Treasury Yields Are Not One Number: A guide to T-Bills, Bonds, TIPS, XIRR, & Zero Curves

Disclosure: This is a ChatGPT summarized post of a long dialog I had to clarify terminology and analysis strategies for TIPS. The material reflects the key items I believe are important when studying how to analyze US Treasury and TIPs bonds and markets. I have reviewed for correctness; and edited for clarity, and conciseness.

One of the confusing things about U.S. Treasury securities is that the word yield can refer to several different calculations.

A Treasury bill can have a bank discount yield, an investment yield, an effective annual yield, and an XIRR (internal rate of return). A Treasury Note, Bond or TIPS has a conventional yield to maturity. A fitted yield curve introduces zero-coupon or spot yields, par yields, discount factors, and forward rates.

None of these rates are necessarily wrong. Each answers different questions and should be employed in different contexts.

The terms yield and rates can be used interchangeably in most contexts. The important distinction is the modifier.

The key to proper usage depends upon understanding the underlying cash flows and the convention being used so they may be applied appropriately.

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Start With Price and Cash Flow

Suppose an investment costs $𝑃$ (price) today and ultimately pays $\mathrm{𝐹\; (face)}$.

For a Treasury bill, there are no coupon payments so the cash flow is very simple:

$$-𝑃⟶𝐹$$

For a Treasury Note or Bond, there are multiple cash flows:

$$-𝑃,{𝐶}_1,{𝐶}_2,\dots ,{𝐶}_{𝑛}+𝐹$$

For a TIPS, the same basic structure exists, except that principal and coupon payments are adjusted for inflation at each instance.

Everything that follows is simply different ways to describe the relationship between:

• prices,
• cash flows,
• and their dates.

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Treasury Bill Yields

Treasury bills are zero-coupon securities. The investor buys the bill below its maturity value and receives the maturity (face) value at maturity.

The U.S. Treasury reports both bank discount and coupon-equivalent rates for bills. [home.treasury.gov], [ecfr.gov]

Let:

• $𝑃$ = purchase price
• $𝐹$ = face or maturity value
• $𝑑$ = days to maturity

1. Holding-Period Return

The most basic return ignores annualization:

$$𝐻𝑃𝑅=\frac{𝐹-𝑃}{𝑃}$$

If a $100 bill costs $99:

$$𝐻𝑃𝑅=\frac{100-99}{99}=1.0101\mathrm{\%}$$

This tells us what was actually earned during the investment period.

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2. Bank Discount Yield

The traditional T-bill quotation is based on the discount from face value rather than the amount invested.

A common expression is:

$$𝐵𝐷𝑌=\frac{𝐹-𝑃}{𝐹}\times \frac{360}{𝑑}$$

Notice two unusual conventions:

1. The return is divided by face value, not the price paid.
2. It uses a 360-day year.

Consequently, bank discount yield is primarily a quotation convention, rather than the most natural description of the investor's return.

Treasury regulations separately define the calculations for T-bill purchase price, discount rate, and investment rate. [ecfr.gov]

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3. Investment Yield

An Investment Yield (aka Simple Rate) instead relates the gain to the money actually invested:

$${𝑟}_{𝑠𝑖𝑚𝑝𝑙𝑒}=\frac{𝐹-𝑃}{𝑃}\times \frac{𝑌}{\mathrm{𝑑<span><br></span>}}$$

where $𝑌$ represents the applicable annual day basis.

This is much closer to the economic question:

What annualized rate did I earn on the money I actually invested?

For short holding periods, this is essentially a simple-interest annualization.

The annual day basis has various conventions such as 365, 3665.25, or actual days (e.g., 366 in periods with a leap day). It is important to understand the convention of the specific application. [add reference to Treasury description of rules]

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4. Effective Annual Yield

A different question could:

 What annual return would result if the holding-period return compounded over an entire year (Y):

$$𝐸𝐴𝑌={\left(\frac{𝐹}{𝑃}\right)}^{𝑌\mathrm{/}𝑑}-1$$

Since:

$$\frac{𝐹}{𝑃}=1+𝐻𝑃𝑅$$

This can also be written:

$$𝐸𝐴𝑌=(1+𝐻𝑃𝑅{)}^{𝑌\mathrm{/}𝑑}-1$$

This differs from a simple annualized yield because it assumes compounding. This is evident by the exponent based upon time.

