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# Duration of inverse floater

#### Eveline

##### New Member
Hi, can someone help explain the principles behind this question?

FRM Exam 2004 Qn 45
With LIBOR at 4%, a manager wants to increase the duration of his portfolio. Which of the following securities should he acquire to increase the duration of his portfolio the most?

a) a 10-year reverse floater that pays 8% - LIBOR, payable annually
b) a 10-year reverse floater that pays 12% - 2 x LIBOR, payable annually
c) a 10-year floater that pays LIBOR, payable annually
d) a 10-years fixed rate bond carrying a coupon of 4% payable annually

Ans b
The duration of a floater is about zero. The duration of a 10-year regular bond is about 9 years. The first reverse floater has a duration of about 2 x 9 = 18 years, the second, 3 x 9 = 27 years.

How do you get 18 and 27 years?

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
It's based on treating the inverse floater as a leveraged bond:
inverse floater = long fixed - long floater; in this case:
$50 MM inverse @ 8% - LIBOR = Long$100 MM fixed @ 4% - Long $50 MM floating rate @ LIBOR (i.e., inverse floater pays$4 MM - $50 MM * LIBOR. Same as other side:$4 MM - $50 MM * LIBOR) since duration of floater = 0, dollar duration of other must equal:$50 MM * duration_inverse = $100 MM * duration_fixed, or duration_inverse = duration_fixed * 100/50 duration_inverse = duration_fixed * 2 can be similarly shown that 2*LIBOR implies 3x duration. but, as shown in handbook (p 188), the key idea is to solve for dollar durations that equate the fixed bond with it's constituents (fixed = inverse + floater, such that inverse = fixed - floater) on the idea that dollar duration (risk) must be preserved. However, this follows Jorion's previous analysis and makes assumptions not in the question (the inverse can be deconstructructed in a different mix). I made up some assumptions above (fixed coupon = current LIBOR). Note, you'll get a higher inverse duration if the mix instead is$20 MM inverse = $100 MM fixed - 80 MM floating. So, it's actually more like a "rule of thumb" here were the precise answer varies based on information not given ...David #### maxima20 ##### New Member Dear all, I am wondering if anyone can help me to calculate prices and durations of below bonds (incl. inverse floaters) An investor owns a portfolio consisting of following 4 bonds: a) 10 million nominal value of 4-year fixed rate bond carrying a coupon of 5% paid semi-annually b) 10 million nominal value of 3.5-year floating rate bond that pays 6M LIBOR paid semi-annually c) 10 million nominal value of 3-year floating rate note that pays 85% * 6M LIBOR + 1% paid semi-annually d) 20 million nominal value of 4-year inverse floater bond that pays 7% - 6M LIBOR paid semi-annually How to calculate prices if I have quotes of FRA contracts spot 6M LIBOR 3,4% FRA 6x12 quoted at 3% FRA 12x18 at 3,5% FRA 18x24 at 3,7% FRA 24x30 at 3,9% FRA 30x36 at 4% FRA 36x42 at 4% FRA 42x48 at 4% ? and how to calculate duration of bonds C and D? is the duration of bond B equal 0,5? Thank you in advance Regards Tracy #### maxima20 ##### New Member can anyone help on the issue above? #### Liming ##### New Member Dear David, May I check with you if my following calculation for the inverse floater with leverage of 2 is correct?$50 MM inverse @ 8% - LIBOR = Long $150 MM fixed @ 4% - Long$100 MM floating rate @ LIBOR
since duration of floater = 0,
dollar duration of other must equal:
$50 MM * duration_inverse =$150 MM * duration_fixed, or
duration_inverse = duration_fixed * 150/50
duration_inverse = duration_fixed * 3

Thank you very much!

Cheers!
Liming

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi Liming,

Sure (i find these difficult) ... i don't think your initial equality holds:
150 @ 4% = 100*L + 50*(8%-L);
6 = 100L + 4 - 50L;
2 = 50L;
L = 25; ?

okay so if we instead solve for (x) as principal of the floater:
floater + inverse floater = fixed; i.e.,
x*L + (150 - x)*(8% - L) = 150*4%
xL + 12 - 0.08x - 150L + xL = 6
2xL - 0.08x - 150L = -6
L*(2x - 150) - 0.08*x = -6

...I am employing procedure Jorion uses (handbook p. 188)...now since Libor (L) varies, the term:
L*(2x - 150)
...must equal zero
if so, x = 75
i.e., either 2x- 150 =0 or - 0.08*x = -6

so that gets us an allocation of:
$75 to floater plus$75 to inverse floater

where the key test is, have we matched (replicated) the cash flow, so the test is:
does: 75*L + 75(8% - L) = 150*4%? Yes!

so then: inverse duration = duration fixed * 150/75
so I would get x2 instead of x3....

hope that helps, David

#### Liming

##### New Member
Hi David,

Thank you. However, I think your calculation still holds for duration = 2x only. I still don't understand when leverage is 2, how duration = 3x is calculated? Can you kindly illustrate? Thanks.

