Estimating "present value net worth"

[jdw_fire]

On the way home that night you stop by the military recruiter's office and learn that their pensions are adjusted every year for the CPI. If the CPI is not manipulated then you'll be protected against a lifetime of inflation! Your pension dollars have the same buying power every year. If you convert that to a lump sum, the dollars received in the first year of retirement have the same value as the dollars received in the second year… and the third year… and the fourth year… and so on until you die. (Thanks to Rodmail for pointing this out.) If you asked for a lump sum, you'd just multiply your life expectancy by the size of the pension. No one would be arguing about the rate of inflation. No discount rates required.

this is leaving out the fact that there is still a time value of money even if there is no inflation (due to the fact that there is generally a real rate of return available to capitol owners) and the discount rate would be akin to the real rate of return you get with TIPS and alteast that large. so the (net present value of that COLAed pension) is less than (amount of 1st pmt) times (total number of pmts)

 

[Maurice]

Nords - the inflation rate is completely irrelevant when calculating the present value of a known future cash flow. Its called the present value after all.

Putting aside the fact that, in your example, you don't know what the fixed payment will be since its based on earnings some years hence - that's obviously a separate issue than inflation.

Imagine that you had a fixed annuity of 1000 per year for the rest of your life. You would discount each of those payments back to today using discount factors derived from the highest risk free interest rate available to you. No where does the market implied inflation rate come into play (except indirectly, insofar as it affects interest rates).

(to be perfectly accurate, as running man pointed out, you would need to consider each of the cash flows as contingent on your survival, hence you'd multiply each of those payments by your expected survival probability - but let not muddy the waters with this here)


I think you are conflating this with the idea of trying to predict its *future* value in 2009 dollars. To do that you would indeed need to take inflation rates into account.

 

[Maurice]

On the way home that night you stop by the military recruiter's office and learn that their pensions are adjusted every year for the CPI. If the CPI is not manipulated then you'll be protected against a lifetime of inflation! Your pension dollars have the same buying power every year. If you convert that to a lump sum, the dollars received in the first year of retirement have the same value as the dollars received in the second year… and the third year… and the fourth year… and so on until you die. (Thanks to Rodmail for pointing this out.) If you asked for a lump sum, you'd just multiply your life expectancy by the size of the pension. No one would be arguing about the rate of inflation. No discount rates required.

I'm repeating myself here, but to say that the PV of the pension would be (paraphrasing you) the annuity * number of years you receive it is only true if real interest rates were identically zero. You are assuming that the forward inflation rate exactly cancels out the discount factor. In the real world that isn't the case.

 

[Independent]

Nords, I think some of your post #16 says that the value of an annuity (whether it's a pension, SS, or private SPIA) to the retiree could be quite a bit different from the cost to the supplier. The retiree may be thinking about how the annuity performs in the "worst case", while the supplier can think about what it costs on average. This is true - it's why we buy insurance even though we know the insurance company makes profits (and also why we buy lots of other things).

However, in looking for a discount rate for the "present value", I'd use this explanation of PV "The present value of a future payment is the amount you'd need to invest today so that your investment would grow to the payment amount on the scheduled payment date". That definition gets me to the rates that jdw fire and Maurice are using.

 

[Gone4Good]

SOOO, what do I use to discount my future cash flow.... and with the interest rate that went up the last few weeks, did I really 'lose' $50,000 to $100,000?

Nope...

What if those cash flow were from a bond you intended to hold to maturity? Did you lose value when interest rates went up? It's the same principal with a pension. The only difference between the two is that the bond is a liquid asset and the pension is not. An inability to sell an asset doesn't make it more valuable, it just complicates price discovery from revealing that value.

 

[ERD50]

An inability to sell an asset doesn't make it more valuable, it just complicates price discovery from revealing that value.

Astute observation!

Funny how we will price something if we can (and fret over the price), but if we can't we don't (and don't give it another thought)!

I know someone who sold some property, and then was all anxious - "what do I do with all this money now! Where do I invest it, what do I do!".

I told them they had that money last week too, but it was just in the form of the property. Nothing changed, other than you can diversify it now. You should have been more anxious with all your eggs in one basket.


-ERD50

 

[Delawaredave5]

Anybody have a spreadsheet to calculate PV of non-COLA pension ?

I thought Bogle suggested using 8x the first year payout as the PV of pension and to consider it a bond.

The PV of cash flow should also be discounted greater for risk -- I know a guy who was pilot for major airline - had big DB pension - then airline went bankrupt - pension was backstopped by PGC - but at 30 cents on the dollar roughly.

 

[NW-Bound]

I know a guy who was pilot for major airline - had big DB pension - then airline went bankrupt - pension was backstopped by PGC - but at 30 cents on the dollar roughly.