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Treasury Notes, Bonds, and TIPS

Coupon securities have several future payments, making the problem different from a T-bill.

5. Yield to Maturity

A conventional Treasury yield to maturity finds the single yield $𝑦$ that satisfies the bond-pricing relationship:

$$𝑃=\sum _{𝑖=1}^{𝑁}\frac{{𝐶}_{𝑖}}{(1+𝑦\mathrm{/}2{)}^{{𝑡}_{𝑖}}}+\frac{𝐹}{(1+𝑦\mathrm{/}2{)}^{{𝑡}_{𝑁}}}$$​

For regular semiannual periods this is often presented more simply as:

$$𝑃=\sum _{𝑖=1}^{𝑁}\frac{𝐶}{(1+𝑦\mathrm{/}2{)}^{𝑖}}+\frac{𝐹}{(1+𝑦\mathrm{/}2{)}^{𝑁}}$$​

with appropriate adjustments for settlement between coupon dates.

The important idea is that YTM applies one yield to all of the bond's cash flows.

Therefore YTM asks:

What single semiannually compounded rate explains the price of this particular bond?

Yield to maturity is a single summary rate for the entire bond. It is the one constant discount rate that, under the bond's yield convention, makes the present value of all remaining coupons and principal equal its market price. This does not imply that the market actually assigns the same discount rate to a payment six months from now as to one five years from now. A zero-coupon or spot curve estimates separate market discount rates for those different payment dates.

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6. XIRR

XIRR provides another useful way to describe an investment when we know the actual dates of every cash flow.

XIRR finds $𝑟$ satisfying:

$$0=\sum _{𝑖=0}^{𝑁}\frac{𝐶{𝐹}_{𝑖}}{(1+𝑟{)}^{({𝑑}_{𝑖}-{𝑑}_0)\mathrm{/}365}}$$​​

where:

• $𝐶{𝐹}_{𝑖}$​ = cash flow
• ${𝑑}_{𝑖}$ = date of that cash flow
• ${𝑑}_0$​ = initial investment date
• $𝑟$ = annual effective return

Microsoft describes XIRR as the return calculation for a schedule of cash flows that is not necessarily periodic. [support.microsoft.com]

This makes XIRR particularly useful as a common comparison framework:

• T-Bill→𝑋𝐼𝑅𝑅
• Treasury Note→𝑋𝐼𝑅𝑅
• Projected nominal TIPS cash flows→𝑋𝐼𝑅𝑅

Unlike conventional Treasury quotations, all of these can be passed through the same mathematical framework for apples-to-apples yield comparisons of a yield for different types of securities.

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Zero-Coupon or Spot Yields

Suppose a five-year Treasury pays coupons every six months.

Its YTM uses one rate to price everything:

$${𝐶}_{0.5},{𝐶}_1,{𝐶}_{1.5},\dots ,{𝐶}_5+𝐹$$

But the market may value a dollar received six months from now differently from a dollar received five years from now.

That leads to the zero-coupon curve. A zero-coupon yield describes the return associated with a single payment at a specific future time.

Let:

$$𝐷(𝑡)$$

be the discount factor for time $𝑡$.

It answers:

What is $1 paid at time $𝑡$ worth today?

The Federal Reserve's Gürkaynak-Sack-Wright yield-curve work describes the discount function as the fundamental building block used to value future Treasury payments. [federalreserve.gov], [federalreserve.gov]

7. Estimating the Zero Curve: Nelson-Siegel and Svensson

Zero-coupon yields are conceptually simple, but most Treasury securities are coupon-bearing securities rather than zero-coupon bonds. As a result, the complete zero curve cannot simply be read from a list of Treasury yields. It has to be estimated from market prices.

Two commonly discussed parametric approaches are Nelson-Siegel (NS) and its extension, Nelson-Siegel-Svensson (NSS), often simply called the Svensson model.

