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One could say that a CDS price is determined by the physical default probability and the risk premium. The physical PD (PPD) is the actual probability of company defaulting within the given period of time. It is purely a theoretical concept as no one really knows what this probability is. We could estimate it using some models or credit ratings, but those ...

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A swap does not require a model because its price can be derived from the yield curve without any assumptions about how the yield curve may move in the future. The PFE however is an indication of by how much the swap's mark-to-market may move between now and a moment in the future. It is of course influenced by how volatile rates are. The more volatile ...

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If you look in Chapter 7.1 where you find equation (1), you will see just below that: $$\frac{d\mathbb{Q}^T}{d\mathbb{Q}} \mid_\mathcal{F_t}=\mathcal{E}_t(v(\cdot,T) \bullet W^*)$$ where $W^*$ is a $\mathbb{Q}$-Brownian motion. Besides, you'll notice on the book that your equation (2) is described as a way to induce the probability measure ...

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Forward interest rates are negative whenever the yield curve is negatively sloped. The US term structure was inverted most recently around 2007. Hard to find bank deposits that have negative yields (find countries experiencing deflation and you may find it), however, treasury bills during recent times of financial stress have yielded a negative rate. The ...

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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 ...

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Quantmod package claims to support downloading data from Federal Reserve Bank of St. Louis Economic database, which contains plenty of rates time series. It should be pretty simple to get them into R using function getSymbols, in the same manner as getSymbols("DEXJPUS",src="FRED") # FX rates from FRED

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Dirty bond price refers to the price of a bond that reflects the interest that has accrued since the issuance of the bond or last coupon payment. It has nothing to do with how you discount cash flows but just whether accrued interest is priced in or not. Thus, dirty and clean bond prices apply to all bonds that pay intermittent cash flows.

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It's very common to work in spreads rather than price for this calculation. The simplest approach would be to get an implied spread for each bond, and then allow the spreads to vary in simulation according to an equity-style factor model. Each spread simulation can then be mapped back to bond prices by reversing the formula. A few points: If you can, ...

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There are two flaws in the argument. The simpler one is that expectations give information about probability distributions (premise D). I think this is what John was referring to in his comment. The fact that the expectation of a forward rate in period X is Y% tells us nothing about the implied probability distribution in that period, and certainly doesn't ...

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Not all bonds have coupons, of course, especially in the world of convertible bonds (where puts are more common). I have yet to see a bond with a put date other than a coupon date. It is typical to forfeit a coupon in case of a put, but that's not universal. As to coupon dates, there's usually a "next business day" clause. Generally speaking, the world ...

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A concrete example of negative forward rates is provided by the 3M CHF LIBOR futures. They're all trading above a price of 100, which implies negative forward rates. See the prices here. Despite the prices of the forwards, CHF libor hasn't actually fixed negative yet. But the forwards are certainly all below zero. Also, your formula for the forward rate ...

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I have come across 2 markets where rates can be negative: Inflation protected bonds. These bonds are pricd with real interest rates. You can think of them as (this is the Fisher equation: $$r = n - i$$ where $r$ is the real interest rate and $n$ is then nominal interest rate (the normal one) and $i$ is the (estimated or priced) inflation. Real rates for ...

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There are some interest rates accessible through the R package YieldCurve. I don't think they contain the latest available interest rates. Another alternative is the package FRBData. Using the function GetInterestRates you can download various of the latest interest rates published by the FRB.

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The problem is more that the article you read uses language that is not consistent with the way most people in finance talk. People typically call the difference between the nominal Treasury yield and an inflation-linked bond the breakeven inflation rate. When people look at the difference between the earnings yield and the nominal interest rate, they might ...

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Maybe because the underlying portfolio's notional may decrease over time? Maybe because the loans are part of a private transaction in which the deal stipulates that notional is paid off over time? Maybe its a pay-through structure in which the original mortgage loan notional is paid off over time and the notional portions are passed down the structure. ...

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The simple but accurate answer should be that Macaulay Duration is the weighted average maturity of cash flows (in years). That is how it is defined in almost every text book and looked at by most market practitioners. That is why its quoted in years and it gives an indication of when, on a weighted basis, cash flows are paid out (mature). For example, in ...

