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15

I like to present to you a slightly different approach: Historically, only one single yield curve was derived from different instruments, such as OIS, deposit rates, or swap rates. However, market practice nowadays is to derive multiple swap curves, optimally one for each rate tenor. This idea goes against the idea of one fully-consistent zero coupon curve, ...


15

To explain why a negative sloping yield curve is bad, you have to start with a theory of the yield curve. The dominant theories for the term structure of interest rates are the rational expectations, liquidity preference, and market segmentation. (The first two theories are quite compatible with each other and have more standing so let's assume that view.) ...


7

There are two parts to your question and I'd like to answer them separately. Curve Construction On a daily basis, you can observe prices on a large variety of instruments, whose prices are driven by news and trading flows. Based on market prices of these instruments, there are a number of ways to create discount curves/forward curves. At a very high level ...


6

Your observations are pretty much correct. The groupings are because of the fine print "Note how I have expanded the drift and volatility terms at $t = T$; in the above these are evaluated at $r$ and $T$." on the same page (p.528). Basically, $w$ is a function of both $r$ and $t$. Since we want to use $w(r,T)$ instead of $w(r,t)$, we taylor expand ...


6

There's no class at this time to add two curves as you want, but it won't be much difficult to write it. The closest you'll get in the library is the ZeroSpreadedTermStructure class, that shows the general idea: it inherits from YieldTermStructure (by way of ZeroYieldStructure) takes a YieldTermStructure and a spread (constant, in this case) and override ...


5

Ok, I've done some digging in the code. It's an issue with the LogLinear interpolation; while trying to find the correct rate for the 1-week node, the bootstrapper wanders unchecked into a region of negative rates and the logarithms blow up. At this time, I'm afraid the workaround is just to use some other interpolation. Or recompile the library and the ...


5

The answer to your first four questions is affirmative. Option-adjusting the spread makes an equivalence between everything theoretically possible, but the quality of results depends significantly on the quality of your interest rate model and its calibration. My personal opinion, though, is that the results need to be treated carefully because the OAS ...


5

The short answer is that Libor swap rates come from the market. They represent a series of cashflows in the future whose value is determined by the fixing, which the market participants have their own valuations of. Since the actual cash flows are now discounted using a separate funding curve, the swap prices embed both a prediction of future fixings and a ...


5

You should use the full yield curve, discounting cash flows at specific dates using the appropriate zero-coupon interest rate. As to which yield curve, that is often a matter of convention. Generally one uses the LIBOR/swaps curve for all but the most liquid products (in which case you use the treasury curve). The curve is constructed from LIBOR/Eurodollar ...


5

The Macaulay duration is a measure of how sensitive a bond's price is to changes in interest rates. Duration is related to, but differs from, the slope of the plot of bond price against yield-to-maturity. The slope of the price-yield curve is $-\frac{D}{1+r}P,$ where $D$ is Macaulay duration, $P$ is bond price, and $r$ is yield. Here's how the definition ...


5

Quantlib supports multi-curve framework (to the best of my knowledge). By the way, there's a "newer" version of that paper (authored by Pallavicini & Brigo). http://arxiv.org/abs/1304.1397 This paper might also be useful for you, very practical and basically answers any question you could have. Also see this discussion about multi-curve discounting ...


4

The risk implied by Euribor or EONIA (or their swaps) is for lending to another prime rated bank. These rate indexes represent where contributor banks are offering funds to each other in the interbank market. Contributing banks are mostly rated P-1 (Moody’s) or A-1 (S&P). You wouldn’t use these rates for govt discount curves because the risk doesn’t ...


4

(In addition to the answers of Freddy and Phil H): With "modern" multi-curve setups: You have to distinguish between discount curves (which describe todays value of the a future fixed payoff (e.g. a zero coupon bond)) and forward curve, which describe the expectation (in a specific sense) of future interest rate fixings. Swaps pay LIBOR rates and are ...


4

To elaborate on Freddy's answer: These days you need to maintain a separate funding (usually OIS) curve to your Libor* type curves. Once you have this discounting curve, you can calculate from Libor instrument market data what the market estimations of that Libor are: 3m instruments like Interest Rate Futures, IRS with a 3m float leg, 3m FRAs can be used to ...


4

Is the author taking logs (and dividing by (T-t) etc) of our previous Z expansion from the previous page? He does, as you will see if you try to do the computation. What did you prevent to find this out by yourself? (I am trying to be constructive.) Mathematically, it doesn't add up to what the author provides as the answer. What am I missing here? ...