It may be a good example of futility of hair-splitting analysis; this guy is pummeled by a personal "black swan". Do you know how he is coping?

 

[Maurice]

Anybody have a spreadsheet to calculate PV of non-COLA pension ?

I thought Bogle suggested using 8x the first year payout as the PV of pension and to consider it a bond.

The PV of cash flow should also be discounted greater for risk -- I know a guy who was pilot for major airline - had big DB pension - then airline went bankrupt - pension was backstopped by PGC - but at 30 cents on the dollar roughly.

Yeah, you could make this pretty complicated if you assume counterparty credit risk and contingent payments from the PBGC. You'd have to calculate the implied default curve for the annuity issuer from their CDS spreads. Then you'd extract a survival probability for the issuer at each cash flow date. You would then compute the expected cash flow at each scheduled payment date as: Scheduled Payment Amount * Prob. of Survival of issuer at Scheduled Payment Date + Guaranteed Pension Amount * [1 - survival probablility of issuer at scheduled payment date].

You'd then have to multiply that entire sum by your own survival probility at that secheduled payment date, since the cash flows cease upon your own death.

You'd do that for all payment dates, then discount each of those expected cash flows back to today using an appropriate discount curve.



Re spreadhseets, I have stuff like this at work - not designed for pension payments but more general sets of contingent cash flows... But unfortunately they use proprietary excel functions so I can't share them externally.

We also use a method of constructing a yield curve that is appropriate for a AA rated bank, but not so much for a retail investor. (we use libor on the short end, futures in the middle, and swap rates on the long end of the curve - and we use the same discount curve for outgoing payments as well as incoming)


Re Bogle's suggestion - its probably reasonable given a set of assumptions about interest rates and your own longevity, but it would be interesting to know what assumptions he made.

 

[Delawaredave5]

It may be a good example of futility of hair-splitting analysis; this guy is pummeled by a personal "black swan". Do you know how he is coping?

He's doing fine - because he's still working - but will have to work longer than he wanted.

I'm not sure it is a "black swan" event. I think we've seen waves of pension failures - first steel industry, then airline industry. Auto industry is next.

I'll soon have 30 years of DB, non COLA pension eligibility. That's the good news. The bad news I'm not convinced my company will be around as long as I need the pension.

So there's a risk there - hard to say what the appropriate discount rate to use when quantifying it as a bond. I've been using 8x the payment - I think that's conservative.

 

[Nords]

Nords - the inflation rate is completely irrelevant when calculating the present value of a known future cash flow. Its called the present value after all.

True, although one factor affecting the size of the discount factor is everyone's assumptions about inflation. I think one issue is that I'm selectively ignoring several discount factors (which would be very important in some situations, not so much in this one) and using proxies in other situations (like CPI instead of risk-free rates).

Putting aside the fact that, in your example, you don't know what the fixed payment will be since its based on earnings some years hence - that's obviously a separate issue than inflation.

Yep, that's the simplifying assumption of ECI = CPI for 13 years.

Imagine that you had a fixed annuity of 1000 per year for the rest of your life. You would discount each of those payments back to today using discount factors derived from the highest risk free interest rate available to you. No where does the market implied inflation rate come into play (except indirectly, insofar as it affects interest rates).

However, in looking for a discount rate for the "present value", I'd use this explanation of PV "The present value of a future payment is the amount you'd need to invest today so that your investment would grow to the payment amount on the scheduled payment date". That definition gets me to the rates that jdw fire and Maurice are using.

Deriving that risk-free rate isn't so simple, is it? I'd be happy to use the interest rate for I bonds, except that the current fixed interest rate is zero. I'd try it with TIPS, but their auctions don't always offer recent data since they don't match expected longevity. Another factor is that an annuity without survivor options stops when the annuitant dies. When the owner of a portfolio of bonds dies, the heirs are still left with a portfolio of bonds. That has to affect the interest rate, although it's probably only relevant to the owner when he's setting things up. But leaving an inheritance has some definite value that would affect the discount factor.

In college-expense planning the "invest today" amount used to be accomplished using zero-coupon bonds. I don't even know if Treasuries offer that option, although I'd be astounded if some investment bank hadn't already created them.

Although I agree that there are a number of not insignificant issues to be considered, I'm hoping that the assumptions have adequately simplified the asset allocation results without making subsequent decisions too dangerous. From a "black swan" perspective it probably doesn't matter if I've chosen to count the equity portfolio as 10% or 20% of the total-- the swan would wipe it out in either case and could only be avoided through some sort of insurance. But assumptions that see no difference in going from 10% to 50% is probably an indication that I'd need better assumptions.

Who wants to take a lot of risk with their assets when they are 95 years old?

