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6

Assume a number of bonds with three constant variables, par value $par$, coupon value $C$ (paid annually), and interest rate $r$, and one changing variable, time to maturity $n$ First off, the relevant formulas: The price $P$ of each bond, as you've already written it, is $$ P=C*[\frac{1}{r}-\frac{1}{r}*\frac{1}{(1+r)^{n}}]+\frac{par}{(1+r)^{n}} $$ The ...


6

Yes, you are correct. Duration is additive, so your aggregate portfolio duration is the weighted average of your individual durations as you present in point 2. That holds assuming a close to flat yield curve and parallel (additive) shifts. If that's not the case, the situation gets a bit more complex. Unfortunately, right now I couldn't find any ...


5

After struggling through the Pianca paper due to its poor proofing ($F$ is never defined but appears to be face value, and $n$ is implied to be the number of periods remaining but is instead maturity), I seem to have it worked out. Using the lambertW function in gsl, I have it replicated in R: # Estimate duration using various closed-form formulae # ...


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 ...


4

Yes, it is definitely possible to do so. With a long fixed-income portfolio, you'd typically be buying puts on treasury futures or writing calls on them (writing calls may not be feasible if you're an institutional investor due to regulatory reasons). In general, duration for long puts/short calls would be negative. However see caveats below: Typically, ...


4

If you're able to work with the results from the paper cited (Pianca, Maximum Duration of Below Par Bonds: A Closed-Form Formula), congratulations! You have the hard part done! Maximum durations for par and premium bonds are trivial. Here is a figure directly from the cited paper: Some points about the figure: the market interest rate used is $i=10\%$...


4

You don't say which duration, but it's generally okay to use effective duration: $$ duration (eff) = \frac{-1}{P(r)}*\frac{Price(r+b) - Price(r-b)}{2*b} $$ where $r$ = rate and $b$ = yield shock. Although, to address Brian's point, the mortgage contains an embedded call option that creates negative convexity, so the three re-pricings, $P(r)$, $P(r+b)$, $...


4

Duration is not linear. It is the weighted average of the duration of the underlyings with the weightings being their values. To get a linear system multiply the durations by the associated pvs and match that quantity instead.


3

DV01 is the dollar variation in a bond's value per unit change in the yield. https://en.wikipedia.org/wiki/Bond_duration IR DV01 is the dollar value change for a 1bp upward parallel shift in interest rates. http://dataforthoughts.blogspot.it/2009/09/economics-of-negative-bond-cds-basis.html


3

Macaulay duration is simply a weighted average. $MacD(A,B)=\frac{V(A) \cdot MacD(A)+ V(B) \cdot MacD(B)}{V(A)+V(B)}$


3

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 ...


3

The intuition behind Macaulay Duration is the average time it takes to get all the cash flows from a bond. Think of it as computing the centre of gravity for a see-saw. You can find the image depicting the same here: This should immediately tell you that Macaulay Duration for Zero coupon bond is the maturity of the bond. In continuous discounting ...


3

Like Aksakal already mentioned in his comment it might depend on the duration formula you use. (see e.g. the wikipedia page or here) It can also depend on the type of instrument as mentioned by Richard. This topic has also been already discussed on the Wilmott Forum (their proposed solution is a reverse floater) Theoretically bonds with embedded options (...


3

The change of the price $P(y)$ if the yield changes from $y$ to $y+\Delta y$ is $$ \frac{P(y+\Delta y) - P(y)}{P(y)} = - D \Delta y + \frac12 C \Delta y^2, $$ where $D$ is the duration and $C$ is convexity. For small $\Delta y$ the square is much smaller. Thus the duration component dominates.


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

There are (at least) two factors here. One is the difference in convexity between the vanilla bond and the cheapest-to-deliver underlying the futures. The second is potential changes in which bond is cheapest to deliver. The former is simple enough to calculate, and you will need to dynamically hedge with futures to offset that risk For the latter, you will ...


2

Duration is also additive if you are dealing with key rate durations. In this case, Effective Duration is the weighted average of your key rate durations.


2

I'm not a bond trader and haven't looked at this in years, so my quantities may not be defined exactly as per convention, but it is generally correct. To answer your question, you should restate the present value of the bond using exponentials. This new formulation is exactly equivalent to what you wrote but much more tractable algebraically (note that my ...


2

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 ...


2

If I understand correctly the question, you wish to completely hedge the interest rate risk (defined as a parallel shift in the yield curve). If that is the case, you should use modified duration, which is the price sensitivity, rather than the MacAulay duration. They are usually close in value, but not quite the same. Fortunately, you can easily transform ...


2

It is useful in risk reports because it tells a trader the interest rate risk of each bond in his portfolio. A trader then only needs to multiply the duration by the expected yield change to calculate the price change. Scenario analysis is then easier. For standard bullet bonds (ones with fixed regular coupons) the duration can be calculated analytically ...


2

The price-yield relationship is negatively correlated; when prices go down, the implied yield goes up. The minus sign allows the modified duration to be positive for a normal bond.


2

If you believe that the fundamental economic relationship is $$ r_{\text{Spread}} = \beta \, r_{\text{Market}} + \text{const} $$ Then in order to obtain the beta of a credit index $I$ with CD01 $c$ to the market you would write $$ r_I = c \, r_{\text{Spread}} $$ and thus $$ r_I = c\, \beta \, r_{\text{Market}} + \text{const} $$ Now you need to ...


2

I calculate duration in Python using numpy, it's nice and simple: def durations(cfs, rates, price, ytm, no_coupons): import numpy as np mac_dur = np.sum([cfs[i]*i/np.power(1+rates[i],i) for i in range(len(cfs))])/price mod_dur = mac_dur/(1+ytm/no_coupons) return mac_dur, mod_dur


1

Go talk to Fincad. Here is their page on integrating with scripting languages: http://www.fincad.com/news-events/assets/pdfs/mar07/using-fincad-developer-scripting-languages.pdf Their analytics libraries include bond analytics, and they have a spreadsheet product so you can test methods and results before implementing them. Disclaimer: I work for a ...


1

Do you know the concept of duration? It is an approximation of how much the price of the bond changes if the interest rate (appropriate for the market in which the bond trades) changes. This is the interest rate is used to discount cash flows. It is common to all the bonds in the same market (e.g. German govis). For various reasons (liquidity, credit risk, ....


1

With a dedicated portfolio all interest rate risk has been eliminated, since you hold ZCB's that deliver exactly the cash you need when you need it. So it is a perfect hedge against i.r. risk. With an immunized portfolio you have reduced i.r. risk (by matching duration and convexity) but you still have a probability distribution of outcomes. Yields at ...


1

You are right that if we exactly want to know the price of a bond after a change in the yield curve, we have to calculate it - and we can. What we can say about duration: it is a linear approximation of the price change if yield change, this works rather fine with plain vanilla bonds but things get more difficult e.g. with callable bonds. keeping the eye ...


1

There are a couple limitations to bond portfolio immunization. Let's start by analyzing cash-flow matching which is a dedication strategy that is the alternative to immunization. Cash-flow matching can completely eliminate interest rate risk. The cash-flow match is setup such that the liabilities (outflows) are precisely offset by portfolio inflows on the ...


1

The purpose of bond portfolio immunization is to protect against large interest rate movements. Hence what you describe is really not immunization (a form of hedging), but rather speculation. Of course, the speculative bet you've laid out can be done with options, and the only limit is in the lower liquidity in deep out of the money options, such as would ...



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