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Modified duration with zero-coupon bond yields

I want to calculate the modified duration for a series of non-fixed cashflows (monthly). I have a series of monthly discount factors. To do this I have been calculating the Macauley duration as ...
Actuary_11234's user avatar
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1 answer
67 views

Proving that Convexity approx. equals Duration squared but something goes wrong?

I am trying to derive a formula for bond convexity that I saw in a textbook which states that $$\text{convexity} = \frac{\text{Macaulay duration}^2 + \text{Macaulay duration} + \text{dispersion}}{(1+\...
Milan's user avatar
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Modified duration of T-Bill and zero coupon bond

How to calculate the modified duration of T-bill (discount instrument) and europeans bills (zero coupon instrument). I couldn't find how Bloomberg is calculating those values on YA
Edson Almeida's user avatar
1 vote
2 answers
230 views

Is the "$\textit{theoretical}$" $DV01$ of a bond an accurate estimate?

Dollar duration $DV01$ is defined as negative of the price of the bond wrt yield: $$DV01 = - \frac{\partial P}{\partial y}.$$ As we know that $P = \sum_{t=1}^{n} \frac{CF_{t}}{(1+y)^{t}}$, then $$DV01 ...
Sane's user avatar
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Spread Duration of a Fixed Rate Corporate Bond, with offsetting Futures Position

My question is relatively simple with respect to the below scenario: I take a $5m long position in a vanilla fixed-rate corporate bond with a spread of 1.50% for a YTM of 5%. These coupons are paid ...
fixedincome94's user avatar
1 vote
0 answers
58 views

How long to hold a bond fund to be guaranteed a certain return? (mathematical proofs)

Is there a mathematical proof of how long one must hold a bond fund or ETF to be guaranteed a positive return? And of how long to be guaranteed a certain return? For simplicity: let's consider ...
Pythonista anonymous's user avatar
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1 answer
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FRN duration when discount curve and projection curve have non-perfect correlation

The textbook example assumes that discount curve and projection curve are the same (or have a perfect correlation). What happens with the FRN's duration when it is not the case? For example, there are ...
Ragewave's user avatar
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Macaulay Duration of a Callable Bond [closed]

I could not find any formula of Macaulay duration for a callable bond in the literature. Can anybody show how to derive it or give a reference where it is already obtained. EDIT My goal is to find a ...
Sane's user avatar
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1 vote
2 answers
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mathematical proof of the hedge ratio formula for bond futures

We know that the hedge ratio ϕ_F that we should use in order to to the duration-hedging through bond futures is: $$ϕ_F= -(DV01_B / DV01_{CTD} )\cdot CF_{CTD}$$ Where $\textrm{DV01}_B$ is the dollar ...
luca dibo's user avatar
2 votes
1 answer
583 views

Using CME SOFR Futures in practice

I'm looking for numerical examples on how CME SOFR futures contracts are used in practice for hedging purposes. Book references containing this discussion are appreciated. Bloomberg's FLDS function ...
SuavestArt's user avatar
4 votes
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Portfolio immunization in time

The company will have to pay out an amount of liabilities $13594$ at the moment $t=9$. In $t=0$ they want to cover it with 4-years zero-coupon bonds and yearly perpetual annuity that is due in arrears ...
Miłosz 's user avatar
2 votes
2 answers
562 views

Quantifying the impact of rates change on bond prices

How can I quantify the impact of a change in interest rates on bond prices? I know that in a classical textbook setting the answer would be to compute the modified duration of the bond and, to account ...
Peter's user avatar
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1 answer
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What is $\sum_{i=1}^n i^2 \frac{CF_i}{CF_{Total}}$

I know that $$\sum_{i=1}^n i \frac{CF_i}{CF_{Total}}$$ is Macaulay Duration, but what is $$\sum_{i=1}^n i^2 \frac{CF_i}{CF_{Total}}$$. I have given it a thought like it is a second moment of something ...
actsci stud tries2learn math's user avatar
2 votes
1 answer
1k views

Simplified formula for duration of interest rate swap

Lets consider the simple interest rate swap instrument as 5-year maturity interest rate swap. I found an interesting simplification to calculate the duration of such swap as, $\frac{\left(1 - e^{-r_t *...
Brian19931's user avatar
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2 answers
230 views

Are there names from the third term onwards in the Taylor approximation for bond pricing?

The first terms are duration and convexity, but are there common names for the terms beyond this?
ltrozzo's user avatar
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1 vote
2 answers
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Is duration of a bond a convex function?

