# Questions tagged [yield-curve]

A yield curve is a plot of yields for various bonds (often government bonds) versus the bonds' maturities. We also often plot swap and other LIBOR rates to get the (related) swap curve.

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### What is the Swap Curve?

What is the so-called Swap Curve, and how does it relate to the Zero Curve (or spot yield curve)? Does it only refer to a curve of swap rates versus maturities found in the market? Or is it a swap ...
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### how to derive yield curve from interest rate swap?

According to some textbooks, to derive the yield curve, quote overnight to 1 week: rates from interbank money market deposit, 1 month to 1 year: LIBOR; 1 year to 7 years: Interest Rate Swap; 7 ...
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### Why is an inverted yield curve a problem?

Immediately preceding the worst of the financial crisis, my professors all pointed out to me that the yield curve had inverted -- short-term yields were more risky than 20-year or 30-year Treasury ...
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### Why isn't the Nelson-Siegel model arbitrage-free?

Assume $X_t$ is a multivariate Ornstein-Uhlenbeck process, i.e. $$dX_t=\sigma dB_t-AX_tdt$$ and the spot interest rate evolves by the following equation: $$r_t=a+b\cdot X_t.$$ After solving for $X_t$ ...
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### Implied term structure from risky discount curve: does it make sense?

We know that, taken every discount curve, it's possible to calculate its forward rates according to our tenor preferences. We know also that it's actually possible to extract an implied term ...
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### What does instantaneous forward mean?

Could you please help me to understand meaning of instantaneous forward rate? I mean economic interpretation at basic level. What is it used for? How can i derive it from zero rate/price? Thanks
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### Bond curve extrapolation

What are the best methods to extrapolate bond yields from an existing curve that doesn't extend quite this far? For example, how would one come about finding a theoretical bond yield for a 40 or 50 ...
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### Deriving Interest Rates

I am trying to teach myself about interest rate swaps, how they are priced, etc... Easy enough - just comparing cash flows of fixed and floating rate bonds. However, what I'm struggling with is how ...
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### Musiela parameterization

I have a question regarding the proof of the Musiela parametrization for the dynamics of the forward rate curve. If $T$ is the maturity, $\tau=T-t$ is the time to maturity, and $dF(t,T)$ defines the ...
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### Shape and geometry of the yield curve

Let the initial yield curve $T\mapsto y(0,T)$ be given for a term structure family of bonds $B(0,T)$ having different maturities. I am trying to figure out the geometric properties of the yield curve. ...
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### How to build the short end of a zero coupon curve for non-core Eurozone countries?

I am in the process of building zero coupon curves for some countries in the Eurozone. I have the following data sets: Euribor and EONIA Swap rates Bond price and yields The bond prices (and thus ...
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### Is trading mean reversion of small principal components of prices profitable?

Many have told me that it is a good idea to look at the third principal component (PC) of yield curve movements, as well as third and fourth PC of G10 currencies. They claim these PCs represent "...
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### Where do swap rates and/or long-term forward rates come from?

I apologize if this is supposed to be obvious, but ... . Libor spot rates are quoted up to a year, beyond that one can use Eurodollar futures to continue to build the curve. Let's say up to 3 years. ...
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### Best method for interpolating yield curve? [Multiple questions]

I'm building a spot curve for US Treasuries. My original selection of cash treasury include all the on-the-run bills, notes, bonds from 6 months to 30 years, as well as some selected off-the-run ...
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### Calculating instantaneous forward rate from zero-coupon yield curve

I have a big dataset containing zero-coupon bond yields with different relative maturities. I fix a time horizon on my dataset and I want to calculate instantaneous forward rate. I'm going to write ...
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### Is Duration really the slope of the Price-Yield curve?

