Actuarial Outpost Deferred Interest Rate Swap
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 Financial Mathematics Old FM Forum

#1
05-04-2018, 09:33 PM
 kianalf SOA Join Date: Jan 2018 College: University of California, San Diego Alumni Posts: 7
Deferred Interest Rate Swap

I'm confused as to the answer to Exercise 9.11 in the Actex Spring 2018 manual.

ZYX Corp. enters into a 4-yr loan agreement which pays 400 M at time 0 and ZYX has agreed to pay interest at the 1-yr spot rate and also repay 100 M of principal at the end of each of the 4 yrs. Interest rate swap covers years 2, 3 and 4. Find the swap rate for this deferred amortizing swap.

On the date of the loan, the yield curve is as follow:
Years to Maturity: Spot Rate
1: 6%
2: 6.5%
3: 7%
4: 7.25%

Letting P_i be the i-yr PV factor
P_1: .9434, P_2: .88166, P_3: .81630, P_4: .75581

and Q_j be the notional amt during the j-th settlement period (in mills)
Q_2: 3, Q_3: 2, Q_4: 1

I did
R = [ Q_2 * (P_1 - P_2) + Q_3 * (P_2 - P_3) + Q_4 * (P_3 - P_4) ]
/ [ Q_2 * P_2 + Q_3 * P_3 + Q_4 * P_4 ]
= .07476309

However, the answer in the book is .074786

I also tried it with the notional amounts going from 4, 3, 2 but that didn't give me the correct answer either.

I'm not sure what I'm missing here? Is it just a negligible rounding error?
#2
05-04-2018, 09:49 PM
 Breadmaker Member SOA Join Date: May 2009 Studying for CPD - and nuttin' else! College: Swigmore U Favorite beer: Guinness Posts: 3,742

I initially thought you left out year 1 by mistake, but I guess you're treating year 1 separately and years 2-4 as an amortizing swap. Mebbe I'll give this a go in Excel.

Are the spot rates effective annual or nominal semi-annual?
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Last edited by Breadmaker; 05-04-2018 at 09:55 PM..
#3
05-05-2018, 12:12 AM
 kianalf SOA Join Date: Jan 2018 College: University of California, San Diego Alumni Posts: 7

Okay thanks! Yes maybe I didn’t mention it before but it’s 1 yr deferred swap. It doesn’t specify in the question but I believe they’re effective annual.
I pretty much just used the template the book used in previous questions just changing the notional amounts.
#4
05-05-2018, 12:47 AM
 Breadmaker Member SOA Join Date: May 2009 Studying for CPD - and nuttin' else! College: Swigmore U Favorite beer: Guinness Posts: 3,742

I did some work in Excel and got 7.74786 as the swap rate. The key thing is that the rate for yr 1 is 6%, followed by 7.74786% for years 2-4. You do have to take into account the CF at time 1 when doing the actuarial equivalence. This is with no rounding anywhere in the PV factors.
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#5
05-05-2018, 01:02 AM
 kianalf SOA Join Date: Jan 2018 College: University of California, San Diego Alumni Posts: 7

Ah I see, I think I need to work a little harder with my calculator keystrokes as well.

Thank you!
#6
05-05-2018, 09:55 AM
 Breadmaker Member SOA Join Date: May 2009 Studying for CPD - and nuttin' else! College: Swigmore U Favorite beer: Guinness Posts: 3,742

You have to account for the cash flow at time 1: otherwise, your pv factors and cash flows will not line up. If you include it in your formula, it should all work out.
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