I understand that in general, the NAV of a bond is a convex function. However, I am not too sure if the same can be said for its duration. Are there references on this? Thanks
Preston Lui's user avatar
1 vote
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70 views

Modified Duration as interest risk [closed]

I am new to bond pricing and I am studying the sensitivity measures of a bond (with discrete compounding) and even though I understand the mathematical concepts of modified duration and convexity ...
user avatar
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2 answers
738 views

ATM interest rate swap dv01 vs off-market swap dv01

How significant is impact on dv01 of an at-the-money swap if rates move significantly? For example: lets say a 5Y USD at the money swap has 4.6 duration. now, if the rate curve move by 150-200bps, ...
toing's user avatar
  • 233
2 votes
1 answer
265 views

Macaulay Duration - Liability matching

Can someone provide a detailed example to prove the following statement: "When the investment horizon is equal to the Macaulay duration of the bond, coupon reinvestment risk offsets price risk.&...
LGE123's user avatar
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1 answer
267 views

Bond value as a function of spread change and duration/maturity

I am trying to calculate the change of value in a universe of bonds given a series of shocks to the credit spread of each bond. As a constraint, the initial dataset only contains the spread change for ...
sets's user avatar
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Formula to calculate the sensitivity of a bond to curve steepening

https://github.com/jerryxyx/TreasuryFutureTrading/blob/master/README.pdf On page 2 of the pdf above, the sensitivity of a bond to increases in the slope of the curve, is given as $\ln(T)$, where $T$ ...
Bazman's user avatar
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1 answer
319 views

Convert spreads to prices for bonds via duration

I have the price of a bond and would like to convert it to spreads. Is this possible by just having dollar duration? Secondly, if I just care about the relative spreads of multiple bonds, is it enough ...
Nickpick's user avatar
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2 answers
707 views

"The five year swap has the same dv01 as a par five year treasury bond" Why?

Am reading a book (The Complete Practitioner's Guide to the Bond Market by Steven Dym, 2009) where the author gives an example of someone buying a 5 year par 4.65% treasury and someone else entering ...
filifunk's user avatar
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1 answer
212 views

What is the definition of horizon current coupon spread duration

Trying to understand the meaning of current coupon spread duration? Is this same as empirical mortgage spread duration?
nsivakr's user avatar
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1 vote
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Calculate the duration of group of Bond ETFs

Is it correct to get the weighted average of a bunch of bond ETFs to get the duration? Is it theoretical correct to say that. I have 6M AGG (duration 8.39), 30M BND (duration 8.7), 60M SHY (duration 1....
the_brass_bottle's user avatar
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1 answer
90 views

Duration and convexity of an open term loan/bond!

Imagine an open term loan with monthly interest payments of [x]% and the principle due when the loan is closed. Both the lender can call the loan, and the borrower can return the loan (with no penalty)...
AnonnonA's user avatar
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0 answers
95 views

Floating rate bond valuation and Duration

I would like to know how a floating rate bond will be valued when LIBOR is used to compute the coupon cashflows and the bond is discounted using an OIS rate. Normally I have seen valuation of a FRB ...
Jaya Mohan's user avatar
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0 answers
166 views

Hedging Curve Risk with Futures

Please help confirm/correct my understanding of hedging curve risk using futures. For example, if I purchased a 9y bond and hedged the duration risk (parallel shift) by shorting 10y treasury futures ...
user58876's user avatar
0 votes
1 answer
1k views

Does credit default swaps have interest rate duration and credit duration?

Will a CDS have interest rate duration and credit duration? It does seem likely that the value of the CDS would depend on the underlying interest rate, or the spread. But when I try to Google this I ...
user394334's user avatar
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1 answer
157 views

Spread duration curve by issuer or by sector

I was surprised to see that spread duration was not offered as a curve in Bloomberg. As a result, I'm trying to find a curve (in Bloomberg) or build a curve (maybe using the Excel API) which ...
mogwai's user avatar
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1 answer
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Duration of Floating rate bond

I have read in several places that the duration of a floating rate bond is simply the time until next coupon payment, because the upcoming coupon payments are not known. I have 2 questions here: Why ...
pussret's user avatar
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1 answer
159 views

Bond Future and Bond Yield relation

I recently read, that yield changes during a short time window can be approximated by dividing the returns of futures on the bond by its Duration. Has anybody heard this before and can shed some light ...
mindandfields's user avatar
1 vote
2 answers
279 views

Is this simple model used to calculate the interest rate duration and credit duration of a floating rate note? Other models?

I found this model for floating rate bonds in a book I am reading and I am wondering if it is used anywhere in practice? $$MV=\frac{\frac{(Index+QM)\cdot FV}{PER}}{\left(1+\frac{Index+DM}{PER}\right)^...
user394334's user avatar
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0 answers
167 views

Bond yield with known DV01

Is there a way to calculate the yield of a bond knowing the DV01 and duration (numerical values) and knowing all other parameters of $V$ (maturity, coupon, etc.)? $$DV01=-\frac{\partial V}{\partial y}$...
SimonCello94's user avatar
3 votes
3 answers
2k views

How can a deep discount bond with a longer time to maturity have a LOWER duration than an otherwise identical bond with a shorter time to maturity?

