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:

ANSWER C

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?

David Please help I am sure I am not the only one who would like to know the answer for this.

Thanks!

DavidM

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

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

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

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.