All of us (including me) would nod our heads. But if you don't "need" the money then it doesn't matter whether you're 45 or 95. That "need" attitude could be tested by donating it to a charity, and at that point most of us (including me) would decide that the money could be a vital component of a long-term-care contingency. So we're not ready to give up control of it yet. Spouse and I felt the same way about our portfolio losses-- the money wasn't "needed" but it could've done a lot of good. Not very effective stewardship on our part.

One of the reasons I spend so much time & effort on high-risk investing is self-defense against grass-is-greener syndrome. When I'm 75 years old and someone proposes a "great investing idea", I want to be able to say "Yeah, I tried that in my 40s and here's five reasons this could end badly".

 

[Gone4Good]

Anybody have a spreadsheet to calculate PV of non-COLA pension ?

Excel Function . . . NPV(RATE,Value1,Value2, . . . )

 

[Gone4Good]

True, although one factor affecting the size of the discount factor is everyone's assumptions about inflation.

But the treasury yield curve already has an inflation assumption embedded in it. For non-cola pensions you'd discount the nominal cash payment against the nominal treasury yield for the year in question. A cash payment 10 years in the future would be discounted against the 10 year treasury yield and a 30 year payment against the 30 year bond. For longer dated cash payments, the 30yr yield is still a useful discount rate because bond term premiums are pretty flat at the long end of the curve. 40 year bonds don't yield much more than 30 yrs. Neither do 100 year bonds.

Deriving that risk-free rate isn't so simple, is it? I'd try it with TIPS, but their auctions don't always offer recent data since they don't match expected longevity.

Treasury yields are always used as the "risk free rate". The only time you'd use TIPS yields is when you're PVing an inflation adjusted cash flow stream. In that case you'd use the "real yield" as a discount instead of nominal treasuries. But when doing this, don't adjust the cash flow for some inflation assumption. If you're going to get $100 in inflation adjusted dollars in some future period, discount $100 by the TIPs yield.


I'm probably not following your point, by why wouldn't you just use the real TIPS yields based on secondary market trades?

Another factor is that an annuity without survivor options stops when the annuitant dies. When the owner of a portfolio of bonds dies, the heirs are still left with a portfolio of bonds. That has to affect the interest rate

I don't think it would. You'd use the same discount rate to PV a cash flow stream with no bullet maturity as you would a normal bond with a bullet payment. Your pension with no survivor benefits is the same as a cash flow stream with no bullet maturity.

 

[Maurice]

Deriving that risk-free rate isn't so simple, is it? I'd be happy to use the interest rate for I bonds, except that the current fixed interest rate is zero. I'd try it with TIPS, but their auctions don't always offer recent data since they don't match expected longevity.

Again, you wouldn't use an inflation-linked bond for a non COLAd annuity.

Like independent suggested, the way to think of the present value of a future cash flow is by pretending you had to buy that individual cash flow in the market. Whats the cheapest price you can find for that cash flow at the same level of credit risk?

Well, as a retail investor, its probably bank CD rates at the short end of the curve and treasurys at the long end. In other words, the cheapest seller of cash flows 1 or 2 years hence is probably an FDIC insured bank. The cheapest seller of cash flows 7-10 years out is probably the US treasury. For the typical retail investor's purposes, the credit risks are the same.

So the way you value a fixed annuity is by determining how much you'd have to pay to buy the individual cash flows in the marketplace. So you build a curve consisting of the best interest rates available to you (at that equivalent credit risk) and compute the discount factors from there - inflation doesn't come into play at all. (except indirectly of course since inflaiton expectations effect interest rates)

Of course in practice you get an incomplete set of data points across the term structure and have to do some interpolation between them.. but its entirely doable with some staightforward arithmetic.

Another factor is that an annuity without survivor options stops when the annuitant dies. When the owner of a portfolio of bonds dies, the heirs are still left with a portfolio of bonds. That has to affect the interest rate, although it's probably only relevant to the owner when he's setting things up. But leaving an inheritance has some definite value that would affect the discount factor.

Yeah, if the payments are contingent on your survival that adds a second layer of complexity. The way we** handle this is not by adjusting the discount factor per say but by adjusting the future 'expected' payment amount before discounting.

For example, pretend you have a single payment of 1000 due to you in one year, if and only if you are alive at that time. Say that we consult the actuarial tables and determine that you have a 98.5% chance of being alive in one year. Therefore we consider the 'expected' payment amount to be $985. We would then discount the 985 back to today using the standard discounting curve I described above. You would continue that process with each future cash flow, using the appropriate 'survival probability' to compute the 'expected value' of each payment, prior to discounting it back to today.

(That process is identical with the process for incorporating credit risk into the equation, its just tht the 'survival probablity' one uses for a corporate issuer is derived from CDS spreads not from actuarial tables)




** when I say 'we', I should point out that my day job is to build systems that value fixed income derivatives of all kinds - FX derivatives, interest rate products, credit derivatives, even various derivatives based on life insurance products

 

[bsmup]

You know what they say about assumptions....

