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# How did they solve for duration in this question? GARP paractice Exam

#### DavidM

##### New Member
Hi David,

This question is from GARP Practice Exam 2, 2006 (Question 61)

How do they find the Duration input of the 6% coupon bond?, in the following question:

What is the best estimate of the market value of a portfolio of USD 100 Million invested in recently issued 6% 10 year bonds and USD 100 million of long 10 year zero coupon bond if intrest rates decline by 0.50%?

a) USD 219 million
b) USD 195 million
c) USD 209 million
d) USD 206 million

The solution says:

To calculate the best estimate of the market value of the portfolio if interest rates decline 0.5%, one needs to calculate the change in the market value of each bond using duration. The duration of the 10 yr zero coupon bond is 10.Thus, the change in the value of this bond equals 10x0.005x100,000,000 which equals 5 million.

The duration of the newly issued 6% bond is 7.802 assuming that the price of the bond is par. Given a duration of 7.802 assuming that the price of the bond is par.

Given a duration of 7.802 the change in the value of the bond equals 7.802x0.005x100000000=3.91million.

I understand there rational of assuming that the bond is at par because it is recently issued. However, I STILL DONT SEE HOW THEY SOLVE THE DURATION TO BE 7.802 FROM THAT ONE ASSUMPTION!!

*I have looked at all the duration formulas and I can not make the connection!

Secondly, not as important- but why cant one solve for portfolio duration and then work out the Market value estimate?

Thanks!

DavidM

#### David Harper CFA FRM

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

Uggg ... sample questions like this sincerely disappoint/frustrate me ... because you are absolutely correct. No duration for the coupon bond can be retrieved from the information given: you need the yield. (Modified) duration is not invariant to yield; higher yield implies lower duration. Therefore, the 7.802 assumes some unstated yield (further, there is an unstated compound frequency; semi-annual? annual?). In short, this is an awful, awful question

Note another error: it's true the Macaulay duration is 10 for a 10-year zero, but the modified duration is < 10; i.e., Mod duration = 10/(1+y/k). Technically, we do not use Macaulay duration, we want the sensitivities (ie.., Modified, dollar duration or DV01). Please note the question also misuses the Macaulay duration as a sensitivity.

Re: "but why cant one solve for portfolio duration and then work out the Market value estimate?"
Absolutely you can, that is how i would; related, if you aggregate to portfolio duration, you want to sum their dollar (aka, value) durations ...
e.g., dollar duration of bond 1 + dollar duration of bond 2 = portfolio dollar duration
(you wouldn't sum Macaulay and you wouldn't sum the sensitivities per se)

Thanks, David

#### concepts

##### New Member
Hi David,

Here is another exam of such question from FRM 2007 question 74 practice exam:

Hong Kong Shanghi Bank has entered into a repurchase agreement with a client where
the client will sell a 10-year US treasury bond to the bank and repurchase it in 10 days.
The bond has a notional value of USD 10m, trades at par with the yield volatility for a 10-
year US treasury 0.074%. The swap’s maximum potential exposure at a 99% confidence
level is closest to:

a. USD 320,000
b. USD 380,000
c. USD 550,000
d. USD 1,200,000

CORRECT: B

The approximate duration for a 10 year bond is 7.0. The volatility of the swap value over
10 years is calculated as follows:
σ(V) = [market_value * duration * yield volatility *(10)0.5]
= 10,000,000 * 7.0 * 0.00074 * 3.16 = 163,806.
To get the 99% confidence interval, we multiply σ(V) by 2.33, which gives approximately
\$380,000.
Reference: John Hull, Options, Futures, and Other Derivatives, 6th ed. Chapter 7

What I don’t understand is that statement “The approximate duration for a 10 year bond is 7.0”

My argument is that if you are asking me to calculate duration for a fixed income production, please give me the coupon rate! Somebody might say “the question says “it is trading at par” which is fine but a 10- year 3% coupon par bond has a modified duration of 8.58, a 10-year 10% coupon par bond has a modified duration of 6.23, also a 10-year 7% coupon par bond has a modified duration of 7.106

How many bonds duration does one have to calculate in order the get the average duration of 7.00 as given by GARP?

