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Given a forward rate, for example: $ F(t, T, T+\delta)$ The instantaneous forward rate $f(t,T)$ fixed in $t$ is the limit when $\delta \rightarrow 0$ of your forward rate. If the relation between forward rate and zero coupon bond is: $F(t,T,T+\delta) = \frac{p(t,T) - p(t,T+\delta)}{\delta p(t,T+\delta)}$ We have, \begin{equation} f(t,T) = \lim_{\...


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1. Observable instruments, spot rates, and forward rates First remember that something observable means that you can observe/find the rate in the market by looking at traded rate instruments or fixings. 1.1. Observed spot rates For simplicity, assume Zero Coupon Bonds (ZCBs) are traded with time left to maturity of 10Y, 15Y and 20Y. Hence, by observing ...


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A forward rate agreement is an agreement to exchange a fixed for a floating rate over one period, with the payment being made at the start of the period. A zero coupon swap (with both legs paid at maturity) is an agreement to exchange a fixed for floating rate over one or more periods, with the payments being made at the end of the final period. So the two ...


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You do not need zero rates to estimate a parametric model of the yield curve, such as Nelson-Siegel. Suppose for instance that you have a cross-section of bond prices. Then: For given parameters for your yield-curve model, compute yield curve; with this yield curve, calculate theoretical bond prices; compute discrepancy between theoretical bond prices and ...


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In general futures contracts are leverage instruments. They never require the investment of principal. They do however require margin: you need to fund your account at a futures exchange so that they have insurance against any losses you incur, as an example this might be 2 days standard volatility. On 1 ED contract for 5bps a day thats probably 10bps margin ...


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(1) corresponds to simple interest and (2) to compound interest. For instance, Canadian treasury bills are based on simple interest (see Broverman's book Mathematics of investment and credit).


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In case you're still looking: https://data.snb.ch/en/topics/ziredev#!/cube/rendoblid


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By definition of the $T$-forward measure $P_T$, the process $\Big\{\frac{P(t,S)}{P(t,T)} \mid t\geq 0\Big\}$ is a martingale under the measure $P_T$, without assuming any specific models of the short rate $r_t$. That is, this martingale property is model independent. However, as a good exercise, you can also do the following: Given the CIR interest rate ...


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In part (a) use discount rate $e^.07 -1 = .072508181$ to get the right answer. For part (b) I am just giving you hint: Calculate bond price at the end of 1st year and 2nd year in the same way as you did in part (a). Use the above calculated price to buy bond from the dividend at the end of first and second year. You may assume bond can be purchased in ...


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Who knows what the 5 year zero coupon rate is in that case, there could be an event 4.5 years out that will have serious interest rate implications that we don't know about. The only thing you can do with these three numbers is extrapolate and say the rate should 9%. You should be aware of what assumptions you're making when you do something like that, but I'...


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Here is a general proof for all parameters in an open domain. $$dr = adt+bdW:=r\big(k(\theta-x)+\frac12\sigma^2\big)dt+\sigma rdW.$$ Let $$u(r(s),s):=e^{-\int_t^sr}B(r(s),s,T)=:\phi(s) B.$$ Then $$u(r(t),t)=\mathbf E\big[u(r(s),s)\big|r(t)\big],\, \forall t<s. \tag{1}$$ So, by Ito's Lemma, \begin{align} du(r(s),s) &= Bd\phi +\phi dB \\ &= \phi \...


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The CMT yields published by the Fed/US Treasury are par yields calculated using a cubic spline model. In other words, these are the yields to maturity as well as coupon rates on bonds whose theoretic prices are 100. With this information in mind, you can linearly interpolate between these yields, or use a cubic spline to fill in rates at other tenors, ...


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Under the Hull-White interest rate model, the short rate $r_t$ satisfies a risk-neutral SDE of the form \begin{align*} dr_t = (\theta(t)-a r_t)dt+ \sigma dW_t. \end{align*} The price at time $t$ of a zero-coupon bond with maturity $T$ and unit face value is then given by \begin{align*} P(t, T) &= A(t, T) e^{-B(t, T) r_t}, \end{align*} where \begin{...