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Maccauly Duration means nothing else than that after the given amount of years, you will have your capital investment back as nominal amount. If you have \$100 invested, and you have a duration of two years, after two years you will have gotten \$100 repaid, not directly dependent of interest rate or payment scheduling (indirectly they are of course!). I ...

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The formula to calculate accrued interest used in Excel is 100% correct (but the day count convention is not). What is not correct and not sufficiently accounted for is the exact day count convention that must be used for the specific asset you try to value its accrued interest on. Here couple points that may cause your errors: I highly recommend you to ...

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What I'm writing is based on the methodology in http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1565134 You observe the nominal and real bond prices/YTMs. Transform them to zero or forward curves. Estimate the multivariate dynamics and project them to your horizon, let's call this $X$. Use the distribution of zero/forward rates to obtain the distribution ...

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Let's approximate the time to maturity to be 3 years and 10 months. Assume that coupon is paid on March 6 each year. Let face value $F=100$ and coupon $c=0.07375F$. Let the discount factor be $d(0,T)=e^{−r T}$ where $r=0.06535$. The price of the bond is ce^{−10/12 \bullet r}+ce^{−22/12 \bullet r}+ce^{−34/12 \bullet r}+(F+c)e^{−46/12 \bullet r}=103.24 \; ...

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You would simply hedge with a floating rate leg. That is the whole idea of swaps though. A price taker is paying fixed and receiving floating then such price taker usually is hedging the risk of interest rates increasing, meaning he is not concerned with the risk of decreasing rates. Generally such participant has floating rate liabilities. Let's say the ...

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A Forward measure is simply a measure such that the $T$-discounted forward bond starting at time $t$ is a martingale. This allows you to work with reference underlying $S(t) = P(t,T_1)/P(t,T_2)$ with $T_1$ and $T_2$ fixed. Flipovic says that $\frac{dQ^T}{dQ} = \frac{1}{P(0,T)B(T)}$ is "called" a forward measure and then proceeds to show that the discounted ...

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Another definition of the $T_{i+1}$-forward measure: Under a $T_{i+1}$-forward measure the forward (LIBOR) rate with (natural) payment in $T_{i+1}$ is a martingale. Since there is a famility of forward (LIBOR) rates, there is a familiy of such measures. I do not have the book at hand, but as far as I understand the situation is as follows. ${Q}_{T_M}$ is ...

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First of all, you should be using OAS, not YTM, as the risk-free interest rate component of YTM is known and should be imposed rather than estimated. Second, rather than regressing OAS against a benchmark, regress changes in OAS against changes in a benchmark OAS. Then calculate a fitted OAS series from the change series, which naturally will not have any ...

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The formula implemented by ACCRINT is documented on office.microsoft.com According to this page it is $ACCRINT = par \times \frac{rate}{frequency} \times \sum_{j=1}^{NC} \frac{A_j}{NL_j}$ where $A_j$ = number of accrued days for the ith quasi-coupon period within odd period. $NC$ = number of quasi-coupon periods that fit in odd period. If this number ...

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The conversion factor associated with each bond the futures' delivery basket is constructed such that the invoice prices of the bonds are identical under the assumption that the yield curve is flat at the level of the futures' notional coupon. Therefore, the bond with the highest duration will be the CTD when yields are above the notional coupon and the bond ...

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First, I am not sure which exact statement was made. Also, you cannot just say "without CF" because you are essentially creating an artificial market with messed-up utility. In summary the cheapest-to-deliver bond is: The bond that results in the smallest loss or greatest profit for the futures seller. Futures sellers have to buy the bonds they are going ...

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The discount rate that you use to compute the price of the bond is a parameter that you use during the computation of both types of valuations (clean or dirty); it is the rate you use to discount the cash flows. The only difference between clean and dirty price is that the clean price removes the accrued interest since the last coupon. Hence, the discount ...

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It's a simple TVM problem - solve for the interest rate. The "current" cost of debt would be market determined, so that's why you use the market value. It ties into how bond accounting works - the premium of the bond is amortized until maturity. The amortization amount would be the difference between the coupon and the interest expense(market rate at ...

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Practically, the best metric is the one your boss wants you to use. Alternately, you can think of the return of the desk like a leveraged security, as described here. This would suggest that the daily performance would be calculated as the profit divided by the basis. If you want to express the return in terms of the capital requirement, then that's one ...

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