The Nelson-Siegel model describes the term structure using four parameters. The Svensson extension adds two more parameters, allowing a second “hump” in the forward-rate curve and greater flexibility at longer maturities. The Gürkaynak-Sack-Wright (GSW) Treasury curve methodology uses the six-parameter Svensson specification for this reason. [federalreserve.gov]

For Svensson, the continuously compounded zero-coupon yield at maturity $𝑡$ can be written as:

$$\boxed{{\displaystyle 𝑧(𝑡)={𝛽}_0+{𝛽}_1\left(\frac{1-{𝑒}^{-𝑡\mathrm{/}{𝜏}_1}}{𝑡\mathrm{/}{𝜏}_1}\right)+{𝛽}_2[\frac{1-{𝑒}^{-𝑡\mathrm{/}{𝜏}_1}}{𝑡\mathrm{/}{𝜏}_1}-{𝑒}^{-𝑡\mathrm{/}{𝜏}_1}]+{𝛽}_3[\frac{1-{𝑒}^{-𝑡\mathrm{/}{𝜏}_2}}{𝑡\mathrm{/}{𝜏}_2}-{𝑒}^{-𝑡\mathrm{/}{𝜏}_2}]}}$$

The six fitted parameters are:

β0​,β1​,β2​,β3​,τ1​,τ2​

The parameters control the level, slope, and curvature of the fitted term structure. In particular, the additional ${𝛽}_3,{𝜏}_2$component gives Svensson the second curvature or “hump” component beyond Nelson-Siegel. [federalreserve.gov]

Once 𝑧(𝑡) is known, the corresponding discount factor follows directly:

D(t) = e^-z(t)t

The important point is that the curve is fitted against bond prices, not merely against the quoted YTM of each bond.

For each Treasury security, the model calculates:

P_model = ∑_i CF_iD(t_i) = ∑_i CF_i e^-z(t_i)t_i

The six Svensson parameters are selected so that these model prices collectively fit the observed market prices as closely as possible. In the GSW methodology, price errors are weighted by the inverse of each security's duration. [federalreserve.gov]

The process can therefore be pictured as:

This distinction is important: the Svensson curve does not convert each bond's YTM into a spot yield. Instead, the model searches for a smooth term structure of zero rates that can jointly explain the prices of many coupon-bearing securities.

Once that term structure has been fitted, it can be used to derive:

• Zero/Spot Yields​
• Discount Factors
• Par Yields
• ​Forward Rates

and to calculate a theoretical price for any bond whose cash flows are known. The GSW paper explicitly describes the Svensson specification as characterizing both the yield curve and discount function across maturities, after which the discount function can be used to price Treasury securities

Difference between Nelson-Siegel (NS) and Nelson-Siegel-Svensson (NSS)

• Nelson-Siegel (NS)
• 4 parameters
• β0 β1 β2 τ1
• one curvature/hump component

• Nelson-Siegel-Svensson (NSS)
• 6 parameters
• β0 β1 β2 β3 τ1 τ2
• adds a second curvature/hump component

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8. Continuously Compounded Zero Yield

If $𝑧(𝑡)$ is the continuously compounded zero yield:

$$\boxed{{\displaystyle 𝐷(𝑡)={𝑒}^{-𝑧(𝑡)𝑡}}}$$​

Therefore:

$$\boxed{{\displaystyle 𝑧(𝑡)=-\frac{\ln 𝐷(𝑡)}{𝑡}}}$$​

This is exceptionally useful because every cash flow can now be valued using the rate appropriate to its own date:

$$\boxed{{\displaystyle 𝑃=\sum _{𝑖=1}^{𝑁}𝐶{𝐹}_{𝑖}𝐷({𝑡}_{𝑖})}}$$​

or:

$$\boxed{{\displaystyle 𝑃=\sum _{𝑖=1}^{𝑁}𝐶{𝐹}_{𝑖}{𝑒}^{-𝑧({𝑡}_{𝑖}){𝑡}_{𝑖}}}}$$​

This is fundamentally different from YTM.

YTM says:

Find one rate that explains this bond.

The zero curve says:

Find the appropriate discount rate for every future cash flow date.

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9. Continuous Zero Yield to Bond-Equivalent Zero Yield

A continuously compounded zero yield can be converted to a semiannually compounded bond-equivalent zero yield.

