# Quiz Question FI

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Bakshi is a Bond Portfolio Manager , who has portfolio of \$40 million under investment which

are partly financed by debt of \$17 million.

Bakshi had a strong belief over the last few years of a decrease in interest rates and hence

also took a long Tbond futures position equivalent to \$8 million of bond contracts.

if ED of bonds is 5.1 , the future has ED of 6.2 and debt has ED of .07 calculate ED of

overall combined portfolio for 100bps decrease in Interest rates

a)5.66

b)6.66

C)10.97

D)11.56

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C?

Just eyeballing it I’m pretty sure the answer’s A, but when I’m calculating it out, I’m getting 5.25.

You sure that .07 for the debt isn’t a typo?

can someone also come up with the formula plz… not sure how to adjust the bond futures portion?

A?

level3aspirant Wrote:
——————————————————-
> can someone also come up with the formula plz…
> not sure how to adjust the bond futures portion?

Duration is additive, so you simply add to the portion on the long bonds.

Then you subtract the duration of the borrowed funds, but I’m still getting 5.25.

OP, you got an answer and an explanation for us?

no way could be A. since you’re using leverage, the duration will always be higher than any individual duration. 5.66 < 6.2, so that’s out.

I got C as well.

D_mod = (D_port * P + D_future * F - D_borrowed * B) / I

where P = amount in portfolio, F = amount of futures, B = amount borrowed, and Durations for each accordingly. I = amount invested, which is portfolio net of leverage.

D_mod = (5.1 * 40 + 6.2 * 8 - .07 * 17) / (40 - 17) = 10.97438. C.

never seen using futures and its ED in leverage calculation. This is new to me.

cfaboston28 Wrote:
——————————————————-
> never seen using futures and its ED in leverage
> calculation. This is new to me.
>

The answer is directly above you.

What a stupid I am.

I get C as well.

I’m not sure I follow the logic of incorporating the the duration of the futures in the calculation of the original position duration. My approach was to first calc the leveraged duration based on

D_mod = (Dp P - Db B)/E

Then taking the weighted average of portfolio and the futures to find the total duration (because duration would be additive)

w Dp + w Df = Dtotal

Of course it gave me an answer not listed there, so can you point out the section in the curriculum that I now need to review?

I am getting the answer C, but the comment regarding “calculate ED of
overall combined portfolio for 100bps decrease in Interest rates ” seems to be misleading. Fixed Income portfolios composed of callable/putable bonds, mortgage backed securities, the duration would change but that is not given in the question. Am I reading too much?

This is from R30, L0S a… formula copied from CFAI text:

Also for levered P, duration of E >> that of unlevered P, or
D(E) = (D(a)A - D(l)L)/E, or
Duration of Equity is [ Duration of Assets * MV, Assets - Duration of Liab’s * MV, Liab’s ] / Equity

Wording “… 100 bps decrease in Interest Rates” is meant to confuse us…. is true for any duration calculation!

@ mp2438 Wrote:

i dont understand the logic of how you adjusted the duration of futures in there? it says “took a long Tbond futures position equivalent to \$8 million of bond contracts” SO why are we not adjusting it in denominator? TKVM

level3aspirant Wrote:
——————————————————-
> @ mp2438 Wrote:
>
> i dont understand the logic of how you adjusted
> the duration of futures in there? it says “took a
> long Tbond futures position equivalent to \$8
> million of bond contracts” SO why are we not
> adjusting it in denominator? TKVM

My guess is that you are not required to put up 8 million in equity in order to gain that kind of exposure through the future. 8 million notional long bond contracts = 80 contracts. I am pretty sure that basic margin on the long bond is \$3,000 per contract, so you could argue that 240,000 should be added to the denominator.

C.

Duration equity (Equity) = Duration assets (assetts) - duration liablilities (liabililities)

Duration equity (17MM) = 40MM (5.1) + 8MM(6.2) - 17(.07)

solve duration equity = 10.97

for 100 bps = 10.97

Alternatively, you could find the equity duration of the portfolio first, and then factor in the futures. It doesnt matter the order. Of you did it that way:

Duration equity (Equity) = Duration assets (assetts) - duration liablilities (liabililities)

Duration equity (17MM) = 40MM (5.1) - 17(.07)

Solve for equity duration and you get 8.81

Then lever up the futures:

Equity (equity duration) + futures notional (futures duration) / equity

23MM x 8.81 + 8MM x 6.2 = 252.33/ 23MM = 10.97….same answer

You dont adjust in the denominator for the future (ie value of equity) because the future is assumed to have a negligible equity (small margin amount).

I think I see what I did wrong… I’m not sure I understand 100%, but it sounds like I need to included the Total Portfolio, the Futures, and the Borrowed funds in the denominator, adjusted by the respective durations; and then divide that whole thing by the Equity….?

