Option Pricing and Actuarial Appraisal Methodologies

[Freebird]

I am trying to understand the "present value of distributable earnings" (PVDE) piece of embedded value reporting. I was looking at an article in Volume 4, Number 1 of the NAAJ, called "Market Value of Insurance Liabilities" (http://www.soa.org/library/journals/...naaj0001_3.pdf) and I don't quite understand one of the statements. I was hoping for some input from forum members.

First of all, here is a statement from page 35 in the article:
"It should be noted that MVA also includes the value of future assets purchased with product cash flow, including premium income on in-force policies, and reinvestment of cash flow from existing in-force assets. If the scenarios used in the appraisal process are arbitrage-free, then the market value of future investment and reinvestment is zero."

I realize this is out of context, but something about the risk neutral valuation bothers me. I think of PVDE in this way, in terms of MVA and MVL:
PVDE = MVA + PV of reinvestments and new premium - MVL

The article above seems to imply that if you are calculating PVDE on a risk neutral basis, then you can calculate PVDE as simply MVA - MVL, and ignore reinvestments and new cash flows, because MV of future investment and reinvestment is zero. I don't see how this can be. Let's say we are running a 40 year projection. Also, let's say we are backing the liabilities with a large 10 year bond just purchased. It seems as if we are netting 40 years of cash flows on the liability side with only 10 years of cash flows on the asset side, assuming a risk neutral valuation, since MV of future investments don't matter in a risk neutal world. Where would distributable earnings come from after year 10, since you can assume no reinvestments on a risk neutral basis for calculating PVDE?

Following this reasoning, you could theoretically calculate PVDE on a risk neutral basis without an asset liability model, using a liability-only projection. You already have the current MVA from your investment department on a risk neutral basis, and the article says you can ignore reinvestment cash flows, so then PVDE = MVA - MVL, and you don't have to model asset cash flows at all.

Comments anyone?

[Freebird]

Hmmm... over 100 views and no comments. I either did not state my question very clearly, which is most likely the case, or it's a tougher question than I thought. I guess it's time to go pay the consultant!

[Eimon Gnome]

The problem is in your restatement of PVDE.

"PVDE = MVA + PV of reinvestments and new premium - MVL"

The market value of the assets includes the pv of all the asset cash flows. There is no reinvestment beyond that. Once you get the cash, you simply discount it back to the present. That is its value today.

Notice that the receipt of future premiums is also an asset. Do the same thing. Discount them back to today. That's the market price today.

I think you're making the problem harder than it is. Don't worry about what happens to the cashflow after it is received. It doesn't matter. You are only after the value of THAT cash flow today.

And you're close on the observation of "only modeling the liabilities". Just add in the future premiums, and you've got the whole shebang (since presumably you have actual, real market values on the invested assets at time zero)

Helpful?

[JMO]

You folks might also be interested in the latest Financial Reporter newsletter. See the article on page 8, "Arbitrage-Free Perspective On Economic Capital Calibration."

[A Student]

I agree with a lot of what is written in the OP. I'm not sure why the premium income on the policies is considered an asset. I would include that as a negative liability and model it with the liabilities since it is linked.

I agree that you can value the liabilities separately, but if the value of the liabilities varies based on asset performance (say a DA crediting rate based on a portfolio rate, or a VA cost based on asset base), then you may need to include assets and figure out a way to model risk neutral values for those items.

[Freebird]

Originally Posted by Eimon Gnome

The problem is in your restatement of PVDE.

"PVDE = MVA + PV of reinvestments and new premium - MVL"

The market value of the assets includes the pv of all the asset cash flows. There is no reinvestment beyond that. Once you get the cash, you simply discount it back to the present. That is its value today.

Notice that the receipt of future premiums is also an asset. Do the same thing. Discount them back to today. That's the market price today.

I think you're making the problem harder than it is. Don't worry about what happens to the cashflow after it is received. It doesn't matter. You are only after the value of THAT cash flow today.

And you're close on the observation of "only modeling the liabilities". Just add in the future premiums, and you've got the whole shebang (since presumably you have actual, real market values on the invested assets at time zero)

Helpful?

Yes, very helpful, thank you. Great explanation!

But another of my issues revolves around Girard's statement "... If the scenarios used in the appraisal process are arbitrage-free, then the market value of future investment and reinvestment is zero." So, anything other than arbitrage-free scenarios results in a non-zero market value. I just don't see it.

I have no doubt that Girard's statement is true, but I'd like to see a demonstration. I guess I'm one of those old school actuaries who would still like to "substitute demonstrations for impressions".

[MathGeek92]

[quote=Freebird;5882196]

But another of my issues revolves around Girard's statement "... If the scenarios used in the appraisal process are arbitrage-free, then the market value of future investment and reinvestment is zero." So, anything other than arbitrage-free scenarios results in a non-zero market value. I just don't see it.

QUOTE]

I would view as a complicated way of saying there exists a portfolio today which replicates your liability cashflows. There is no future investment/reinvestment in a "perfect" price.

[Freebird]

I really appreciate all the responses on my post, but I still am not quite there yet. Since I have a tendency to beat dead horses, I am going to keep going. First of all, here's Girard's statement once again which, as I said, I know is true, and I'll forgo the demonstration:

"It should be noted that MVA also includes the value of future assets purchased with product cash flow, including premium income on in-force policies, and reinvestment of cash flow from existing in-force assets. If the scenarios used in the appraisal process are arbitrage-free, then the market value of future investment and reinvestment is zero."