If ${𝑧}_{𝑐𝑐}$ is the continuously compounded rate:

$$\boxed{{\displaystyle {𝑧}_{𝐵𝐸𝑌}=2({𝑒}^{{𝑧}_{𝑐𝑐}\mathrm{/}2}-1)}}$$​

Going the other direction:

$$\boxed{{\displaystyle {𝑧}_{𝑐𝑐}=2\ln (1+\frac{{𝑧}_{𝐵𝐸𝑌}}{2})}}$$​

This allows the same underlying discount factor to be reported using different yield conventions.

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10. Discount Factor From a Bond-Equivalent Zero Yield

Given a bond-equivalent zero yield $\mathrm{𝑦\; (}$𝑧𝐵𝐸𝑌 above):

$$\boxed{{\displaystyle 𝐷(𝑡)=\frac{1}{{(1+𝑦\mathrm{/}2)}^{2𝑡}}}}$$​

The continuous yield and bond-equivalent yield are simply alternate descriptions of the same discount factor.

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11. Discount Factors From Treasury Bills

A Treasury bill is particularly convenient because it already has a single maturity cash flow.

If a bill will pay $100 at maturity and has a market price of $𝑃$P:

$$\boxed{{\displaystyle 𝐷(𝑇)=\frac{𝑃}{100}}}$$​

So a bill priced at:

$$99.50$$

immediately implies:

$$𝐷(𝑇)=0.9950$$

No coupon stripping or fitted YTM is required.

We can then calculate its continuously compounded zero rate:

$$𝑧(𝑇)=-\frac{\ln (0.9950)}{𝑇}$$​

This makes T-bills especially useful for estimating short-term discount factors.

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12. Simple Rate From a Discount Factor

For short-term financing calculations, we may want a simple annualized rate (aka Investment Yield).

If:

$$\frac{1}{𝐷(𝑇)}=1+𝑟\frac{𝑑}{𝑌}$$​

then:

$$\boxed{{\displaystyle 𝑟=(\frac{1}{𝐷(𝑇)}-1)\frac{𝑌}{𝑑}}}$$​

Going the other direction:

$$\boxed{{\displaystyle 𝐷(𝑇)=\frac{1}{1+𝑟(𝑑\mathrm{/}𝑌)}}}$$​

This becomes useful for carrying a Treasury or TIPS price to a future settlement date.

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13. Forward Discount Factors

Suppose we know:

$$𝐷({𝑡}_1)$$

and:

$$𝐷({𝑡}_2)$$

The discount factor applying specifically between ${𝑡}_1$​ and ${𝑡}_2$​ is:

$$\boxed{{\displaystyle 𝐷({𝑡}_1,{𝑡}_2)=\frac{𝐷({𝑡}_2)}{𝐷({𝑡}_1)}}}$$​

The corresponding accumulation, or carry, factor is:

$$\boxed{{\displaystyle 𝐴({𝑡}_1,{𝑡}_2)=\frac{𝐷({𝑡}_1)}{𝐷({𝑡}_2)}}}$$​

This is particularly convenient for forward pricing because it avoids unnecessary conversion into an intermediate yield convention.

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14. Continuously Compounded Forward Rate

The continuously compounded rate between ${𝑡}_1$ and ${𝑡}_2$​ is:

$$\boxed{{\displaystyle 𝑓({𝑡}_1,{𝑡}_2)=-\frac{\ln [𝐷({𝑡}_2)\mathrm{/}𝐷({𝑡}_1)]}{{𝑡}_2-{𝑡}_1}}}$$​

This is a forward rate, not today's spot rate for maturity ${𝑡}_2$​.

It answers:

What rate between future dates ${𝑡}_1$ and ${𝑡}_2$​ is implied today by the current term structure?

The Federal Reserve notes that fitted Treasury curves can be represented as zero-coupon yields, par yields, instantaneous forward rates, or forward rates over specified future intervals. [federalreserve.gov]

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15. Par Yield

The par yield answers another question:

What coupon would cause a hypothetical bond of a given maturity to trade at par?

Given semi-annual discount factors:

$$𝐷({𝑡}_1),𝐷({𝑡}_2),\dots ,𝐷({𝑡}_{𝑁})$$

and a $100 face value, par requires:

$$100=\frac{𝑐}{2}\sum _{𝑖=1}^{𝑁}100𝐷({𝑡}_{𝑖})+100𝐷({𝑡}_{𝑁})$$

Solving for the annual coupon rate:

$$\boxed{{\displaystyle 𝑐=\frac{2[1-𝐷({𝑡}_{𝑁})]}{\sum _{𝑖=1}^{𝑁}𝐷({𝑡}_{𝑖})}}}$$​

This distinction matters when looking at Treasury CMT rates.