4

Garabedian, Typically, the "swap curve" refers to an x-y chart of par swap rates plotted against their time to maturity. This is typically called the "par swap curve." Your second question, "how it relates to the zero curve," is very complex in the post-crisis world. I think it's helpful to start the discussion with a government bond yield curve to ...


3

You can create the data using the procedure described in the reference manual on pages 31 and 32. The necessary code is copied below: # The following code may be used to generate an empty data set, # which can then be filled with bond data: ISIN <- vector() MATURITYDATE <- vector() STARTDATE <- vector() COUPONRATE <- vector() PRICE <- ...


3

Standard 3m curve interpretation: H, M, U, Z = Mar, Jun, Sep, Dec IMM dates in the futures convention (see SRKX's answer), and 2Y would be just the calendar 2y point. Assuming that what you found was done in 11th July 2011: U1 21 Sep 11 - 21 Dec 11 (IMM = 3rd Wednesday to following IMM) Z1 21 Dec 11 - etc H2 21 Mar 12 M2 20 Jun 12 U2 19 Sep 12 Z2 ...


3

I think they are using the same convention as the future exchanges for delivery months. You can find a complete mapping on the wiki page. The letter corresponds to a month and the number corresponds to the last digit of the year. So for example to understand U1 you find U=>September and 1=>2011 (you have to "guess" the relevant decade, it's quite ...


3

Theoretically, a rising yield curve is compensation for the additional duration risk. An inverted yield-curve is saying that the market thinks that: Next-year's figures for: growth plus inflation is less than Ten years' time's figures for: growth plus inflation Which means that expectations are either of a recession (some negative economic growth; and ...


3

Depends on circumstances - if you just trade futures intraday for yourself, secondary market T-bills (http://www.federalreserve.gov/releases/h15/data.htm#fn3) will be good enough.


3

The original Nelson Siegel paper describes a parsimonious model of the term structure using only four or three (if $\lambda_t$ is fixed). Filipovic (1999) proves that this model can never be used in a arbitrage free context, paraphrasing the abstract: We introduce the class of consistent state space processes, which have the property to provide an ...


3

As @michipilli said, if $Z = 1+ as + bs^2 + cs^3$ (where I have substituted $T-t$ by $s$ for ease of notation and also suppressed the dependencies of $a$, $b$ and $c$) and $\log (1+\zeta) = \zeta - \frac{1}{2}\zeta^2 + \frac{1}{3}\zeta^3 + ...$ then, \begin{align*} \log Z &= (as + bs^2 + cs^3) - \frac{1}{2}(as + bs^2 + cs^3)^2 + \frac{1}{3}(as + ...


3

It's hard to be sure without seeing the inputs, but I'm guessing that the implied curve changes shape because the original curve does (which you can see from your output: except for the 1-year and 5-years points, the actual discounts are different). The reason the original curve changes is probably the different position of weekends or holidays (so that, ...


3

Your overall approach is correct. However to my knowledge it is formally more appealing to work with a parameterized and smoothed yield curve. Basically one assumes that the yield curve can be described by a smooth function $r(t,\alpha, \beta,\gamma)$ (mostly of three parameters) Given a set of market data $Y(t,T_1)\dots Y(t, T_n)$ one looks for ...


3

I think what you wrote is correct. I'll rephrase everything according to my way to give you another point of view. The price of a coupon bond at time $t = 0$ is the sum of the discounted cashflows given by the coupons and the face value: $$ P_0 = F \cdot D(0, T_n) + \sum_{i=1}^{n} 11.04\% \cdot 0.5 \cdot F \cdot D(0, T_i) $$ where $F$ is the face value, ...


2

You are correct: none of the durations are the slope of (the tangent to) the price/yield curve. Rather the slope is the "dollar duration" = modified duration * Price *-1. This will tend to betray rather large numbers; e.g., under continuous compounding the modified/Macaulay duration of a 100 par 10-year zero coupon bond is 10.0 years. The slope (of the ...


2

Samuelson once quipped that the yield curve had successfully predicted 9 of the last 5 recessions. Explanations for why the yield curve inverts have been covered adequately above. But what does it mean for the economy? For the yield curve inversion to predict stock market performance enough to make an actual decision, the bond market would have to be more ...


2

An inverted yield curve basically means that interest rates will be higher for the coming year than for the years following. That means that entities that need do borrow for short term purposes will do so at a greater cost that those borrowing for the long term. That is an unusual and "unnatural" relationship. All other things being equal, that will dampen ...



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