I think one of my problems is that my background is U.S. cash flow testing (CFT) using real-world scenarios in asset liability models (TAS). Thus, I have always concentrated on contributions to surplus, and PV of ending market surplus (PVEMS). Now I am in the Embedded Value reporting unit and trying to relate MCEV to PVEMS. I think the article in the Financial Reporter, "Arbitrage-Free Perspective On Economic Capital Calibration" (thanks, JMO!) helped a lot to point me in the right direction. PVEMS from CFT, using after tax earned yield as the discount rate, is the same as real-world PVDE, so that is a bridge between real-world PVDE and risk-neutral MCEV. In other words, MCEV in the risk-neutral world world, PVDE if you will, is a lot like PVEMS in the real world.

With that said, let me get back to Girard's statement. In MCEV valuation, my understanding is that you can use 40 years of forward rates based on the swap curve at the valuation rate, instead of the average results from 1,000 risk-neutral scenarios, to value a product. But you have to prove that the product is not interest sensitive. It would be a lot like saying that you can price an at-the-money option with only one scenario if you can prove that it always stays at the money no matter what the scenarios are doing. Obviously, this is a ridiculous analogy, but I wanted to get a frame of reference to understand MCEV valuation using the so-called "certainty equivalent" scenario. I don't think there are a lot of life insurance products without any optionality in them where you can use the CE scenario (maybe term or a product with fixed cash flows). So it seems to be of limited use, especially for a full portfolio of life insurance.

Since the certainty equivalent is a single deterministic scenario, not an arbitrage-free set of scenarios, applying Girard's statement means that you can't ignore reinvestments. In other words, under the certainty equivalent scenario, the market value of future investment and reinvestment is not zero. Thus, it seems that you would have to have an asset liability model to even get PVDE from the certainty equivalent scenario. But my understanding is that European MCEV rules allow the certainty equivalent scenario in a lot of circumstances, and also they allow companies to run liability-only models to calculate it.

In my experience with CFT, I would never think of running a liability only model, and not model the asset side also (meaning use the MVA of existing assets at the model start date). Otherwise, I would be missing the reinvestments in my final result. European rules seem to allow this shortcut (liability-only model) for the certainty equivalent scenario in MCEV valuation, which seems to be in contradiction of Girard's statement.

OK, if you have come this far, you are a brave soul! Are there any responders from companies using liability-only models for the certainty equivalent scenario who might be able to put me on the right path?

Actually, any comments at all?

[Eimon Gnome]

Maybe this will help.

You seem to imply that the ability to (re)invest has value. So let's look at that statement.

The cash received n years from now has value. It's value is the NPV @ the risk free rate.

How about the ability to reinvest that money when it comes in? You certainly have that opportunity, and it will produce investment income after you invest it. But you are investing at the then market rate. Is that opportunity worth anything? I'd say no.

You do not need anyone's permission to be able to buy a security in the future at the then market price. That is not an option. It has no value. You don't have to pay anyone today for that privilege. Since the opportunity/option is worthless, the "ability to reinvest then" is really worthless.

btw, I will gladly sell you an option to buy bonds or stocks at the then market price for a nominal amount if you think this is not the case. I will send you an elegant piece of paper stating that Freebird has my full authorization to buy stuff if he has the cash.

[Freebird]

Originally Posted by Eimon Gnome

Maybe this will help.

You seem to imply that the ability to (re)invest has value. So let's look at that statement.

The cash received n years from now has value. It's value is the NPV @ the risk free rate.

How about the ability to reinvest that money when it comes in? You certainly have that opportunity, and it will produce investment income after you invest it. But you are investing at the then market rate. Is that opportunity worth anything? I'd say no.

You do not need anyone's permission to be able to buy a security in the future at the then market price. That is not an option. It has no value. You don't have to pay anyone today for that privilege. Since the opportunity/option is worthless, the "ability to reinvest then" is really worthless.

btw, I will gladly sell you an option to buy bonds or stocks at the then market price for a nominal amount if you think this is not the case. I will send you an elegant piece of paper stating that Freebird has my full authorization to buy stuff if he has the cash.

Yes, I see your point, but I will pass on your generous contract offer!

What you are saying make sense in that crazy risk-neutral world where assets are priced. However, what about when we are talking about the CE scenario used by some companies to calculate EV? CE is a deterministic scenario, so by definition it can't be arbitrage-free, which requires a set of stochastically generated scenarios. If it is not arbitrage-free, then it is a real-world scenario (risk-free of course, since EV requires swap rates). Girard goes on to say the following: "On the other hand, if the scenarios used are not arbitrage-free, then the value of future investment may not be zero. Often it is the practice not to require arbitrage-free scenarios, and if this is the case, future investment will have a nonzero valuation. In this situation, the use of the term market value, or even fair value, is inappropriate."

Suppose a company is running an EV for a 30-year term product, which would not be interest-sensitive, and so the company chooses to use the single CE scenario to calculate PVDE. Girard's statement says you cannot ignore future investment, which in this case would be existing assets maturing or sold, and new premium coming in. Thus, PVDE would not be simply be MVA - MVL, where MVA is the current market value of the company's existing assets on the valuation date. The reinvestment portion of MVA would not be zero in this case, so it seems to me that the company would have to run a complete asset liability model to get PVDE, and they cannot rely on the MVA - MVL shortcut.

Comments anyone?