIPS (Separate Trading of Registered Interest and Principal Securities)

\[\]

Type	Term Lengths	Payment Freq
TIPS	5, 10, 30 years	6M
FRNs	2 years	Quarterly (rate resets weekly)
STRIPS	Derived from notes/bonds	None (zero-coupon; sold at discount)

\[\]

Treasury Inflation-Protected Securities (TIPS)

• Every 6M, the principal is adjusted with the changes in Consumer Price Index (CPI)

• Interest payment is adjusted principal multiplied by a fixed coupon rate

• At maturity, investor receives the greater of original or inflation-adjusted principal

• Coupon exempt from state and local taxes; subject to federal tax

TIPS Example (2 Coupon Periods)

• Assuming

  • Face Value: \(\$1,000\)

  • Coupon Rate: \(1\%\) annually (\(0.5\%\) semiannually)

  • CPI:

    • First 6 months: \(2\%\)

    • Second 6 months: \(1.5\%\)

• First coupon period adjusted Principal

  \[\$1,000 \times (1 + 0.02) = \$1,020\]

• First coupon payment

  \[\$1,020 \times 0.5\% = \$5.10\]

• Second coupon period new adjusted principal

  \[\$1,020 \times (1 + 0.015) = \$1,035.30\]

• Second coupon payment

  \[\$1,035.30 \times 0.5\% = \$5.18\]

\[\]

Period	Adjusted Principal	Coupon Rate	Coupon Paid
First 6 months	$1,020.00	0.5%	$5.10
Second 6 months	$1,035.30	0.5%	$5.18

U.S. Treasury Floating Rate Notes (FRNs)

• Maturity is 2Y

• Interest resets weekly based on 13-week Treasury bill rate + fixed spread

Treasury STRIPS

• Created by stripping interest and principal from Treasury notes/bonds

• Sold at a discount; pay full face value at maturity

• Backed by U.S. government