@ 1morelevel

i understood that why it’s fine even if we dont include futures amount in denominator (the margin will be very small) but now i am wondering why you have taken equity as 17 mn. ???

shouldnt it be 40 (assets)-17(liability) = 23 mn equity

am i missing something else now?

Rustyrudder - you divide by net equity, assets - liabilities.

Typo. How do i fix it?

Level3aspirant - its a typo… i was transcribing from sloppy notes. Not sure how to fix it….

Solution:

COV of bonds = -5.1*.01*40000000
COV of debt = -.070*.01*17000000
COV of futures=-6.2*.01*8000000

Total Cov =2040000+11900+496000=2524100

Using \$(delta value)= - ED * .01 * (40000000-17000000)=2524100

Hence ED=10.97

Got it. thanks :)

1morelevel Wrote:
——————————————————-
> Rustyrudder - you divide by net equity, assets -
> liabilities.
>
> Typo. How do i fix it?
>
> Level3aspirant - its a typo… i was transcribing
> from sloppy notes. Not sure how to fix it….

Captain Barbosa Wrote:
——————————————————-
> Correct answer is C) 10.97
>
> Solution:
>
> COV of bonds = -5.1*.01*40000000
> COV of debt = -.070*.01*17000000
> COV of futures=-6.2*.01*8000000
>
> Total Cov =2040000+11900+496000=2524100
>
> Using \$(delta value)= - ED * .01 *
> (40000000-17000000)=2524100

What is COV ?
>
> Hence ED=10.97

Hi,

A few questions

C.

Duration equity (Equity) = Duration assets (assetts) - duration liablilities (liabililities)

Duration equity (17MM) = 40MM (5.1) + 8MM(6.2) - 17(.07)

solve duration equity = 10.97

==> How are you getting 10.97 here? Don’t know why the arithmetic isn’t working for me. Unless you mean 23MM for equity, which I believe it is.

for 100 bps = 10.97

Alternatively, you could find the equity duration of the portfolio first, and then factor in the futures. It doesnt matter the order. Of you did it that way:

Duration equity (Equity) = Duration assets (assetts) - duration liablilities (liabililities)

Duration equity (17MM) = 40MM (5.1) - 17(.07)

Solve for equity duration and you get 8.81.

==> Same issue here. Equity should be 23MM I think.

Then lever up the futures:

Equity (equity duration) + futures notional (futures duration) / equity

23MM x 8.81 + 8MM x 6.2 = 252.33/ 23MM = 10.97….same answer

You dont adjust in the denominator for the future (ie value of equity) because the future is assumed to have a negligible equity (small margin amount).

—————————————————– This feeling is so so so f x x k ! - qqqbee, August 27, 2010

2040000+11900+496000 = 2547900 and not 2524100 (or My MS Excel is wrong….)

And takes me to a duration of 10.5.

I can’t reproduce 10.97…what’s going on?

Captain Barbosa Wrote:
——————————————————-
> Correct answer is C) 10.97
>
> Solution:
>
> COV of bonds = -5.1*.01*40000000
> COV of debt = -.070*.01*17000000
> COV of futures=-6.2*.01*8000000
>
> Total Cov =2040000+11900+496000=2524100
>
> Using \$(delta value)= - ED * .01 *
> (40000000-17000000)=2524100
>
> Hence ED=10.97

Where is this question from? and please post their guideline answer (if there is more than you gave)

i dont get the way everyone is attempting this.. too much confusion, we need the guideline answer. Looking at it tho, here were my thoughts:

Duration of leveraged portfolio = [(Duration assets*total assets)-(Duration liab*total liab)] / Equity -its a formula straight out of the book about calc leveraged dur.

total assets = 48M (40M AUM + 8M futures)
Duration of assets = 5.283 -> .8333(5.1) + .1666(6.2) = 4.25+1.033 (the .8333 is the weight of AUM(40M), .1666 is weight of futures (8M))

total liab = 17M
Duration of liab = .07

total equity = 23M (40-17)

So in the formula you have:
(5.283*48M) - (.07*17M) / 23M = 10.974

I think many of us are doing very similar things to come up with the same answer.

Nevermind, I was just confused that the 8M futures was a later purchase and couldn’t figure out how it was funded.

thanks. know i know it.

mark:
You dont adjust in the denominator for the future (ie value of equity) because the future is assumed to have a negligible equity (small margin amount).

Thanks for that. I incorrectly had the debt beta at 0.7 instead of 0.07. Without picking up on this discrepancy I saw someone else who’d made the same mistake, and assumed there was something bigger amiss. But there wasn’t - it was just me.

APP.

Thank you guys, this is all so helpful! =) I love it when people approach the same problem in various ways.. it helps to better understand the topic..

1morelevel Wrote:
——————————————————-
> You dont adjust in the denominator for the future
> (ie value of equity) because the future is assumed
> to have a negligible equity (small margin amount).

Can you refer me to the book where it states that?
To me, Asset = 40+8=48; Liability = 17 –> equity = 31.