When looking at the Price-vs-Yield graph for a fixed rate instrument, we are often told that the duration is the slope of that curve. But is that really right? Duration is (change in price) divided ...
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### Looking for a pricing library supporting Mutli-curve Framework

I am looking for a builder of Yield curves by tenors (O/N, 1M, 3M, 6M, 12M) respect to a given discount curve based on multi-curve framework as described below : Interest-rate Modelling with Multiple ...
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### How does one estimate theta in the Ho-Lee model from a yield curve?

I have a yield curve constructed using linear interpolation with data points every 3-months for US treasuries. I would like to use that calibrate a Ho-Lee model, but I can't wrap my head around how ...
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### Which interest rate should I use for the discount rate in real-world pricing?

Suppose I want to compute the time value of money (present value, future value, etc). I need to put an interest rate into the calculation. Which real world interest rate would best be used here, ...
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I have few questions about using OAS as a measure of risk: does OAS allow for comparison between bonds with and without embedded options (e.g. a callable bond against a plain vanilla one against a ...
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### Wrong discount factors when finding Nelson Siegel Svensson model parameters

I am trying to determine the parameters for the Nelson Siegel Svensson model and am solving a Non- Linear Optimization problem to do this. Some of the code I have written is below and this is where my ...
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### What are some simple algorithms for hedging vanilla bonds?

My team will soon be implementing an auto hedger for our bond trading desk which will be integrated tightly with our risk application and I am interested in researching how this may work. Any advice ...
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### Why 10-year versus 2-year spread?

I occasionally see the 10y-2y spread referenced as a recession predictor. See, e.g., https://seekingalpha.com/article/4201787-current-slope-yield-curve-tell-us Why 10y minus 2y? Specifically, why use ...
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### Principal Component Analysis of yield curve change

Following pictures are the Principal Component Analysis for the yield curve change from https://www.coursera.org/learn/interest-rate-models/lecture/ZHMM6/principal-...
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### Use of interest rate swaps in liability-driven investing

You probably have home across recent events in the UK bond markets. The Financial Times article "The reason the BoE is buying long gilts: an LDI blow-up" from Sep. 28, 2022 goes through why ...
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### Bootstrapping spot rates from treasury yield curve

I'm attempting to construct a spot rate and forward rate curve from the 2011 daily treasury yield curve rates provided by the US Treasury. All US Treasury securities (1m, 3m, 6m, 1y, 2y, 3y, 5y, 7y, ...
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### No-arbitrage in term-structure models

I am a bit confused about what the implication of "no-arbitrage" in popular term struchture models (such as affine term struchtre models or HJM models) are? Is it solely a restriction on the cross-...
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### Bootstrap with QuantLib: Fair Swap or zero NPV

In all brevity What is the termination condition used in QuantLib's curve bootstrapping? Can I modify this setup to my needs, e.g. can I tune this to a higher accuracy? Background When bootstrapping ...
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### Python Quantlib yield curve dates different than input

Thanks very much for your help in advance. I am trying to understand the yield curve construction from Python Quantlib. And it seems I cannot get the curve output the same dates(nodes) as my input ...
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### Dual Curve Bootstrapping - When to OIS discount?

I am a new quant and I am trying to understand some of the specifics of dual curve bootstrapping. For concreteness, suppose I want to build a Libor forward curve. From what I understand OIS ...
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### How does the 2-factor Hull White model propagate the forward rates curve?

I've been trying to get a grasp on some of the basics of interest rate modeling, and am looking to simulate rates using the 2 factor Hull White model, which I am aware offers a more realistic model of ...
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### Why does a barbell portfolio have higher convexity than a bullet porfolio

I cannot quite understood absolutely why a barbell portfolio has higher convexity than a bullet porfolio. I can easily understand how the parallel line represents duration but I cannot see what the ...
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### Swap curve construction

I am new to this area so my question might be basic to many but please answer. For valuing interest rate swaps how will we define which curve to take, like sometimes we use usd 3m curve, or usd 3mv 6m ...
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### What curve are you shifting when you calculate DV01 for a swap?