• A par yield describes a hypothetical par coupon security.
• A spot yield describes a single payment at a particular horizon.

These yields are generally not identical.

The par yield 𝑐 is the annual coupon rate that would cause a hypothetical Treasury security of a specified maturity to trade at par when its cash flows are valued using the underlying term structure. The U.S. Treasury's published Constant Maturity Treasury (CMT) rates are par yields read from Treasury's fitted daily par yield curve at specified constant maturities.

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16. Price Residuals and "Cheap" Bonds

Once a zero curve has been estimated, each individual bond can be repriced:

$${𝑃}_{𝑚𝑜𝑑𝑒𝑙}=\sum _{𝑖}𝐶{𝐹}_{𝑖}𝐷({𝑡}_{𝑖})$$

Then define:

$$\boxed{{\displaystyle 𝑅𝑒𝑠𝑖𝑑𝑢𝑎{𝑙}_{𝑃}={𝑃}_{𝑚𝑎𝑟𝑘𝑒𝑡}-{𝑃}_{𝑚𝑜𝑑𝑒𝑙}}}$$​

Under this sign convention:

$$𝑅𝑒𝑠𝑖𝑑𝑢𝑎{𝑙}_{𝑃}<0$$

means:

$${𝑃}_{𝑚𝑎𝑟𝑘𝑒𝑡}<{𝑃}_{𝑚𝑜𝑑𝑒𝑙}$$​

so the bond is cheap relative to the fitted curve.

Conversely:

$$𝑅𝑒𝑠𝑖𝑑𝑢𝑎{𝑙}_{𝑃}>0$$

indicates that the bond is rich to the fitted curve.

This is useful because it compares the actual security against a synthetic valuation having exactly the same cash flows.

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17. Convert a Price Residual to a Yield Residual

Price residuals are mathematically useful, but basis points are often more intuitive.

Calculate:

$$𝑌𝑇{𝑀}_{𝑚𝑎𝑟𝑘𝑒𝑡}$$​

from the observed market price and:

$$𝑌𝑇{𝑀}_{𝑚𝑜𝑑𝑒𝑙}$$​

from the zero-curve model price.

Then:

$$\boxed{{\displaystyle 𝑅𝑒𝑠𝑖𝑑𝑢𝑎{𝑙}_{𝑏𝑝}=(𝑌𝑇{𝑀}_{𝑚𝑎𝑟𝑘𝑒𝑡}-𝑌𝑇{𝑀}_{𝑚𝑜𝑑𝑒𝑙})\times 10,000}}$$​

Therefore:

$$𝑅𝑒𝑠𝑖𝑑𝑢𝑎{𝑙}_{𝑏𝑝}>0$$

means the security yields more than its zero-curve synthetic equivalent and is consequently cheap under this convention.

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18. Bid, Ask, and Which Price to Analyze

A final source of confusion is bid and ask terminology.

The practical rule is simple:

• You buy at the ask.
• You sell at the bid.

Therefore:

$${𝑃}_{𝑎𝑠𝑘}>{𝑃}_{𝑏𝑖𝑑}$$​

and because bond yields move inversely to prices:

$${𝑌}_{𝑎𝑠𝑘}<{𝑌}_{𝑏𝑖𝑑}$$​

When investigating something that looks attractive to buy, the ask price is therefore the economically relevant executable side.

For general fair-value analysis, the midpoint:

$$\boxed{{\displaystyle {𝑃}_{𝑚𝑖𝑑}=\frac{{𝑃}_{𝑏𝑖𝑑}+{𝑃}_{𝑎𝑠𝑘}}{2}}}$$​

can provide a useful analytical price.

For FedInvest data, from the investor's perspective:

• Buy ≈ Ask/Offer
• Sell ≈ Bid

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Putting All the Yield Measures Together

The easiest way to remember the different measures is to ask what question each one answers.

