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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 ...


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

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

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 ...


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 ...


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

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, ...


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

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 ...


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

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 ...


1

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 ...


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 ...


1

you can set up sucha portfolio using options. However I doubt you can make any money unless you are going for some very volatile IR. Using Options you can both buy a call and sell a put at 4% difference to the current IR. You will loose the premiums you paid on the oprions, if IR stay within the 4% premium. Using bonds, it is easy. if IR rise, you will ...


1

The top reference for this topic is Risk Management: Approaches for Fixed Income Markets by Golub and Tilman. The main measures you will want to calculate for hedging the yield curve risks of a bond portfolio are the key rate durations. The wikipedia article gives a brief overview. If you have access to Lehman/Barclays data, they calculate key rate ...



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