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Yes. The time p/l can be found by leaving all the inputs the same and allowing a day to pass. I prefer not to call it theta - that term is used to describe the time decay of options.


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Just adding the valuation date/time to the bonds identifier will make it clearer. See below:


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I think one way to approach the answer is thinking what are these two rates used for. Starting with zero coupon rates, it's aiming for getting the par value back at maturity (similar to a bank's loan, where in the end payments are all up). For forward rates however, is calculated under the risk neutral measure and is mostly used for option pricing in fixed ...


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The website below shows how to price bonds from curves, currently it only supports fixed rate and zero-coupon bonds, but it might give you an idea how to price a floater using similar concept: Goto: https://www.opencminc.com Switch to Yield Curves under the Market Data section Click on any curve point. For example: click on a rate under 10Y https://www....


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The delta is (value of bond) - (value of bond if rates go up 1bp) =5mm/(1.0210)^5 - 5mm/(1.0211)^5 =$2206


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Your answer is correct. You included .5 in the exponent and therefore got an annualized result. 6.118% divided by 2 is your bootstrapped 6 month spot rate.


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You think you make a mistake where you actually don´t make one. The exercise is just like it is. Resulting in $$r_{6m}>r_{12m}$$ The difference in your both answers, based on the same rounding, lays in the different basis for the logarithm. $$r_{6m} = - 2 \log_e \left( \frac{99.8-102.5 e^{-r_{1y}} }{2.5} \right) = \textbf{6.118%}$$ $$r_{6m} = - 2 \log_{...


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If you have a negative yield which cannot be ruled out nowadays the issueprice will be above par.


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There is only one cashflow for the zero-coupon bond. At maturity, it pays the par value. They are always issued below par, as the buyer is paying the NPV for the bond that matures in the future. Here is a brief reference at Investopedia. A zero-coupon bond, also known as an "accrual bond," is a debt security that doesn't pay interest (a coupon) but is ...


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Safest would be FRA to lock the interest rates. Thanks


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You have to be very careful with terminology here. In particular "yield" is being thrown around carelessly by both of you. The textbook is correct if the (meaningless) phrase "at a 6% yield (rate)" is crossed out and replaced by "at a 6% discount rate". And this is how Tbill's are handled when they are issued (the press release by the US Treasury speaks of ...


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If I understood well, your model falls into the generic case of affine models. This reference might help you : http://arxiv.org/pdf/1512.03677v1.pdf


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We shall prove this by contradiction. Let $\theta=0$ and $\sigma=0$. $X_t=X_0e^{-kt}$ and $$B(0,t)=\exp\Big(-\int_0^te^{X_0e^{-ks}}ds\Big).$$ Suppose the contrary that $B(0,t)$ is affine. We should have $$ B(0,t)=\exp{\left(A(0,t)-C(0,t)e^{X_0}\right)}\;\;\ \forall (t,X_0), \tag{1} $$ Differentiate the logarithm of Equation (1) with respect to $t$ side, $...


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The short answer is - you need more data. If you want to build a full zero-rate swap curve, typically these curves go out to 30 years. In general, the front of the curve is made from LIBOR rates, which you have. Typically you don't see practitioners use anything past the 3M point but some will use up to the 6M point. For the 2nd part of the curve, from ...


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Let $r_t$ be the interest rate. Then \begin{align*} B(t, T_i) &= E\Big[e^{\big(-\int_t^{T_i} r_s ds\big)} \mid \mathscr{F}_t\Big]\\ &= e^{\int_0^t r_s ds} E\Big[e^{\big(-\int_0^{T_i} r_s ds\big)} \mid \mathscr{F}_t\Big]. \end{align*} Note that, for $t>0$, unless $r_t$ is deterministic, \begin{align*} E\Big[e^{\big(-\int_0^{T_i} r_s ds\big)} \mid ...


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