Bank Discount Yield

How is this Treasury bill conventionally quoted?

$$𝐵𝐷𝑌=\frac{𝐹-𝑃}{𝐹}\frac{360}{𝑑}$$​

Holding-Period Return

How much did my invested money actually grow during this period?

$$𝐻𝑃𝑅=\frac{𝐹-𝑃}{𝑃}$$​

Investment Yield

What is that return on a simple annualized basis?

$$𝑟=\frac{𝐹-𝑃}{𝑃}\frac{𝑌}{𝑑}$$​

Effective Annual Yield

What annual return corresponds to compounding this holding-period return?

$$𝐸𝐴𝑌={\left(\frac{𝐹}{𝑃}\right)}^{𝑌\mathrm{/}𝑑}-1$$

YTM / Bond-Equivalent Yield

What single conventional semiannual yield explains this particular coupon bond's price?

$$𝑃=\sum _{𝑖}\frac{𝐶{𝐹}_{𝑖}}{(1+𝑦\mathrm{/}2{)}^{2{𝑡}_{𝑖}}}$$​​

XIRR

What single annual-effective rate explains my actual dated cash flows?

$$0=\sum _{𝑖}\frac{𝐶{𝐹}_{𝑖}}{(1+𝑟{)}^{𝑑𝑎𝑦{𝑠}_{𝑖}\mathrm{/}365}}$$​

Zero / Spot Yield

What rate applies to one payment at exactly this horizon?

$$𝑧(𝑡)=-\frac{\ln 𝐷(𝑡)}{𝑡}$$​

Discount Factor

What is $1 at this future date worth today?

$$𝐷(𝑡)={𝑒}^{-𝑧(𝑡)𝑡}$$

Forward Rate

What future-period rate is implied by today's discount curve?

$$𝑓({𝑡}_1,{𝑡}_2)=-\frac{\ln [𝐷({𝑡}_2)\mathrm{/}𝐷({𝑡}_1)]}{{𝑡}_2-{𝑡}_1}$$​

Par Yield

What coupon would make a hypothetical bond at this maturity trade at par?

$$𝑐=\frac{2[1-𝐷({𝑡}_{𝑁})]}{\sum _{𝑖}𝐷({𝑡}_{𝑖})}$$​

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The Big Picture

Separate different needs into: quotation, return, and valuation.

Market quotation

$$\text{T-Bill }\to 𝐵𝐷\mathrm{𝑌, }$$$$𝐼𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛\mathrm{𝑡 }𝑌𝑖𝑒𝑙𝑑$$

Treasury Note/Bond/TIPS→𝑌𝑇𝑀/𝐵𝐸𝑌

These help market participants quote securities using established market conventions.

Investment comparison

$$\boxed{{\displaystyle \text{Actual dated cash flows}\to 𝑋𝐼𝑅𝑅}}$$​

XIRR provides a common annual-effective framework that can be applied to align very different cash-flow patterns.

Valuation

$$\boxed{{\displaystyle \text{Market prices}\to \text{Zero Curve}\to 𝐷(𝑡)}}$$

and then:

$$𝐷(𝑡)\to \{\begin{array}{l}\text{Model Prices}\\ \text{Spot Rates}\\ \text{Forward Rates}\\ \text{Par Rates}\end{array}$$​

These relationships allow analysts to think of the discount factor as the common language of fixed-income valuation.

Once $𝐷(𝑡)$ is known, the choice among continuously compounded yield, bond-equivalent yield, simple yield, par yield, or forward rate often becomes a matter of how to describe the economics rather than a change in the underlying economics.

The rate convention matters. But ultimately:

$$\boxed{{\displaystyle \text{Price today}=\sum (\text{Future Cash Flow}\times \text{Discount Factor})}}$$

Everything else is a form of expressing, comparing, or interpreting that relationship.

Notes and sources

The U.S. Treasury publishes both bank-discount and coupon-equivalent Treasury bill rates, and federal regulations provide the formal Treasury bill price, discount-rate, and investment-rate calculations. [home.treasury.gov], [ecfr.gov]

The Federal Reserve's Gürkaynak-Sack-Wright framework treats the discount function as a fundamental building block and allows the fitted Treasury term structure to be expressed as zero-coupon, par, and forward yields. [federalreserve.gov], [federalreserve.gov]

Microsoft's Excel XIRR function calculates the internal rate of return for cash flows that need not be periodic, making it useful when actual cash-flow dates matter. [support.microsoft.com]