Goldfarb manual (page 495-496)

[bagheera]

In problem 1 on page 495 the expected number of defaults for band 5 are to be determined. Now band 5 has two obligors. Therefore, shouldn't the expected number of defaults (or default rate) be equal to the expected default probability for the band (as obtained in the example for expected number of defaults)*# of obligors? We have two independent Poisson default processes (each corresponding to one obligor), therefore the overall mean should be the sum of individual means.
See Crouhy page 404. It's mentioned toward the end of the page,
n_bar=sigma P_A. I think we have an analogous situation in this example.

[rekrap]

In problem 1 on page 495 the expected number of defaults for band 5 are to be determined. Now band 5 has two obligors. Therefore, shouldn't the expected number of defaults (or default rate) be equal to the expected default probability for the band (as obtained in the example for expected number of defaults)*# of obligors? We have two independent Poisson default processes (each corresponding to one obligor), therefore the overall mean should be the sum of individual means.
See Crouhy page 404. It's mentioned toward the end of the page,
n_bar=sigma P_A. I think we have an analogous situation in this example.

You would be double counting... The expected loss already includes the sum of the two instruments losses. Divide E[L] by the band exposure (L=5) to get the expected number of defaults in that band, the average loss for that band.

[bagheera]

ok got it. Thank you!

[Jy88]

Would like to clarify on this; in the manual page 290, 5th line, it mention that:

An under funded plan is better off investing in bonds to maximize the hedge it offers against the liabilities. An over funded plan though will want to invest in equities.

I reckon this might be interchanged. I thought that under funded plan you should invest in equities so you get more return to fund the plan and vice versa.

Anyone had similar thoughts on this?

[rsgoldfarb]

This will be clearer if you more carefully read the entire section of the pages you referred to. There are two different perspectives that are discussed - one is on the surplus risk and the other is on the change in the surplus.

To minimize the risk, you need bonds to "hedge" and so only if you have extra assets (i.e. over-funded) should you invest in equities. The equity allocation rises as the funding ration A/L rises.

To focus on the change in the surplus, then indeed investing in equities to achieve the larger expected return may be needed if you are underfunded. But this introduces risk (you aren't as hedged as above) and so there is a trade-off that needs to be considered.

The numerical problems should clarify this.

[Jy88]

Thanks Mr.Goldfarb for clarifying that.

So to say that the funding ratio increases, the allocation to equities increases, which means increase in surplus risk. Thereby, for an under funded plan , to hedge, we should invest more in bonds as if we invest in equities, the expected change in surplus would increase but then that would incur more risk. So to have a good hedge, we should invest in bonds.

Whereas, for an overfunded plan we should invest more in equities as the funding ratio increases. This, however, imply risk which can be offset by the diversification benefits between equities and bonds?

Is my understanding on the right track? Thanks again. Really appreciate this.

[rsgoldfarb]

I don't think you have it quite right, but unfortunately I won't be able to respond in detail here. I suggest you refer back to the notes and read those carefully. There are two different objectives that can be set - minimize the risk of the surplus, in which case hedging the liability risk is critical, or maximize the change in the surplus, in which case investing in equities is critical. These two objectives lead to different decisions about how much to invest in equities and so the "choice" of what to do depends on personal preferences for the risk-return tradeoff.

[Jy88]

Thanks a lot for that :-p

[Jy88]

What actually is negative convexity?

Is it increase in rates gives u an increase in duration?
Is it increase in rates gives you decrease in duration?

In the Goldfarb manual; pg 236, under Z-Pac it mention that an incrse in duration as rates rise is known as negative convexity. Later on on page 308, under the futures hedge, it says that negative convesity of futures contracts is becuase the delivery option causes duration to fall as rates rise and vice versa. So which one is the right one?

Also, what is the formulae for convexity; is it
= 1/(P*(1+y)^2) * Sum(T=1)*T*PV(CF). on page 273 of the goldfarb manual, it doesn't seem to have the (1+y)^2, however, on the page 149, under convexity, the formula correspond to the one I state. Anyone can confirm on this?

Thanks a lot :-p

[Jy88]

The question is asking us to calculate a 10-year swap that pays 6% fixed and receives floating can be purchased with any notional amount. The yield curve is falt at 6.5%. Calculate the DVBP.

DVBP = (par amount * (Price + AI) * Modified Duration)/notional amount.

In the answer provided to this question. The DVBP of the fixed rate leg of swap is 694.8 for $1mil notional. However, the answer I got is 703.25 (assumed if payable once per year) or 711.44 (assuming semi-annual)

Anyone can get that answer 694.8?

Thanks a lot.

[Jy88]

For fixed annuities, it contain risk that crediting rate will have to rise to prevent policyholder surrenders. And this can be managed using interest rate swaps.

Therefore, in the manual (pg 321) it says that we can enter into swapoption. So when the interest rate fall, the firm has the option to lower crediting rate and if interst rate rise, it can sell this swap to third party and use the proceeds to manage the risk.


My question is:

For interest rate swap, the insurer would be the buyer then he would be paying fixed and receive floating. So if this is entered into swapoption, the insurer would have the option to exercise this at the delivery date if the interest rate increase, then he will get the floating rate and use that to pay for the higher crediting rates to prevent surrenders. And if interst rate decrease, he will just not exercise the option. This is my understanding. Is that correct?

But how come, he needs to sell the swapotion? And how would that help to decrease crediting rates when the interest rate decrease? And when he sell the swapoption, then he will be paying floating and receive fixed. Isn't that worst in a rising interest rate environment?

Thanks.