Here is a brief excerpt on the fixed income chapter from the 2020-2021 level 1 CFA curriculum: Generally, for the same coupon rate, a longer-term bond has a greater percentage price change than a ...
Pertinax's user avatar
  • 131
-1 votes
1 answer
2k views

QuantLib Python: how to calculate duration and convexity for irregular cashflows? Can I use SimpleCashFlow or must I define a custom bond?

I have 2 questions: If I want to discount a set of irregular cashflows, I can do it using the SimpleCashFlow class, or defining a bond with custom cashflows (thank you to Ballabio and David Duarte for ...
Pythonista anonymous's user avatar
0 votes
1 answer
63 views

What is the duration of a rolling 5 year investment?

I have difficulty with the duration of a 5 year investment (like GIC). In such an investment, the investment (GIC) rate is reset as the current market rates. So the market value is equal to the face ...
xren's user avatar
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1 vote
1 answer
454 views

Modified Duration and how it explains bond price sensitivity to changes in the yield to maturity [closed]

My questions is the following: why is it that the modified duration can explain so well the change in price of bonds? I don't get the mathematical relationship behind this, that is between modified ...
adriano's user avatar
  • 23
0 votes
1 answer
2k views

Gamma/Convexity of a Swap vs a similar bond

As a rule of thumb, how would the duration and convexity of a 30y UST bond paying X% compare to the duration and convexity of a matched maturity vanilla interest rate swap, with a similar fixed rate. ...
CreditNecromancer's user avatar
1 vote
1 answer
495 views

Duration and Convexity

I am searching to estimate the evolution of my portfolio duration following a yield increase/decrease. Can i use the convexity? I mean IR delta x (- convexity) = Duration delta Is it correct? Thanks a ...
Jerome Zerbib's user avatar
1 vote
1 answer
2k views

Duration of a futures contract

I have tried to find an answer to this question but have come up with nothing. So, it is my understanding that to find the duration of a futures contract (assuming no switch risk), all you need to do ...
basisnerd123's user avatar
0 votes
1 answer
136 views

Is the risk the same for two different tenor bonds with the same DV01?

I have two different bonds (for e.g. 1yr and 10yr) that have the same DV01. The notional for 1yr bond is definitely more than the 10yr bond. Is the risk same for the bonds the same because DV01 for ...
lakshmen's user avatar
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0 votes
1 answer
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Hedge Active Duration by Issue Currency or Country of Risk

For example, lets say I own a bond issued by a company in Mexico that's denominated in USD and I want to hedge my duration exposure. I obviously need to hedge duration to the US yield curve. Do I ...
short_vol's user avatar
2 votes
1 answer
3k views

Duration of a floating rate bond with spread

I need to calculate the duration of a floating rate bond with spread. With zero spread the price of the bond is given by: $$p_\tau=(1+c_1)e^{-r(\tau_1) \cdot \tau_1}$$ so the duration is: $$-\frac{\...
Rodrigo Palacios's user avatar
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0 answers
535 views

How to calculate the new price of a bond using duration rule and duration with convexity rule?

A bond with a 30 year maturity, par value of $1000 and is 8% p.a. coupon is selling at an yield to maturity of 8% p.a. The modified duration of the the bond at its yield is 11.26%, and its convexity ...
Chandramouli Raman's user avatar
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0 answers
1k views

Why is the effective duration of a floating rate bond the time to the next payment?

I am wondering why the effective duration of a floating rate note is the time to the next payment. Effective duration is is defined as $$\frac{V_{-\Delta y}-V_{+\Delta y}}{2V_0\Delta y},$$ for a small ...
user394334's user avatar
2 votes
0 answers
303 views

How to optimise Fixed Income portfolio (Yieldbook) based on YTM, duration, rating and exchange rate

I have a fixed income portfolio built up in the Yieldbook and BBG Port. However due to some bad performance of my portfolio compared to the benchmark I would like to build up an optimisation which ...
Luigi87's user avatar
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1 vote
0 answers
174 views

Interest rate hedging using treasury futures – timing and duration

I'm pondering over the following (rather standard) problem: We have \$10 million invested in government bonds and are concerned with highly volatile interest rate over the next six months. We want to ...
user2175871's user avatar
1 vote
2 answers
90 views

Step-up bonds should be more, not less sensitive to market interest rates, shouldn't they?

I keep reading that "a step-up bond provides more protection to an investor in the face of market interest rate fluctuations", that "a step-up bond typically performs better than any ...
BabaYaga's user avatar
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0 answers
134 views

Bond Change in Absolute or Relatively Percentage

Since most of the bonds prices are quoted in price, for example, bond price of 103 means 103% of the principal or face value (ex. $1000). Suppose a bond has modified duration of 4.62 years. If yield ...
M00000001's user avatar
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