Thanks,

Concepts

#### David Harper CFA FRM

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

You are exactly correct: for modified duration you need yield and coupon; if you are given the assumption "trades at par," you still need another number. It is totally inconsistent with our assignments to assume a duration of 7.0 (further, it should specificy modified duration). The only duration we can "eyball" really is for a zero coupon where Mac duration = term to maturity, and Mod duration = Term / (1 + y/k). So even Mod duration needs a yield for the zero. For non-zeros, we need coupon *and* yield as mod duration varies with either. David

#### [email protected]

##### New Member
n cp pv n*pv
1 6 5.660377358 5.660377358
2 6 5.33997864 10.67995728
3 6 5.037715698 15.11314709
4 6 4.752561979 19.01024792
5 6 4.483549037 22.41774519
6 6 4.229763243 25.37857946
7 6 3.990342682 27.93239877
8 6 3.764474228 30.11579382
9 6 3.551390781 31.96251703
10 6 3.350368661 33.50368661
10 100 55.83947769 558.3947769
S 100 780.1692274

MacDur 7.8017 (How Mac D may be calculated) years (can interpreted as analogous to payback period)

Also, For a fixed rate & regular bonds Mod Dur = 7.8017 / 1.06 (since priced at par; Coupon = Yield) = 7.3601

Regards,

NRafanan, PRM

#### concepts

##### New Member
Hi NRafanan,

May be either you did not understand my question or GARP question.
The question did not give any coupon rate! You used 6% coupon rate in your calculation, you cannot assume a rate in the exam. What my question / concern is that GARP should be more precise in writing questions and give as much information as possible the one enable one to answer the questions
Calculating fixed income production when given all the necessary information I think should be at the finger tips of very FRM candidate by now.

Thanks,

Concepts

#### [email protected]

##### New Member
Hi David, refer to text below -

And probably I misundestood it.

Given Question (Original text read as) - What is the best estimate of the market value of a portfolio of USD 100 Million invested in recently issued 6% 10 year bonds !!! ( and USD 100 million of long 10 year zero coupon bond if intrest rates decline by 0.50%)?

A bond when issued must have a stated coupon (6%) and maturity (10 years) [unless a consol or a perpetuals] and usually regular bonds are issued at par or at a premium by the issuer for obvious reason. [The issuer can control these variables]

An experienced risk manager should be able to read in between the lines despite limited data and make logical assumptions. But I guess your right, the examiners should write explicit given/data for the examinees to answer the questions unequivocably.

In the final analysis RM is both a science and an art!

Kind regards,
Rhapsody

#### David Harper CFA FRM

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

I agree the question provides the coupon (6%) but no yield is given; if the question had said "priced at par" then we'd know yield = coupon, but without that information, it's an assumption the bond is priced at par...I totally agree with you that the risk manager, and the FRM candidate, should be able to read in between the lines despite limited data and make logical assumptions
... but in the cirriculum we are assigned precise material (Tuckman in this case) which includes understanding ideas like "duration varies with yield" and, eg., distinctions btwn Macaulay/modified duration ... imprecise questions undermine the material IMO... and a related problem is that bad questions can actually be detrimental to the best candidates: they will sometimes get appropriately stuck if the question lacks vital information ...(and, of course, for practice questions, it's not helping current candidtates to encourage duration guessing!)

in any case, thanks for showing your calculations!

David

#### rvalive

##### New Member
Rhapsody:

Did PRM exam have imprecise questions like this? Just curious.

With 7.36 duration the answer comes to 401007.54 which is closer to 380,000 than any other answer choice so I will choose it. But, I now feel like a contestant in "Price is Right!" . Hmmm.... considering the debacle of the last few years, financial industry players are in the same position except may be with better degrees than the average game show contestant.

Concepts:
I used to get confused in the beginning. May be we should have a common assumptions & "incorrectly specified formulas, like absolute VAR for example" in a thread all by itself to help folks like us.

#### dthigale

##### Member
Hi David,

With the assupmtion of bond is at par, that is yield is 6 %, I tried to calculate the duration with:

change in bond price / change in the yield.

= 100-103.8 / 6-5.5 ( Bond price from calculater with FV= 100 / PMT = 6 (same coupon) / I/YR= 2.75 (semi annual) , N 2= 20)

The duration comes to -7.6 << But according to calculations (Rhapssody) above this is different.

I think you have similar example in 4C ( Bond price goes to 851 from 851.23 from 1 basis point change)

-D

#### jesal27

##### New Member
David, the mac duration of the new issue is 7.8 Yrs. All you need to do is take the weight of each Coupon and add them. Yes, the question assumes the bond is selling @ par as no yield is mentioned.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @jesal27 Yes, agreed. At the time I replied (years ago), I don't think I realized that we can use Tuckman's shortcut for a par bond, where mod duration = [1 - 1/(1+y/k)^(T*k)]/y, which under the assumption of annual compounding (k = 1) here gives the elegant D* = [1 - 1/(1+y)^T]/y = [1 - 1/1.06^10]/0.06 = 7.360 years in mod duration, or 7.360*1.06 = 7.802 in Mac duration (i.e., the result same we get from summation of weighted cash flows).

However, this older FRM question does not meet the modern FRM standards: 1. The par price assumption isn't a necessary assumption (so it really does need to be stated as duration is variant to yield as this dynamic is especially timely in the current ZIRP/negative interest rate environment ) , 2. There is an incorrect (inexact) usage of Mac duration in the solution, when modified duration is wanted, and 3. modern FRM would supply the compound frequency assumption. Thanks,