I understand that a general swap has 4 curves attached to it: the flat forecast curve associated with the fixed leg, the forecast curve associated with the floating leg, the fixed leg discount curve ...
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### How to compute par yield from zero rate curve?

How does one calculate the below two-year par yield given the zero rate curve: Assume the following two-year zero rate curve, with continuous compounding: ...
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### Why is "full" Yield Curve (term structure of interest rates) 3 component based?

I am trying to understand bond-valuation and construction of yield curve. I don't have any exposure to bootstrapping or what-so-ever as of now. So it's appreciated to have an example but not too ...
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### What is Dual Curve Bootstrapping? And how to do it, with an example?

I am starting to explore this area. My ultimate aim is to build a 3 month LIBOR forward curve. I wish to know what exactly 'Dual Curve Bootstrapping' is (If someone could explain it in clear words). ...
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### Yield curve fitting example in Wilmott on Quant Finance p.528

In Wilmott on Quantitative Finance Vol. 2, p. 528, Section 31.4.2, is given a power series expansion for a zero coupon bond $$Z(r,t;T)=1+a(r)(T-t)+b(r)(T-t)^2+c(r)(T-t)^3+\dots$$ then it says to ...
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### pricing in the case where payment currency and collateral currency are different?

I'm asking for the curve construction of the discount curve in the case where payment currency and collateral currency are different. If I refer to BBG, in the case of a USD swap collateralized in EUR,...
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### Will rolling-down-yield-curve bond strategy work if interest rates remain unchanged?

Suppose I have 2 strategies; A) Buying A One Year Bond And Holding To Maturity (Buy & Hold To Maturity) B) Buying A 3 Year Bond and Selling After One Year (Rolling Down The Yield Curve) Assume ...
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### How to manage evaluation date changes in QuantLib while using ImpliedTermStructure Class

I will not attach the whole code 'cause it would be just a huge waste of space and it would be not useful for this question's purpose. What I am going to attach here is a snippet code and its output, ...
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### QuantLib Python Swap Yield Curve Bootstrapping Dates and Maturities

This is somewhat related to the question I asked here but simpler. I am trying to bootstrap a yield curve from swaps, and am having a problem with the dates/maturities that are coming out. The code ...
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### Pricing a bond contract from the yield curve

When giving a particular class in financial mathematics for a student I saw a problem in a list of exercises that says: How to calculate the price at 15 December 2010 of a bond paying a coupon of 11....
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### Using the termstrc package in R

I am attempting to use the function estim_nss from the termstrc package in R to find the spot curve from constant maturity rates published by the Fed. I am using this package because I will need to ...
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### Fitting the Term structure of Discount Bonds with Ho-Lee

I was now reading a book on interest rate modelling, and I am having trouble picturing the practical issues of model calibration with the Ho-Lee model. Apparently, one of the drawbacks of this model ...
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### IR Yield Curve and Fixing Dates

Consider two FRAs. 3x6 , Effective 3 months from now, terminates in 6 months. The floating leg payer pays 3-month LIBOR. Fixing date for LIBOR 40 business days. To price this at par, the fixed leg ...
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### Attributing the change in NII to Shift, Twist and Butterfly

The movement of the zero rate curves can be decomposed into a shift movement (the level of interest rates) and a twist movement (the slope of the curve) and butterfly (the curvature of the curve). If ...
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### Derivation of the Nelson-Siegel model and proof of arbitrage

1. I am looking for a derivation of the Nelson-Siegel model $y(m)=a+b\left( \frac{1-e^{-\lambda m}}{\lambda m}\right)+c\left( \frac{1-e^{-\lambda m}}{\lambda m} -e^{-\lambda m} \right)$ It is ...
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### Repricing SOFR Quotes and Non-Zero NPV

I generated/calibrated a SOFR Curve using Quantlib Python and would like to know why when repricing the swaps have non-zero NPVs. Appreciate any assistance. Thank you. Parameters ...
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