Cheers!
Liming

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi Liming,

I don't follow, I can't get 3x out of it.

But if instead the floater coupon were: 12% - 2*LIBOR, then you would have duration @ 3X, as:
100*LIBOR + 50*(12% - 2*LIBOR) = 150*4%

then, 150/50 would get you 3x the duration of the fixed....David

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi Liming,

...sorry, maybe you refer to finding the mulitiple?... find the dollar duration relies on the key idea that risk (duration) must be preserved in the replication:

risk (fixed coupon) = risk (synthetically equivalent floater + inverse)
i.e., if the right hand side replicates than risk should be the same, and we want to use dollar duration to aggregate (not duration per se)

duration(fixed coupon) = duration (floater) + duration(inverse)
..then simplifying assumptino that, since duration of floater is only "time to next coupon," that is near enough to zero, round down (although we can include the small duration to be more precise)

150*Mod Duration of fixed = ~0 + 75*Mod Duration of floater
so, Mod Duration of floater = Mod Duration of fixed *150/75

...but it seems maybe you know that part...David

#### Liming

##### New Member
Dear David,

Thank you for your reply. Sorry I need to bother you again with another question that has been puzzling me for long. This is about the option-adjusted spread for mortgage back securities.
According to page 181 of FRM handbook 5th edition, '... OAS = Static spread - Option cost. During market rallies, long-term Treasury yields fall, ... pushing up the yield spread.' Although I understand the equation a bit, I don't quite follow the logic of what happens during market rallies in the book's description.
1) why does market rallies imply Treasury yield fall? Does this rally refer to rally in Treasury bond market only? From my understanding of the market, the recent credit crunch has lead to a overall market liquidity freeze, associated with a fall in Treasury yield, due to investors' flight to quality. Therefore, I'm confused about whether a rally indicates yield fall or a market liquidity contraction leads to yield fall?
2) In the described scenario, the bond is likely to be prepaid early. I think this means that the option sold implicitly to bond issuers is more 'in - the - money', thereby increasing the cost of writing the option. However, I don't quite understand why as the book describes, ' their option cost increases, pushing up the yield spread'? I can't see the necessary link between the option cost and the yield spread. I'm just guessing that maybe it's because OAS should maintain stable, therefore requiring the 'Static spread' to increase so as to adjust to the higher option cost?

Cheers!
Liming

#### maxima20

##### New Member
Hi David,

An investor owns a portfolio consisting of following 4 bonds:
a) 10 million nominal value of 4-year fixed rate bond carrying a coupon of 5% paid semi-annually
b) 10 million nominal value of 3.5-year floating rate bond that pays 6M LIBOR paid semi-annually
c) 10 million nominal value of 3-year floating rate note that pays 85% * 6M LIBOR + 1% paid semi-annually
d) 20 million nominal value of 4-year inverse floater bond that pays 7% - 6M LIBOR paid semi-annually

How to calculate prices if I have quotes of FRA contracts
spot 6M LIBOR 3,4%
FRA 6x12 quoted at 3%
FRA 12x18 at 3,5%
FRA 18x24 at 3,7%
FRA 24x30 at 3,9%
FRA 30x36 at 4%
FRA 36x42 at 4%
FRA 42x48 at 4% ?

and how to calculate duration of bonds C and D?
is the duration of bond B equal 0,5?

Thank you!

#### flex

##### Member
Hi, @David Harper CFA FRM, Hi all !
Let me to deviate from topic's core concept.

so, if comment 's problem claim price (price component) calculation given (over) FRA-%-quotes (like:

How to calculate prices if I have quotes of FRA contracts
spot 6M LIBOR 3,4%
FRA 6x12 quoted at 3%
FRA 12x18 at 3,5%

FRA 18x24 at 3,7%
...), allow that quotes directly calculate d: d(0.5)=1/(1+z0/2); d(1)= 1/{(1+z0/2)(1+'FRA 6s12m'/2)} ,

d(1.5)= 1/{(1+z0/2)*(1+'FRA 6s12m'/2)*(1+'FRA 12s18m'/2) } etc ? i.e. f(.5;1)=='FRA 6s12m',
and f(.5;1) calculation no need extra compounding/decompounding?

ps: my qstn is excited by 'spot 6M > FRA 6m' relation
Thanks, flex

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