 

[Rustic23]

Maurice,
Inflation is one of the core parts in building a 'discount rate', Cost of capital, Inflation, Risk. There is a good article on building a discount or cap rate:

The art and science of business ... - Google Book Search

While this is for valuing property, business, or your own NPV, the same principles apply. You are discounting a future stream of income. I concur that tips or I bonds are not the rate to use, but treasuries is a good starting or safe rate and as explained in the article, inflation is accounted for in it's rate.

I still profess that the best way to arrive at the answer Nords was seeking is to do a discounted cash flow for each of his streams of income and add the results together. There may be different discount rates for each stream based on risk, but not on inflation. It matters not if the income stream is cola or not as you will place a different amount in each year based on weather it is or is not cola. Then the stream is discounted back at a safe rate + an allowance for risk. Therefore the military pension stream should have a lower discount rate than his expected stream from investments.

Based on the fact that his cola'd pensions come from the government, and therefore, IMHO, could be discounted at a safe rate, and his investments would be higher, I believe his conclusion that the current down turn in the market does not have as big of an effect on his NPV as one might expect by looking at his statements or listening to the news. If you want to quibble that his SS has a risk factor, I would not put up much of an argument. That is for each individual to decide.

 

[Texas Proud]

For example, pretend you have a single payment of 1000 due to you in one year, if and only if you are alive at that time. Say that we consult the actuarial tables and determine that you have a 98.5% chance of being alive in one year. Therefore we consider the 'expected' payment amount to be $985. We would then discount the 985 back to today using the standard discounting curve I described above. You would continue that process with each future cash flow, using the appropriate 'survival probability' to compute the 'expected value' of each payment, prior to discounting it back to today.

I would agree with you if this was a portfolio... but like Schrodinger's Cat, you are either alive or dead.... you can not be 98.5% alive... (well, I guess that can be debated, but you either are getting the payments or you are not)

So, I think you have to make an assumption (yes, I know )... that you are either dead or alive.... and if you are a bit younger, it really does not make that much difference in the outlying years....

 

[wcv56]

I can see in the future government pensions carrying a time limit of 25 or 30 years
in the future. State and federal gov cannot possibly afford what will happen in the
next decade for retirement. Social Security will most likely be moved back to at least
65 for early withdrawal and 68 or something of the sort for full benefit.
JMHO

 

[Maurice]

Rustic - your book says to take into account 'inflation + real interest rates'. Sounds like nominal interest rates to me.

 

[Maurice]

I would agree with you if this was a portfolio... but like Schrodinger's Cat, you are either alive or dead.... you can not be 98.5% alive... (well, I guess that can be debated, but you either are getting the payments or you are not)

So, I think you have to make an assumption (yes, I know )... that you are either dead or alive.... and if you are a bit younger, it really does not make that much difference in the outlying years....

That's true, but then a high yield bond is either going to default or it isn't. But before the fact, one can't know which will happen, so the market prices it based on probabilities. Calculating PVs is all about calculating market prices, is it not?

Having said that, of course for ER planning purposes one should assume the 'worse case' scanario that one lives a long time...

 

[Maurice]

I can see in the future government pensions carrying a time limit of 25 or 30 years
in the future. State and federal gov cannot possibly afford what will happen in the
next decade for retirement. Social Security will most likely be moved back to at least
65 for early withdrawal and 68 or something of the sort for full benefit.
JMHO

I think before they put a time limit on them they'll be means tested. That's one reason SS doesn't show up in any of my spreadhseets.

 

[Gone4Good]

I would agree with you if this was a portfolio... but like Schrodinger's Cat, you are either alive or dead.... you can not be 98.5% alive... (well, I guess that can be debated, but you either are getting the payments or you are not)

Maurice is right.

The calculation is no different than asking "What is the fair value of a 50% chance to win one dollar?" The correct answer is $0.50. If you were then to ask, "What is the fair value of a 50% chance to win one dollar one year forward" the correct answer would be .5 / (1 + i) where "i" equals your discount rate.

While you're right that you'll either end up with one dollar or zero, not $.50, the market would value that cash flow stream as described. Any other valuation would result in arbitrage profit opportunities (in a robust and efficient market).

 

[Rustic23]

Discounted Cash Flow

attached is a simple 25 year discounted cash flow spreadsheet. I think I used an assumption of 3.5% constant inflation for the numbers, and a 7% constant return on the investment numbers. Not real sophisticated but it should give you an idea how to build one to suit your needs.

 

[Maurice]

Again, you don't use inflation to discount a known cash flow. Use an appropriate risk-free interest rate.

 

[Maurice]

Here's Wiki's take, if anyone's interested.

Discrete cash flows

The discounted cash flow formula is derived from the future value formula for calculating the time value of money and compounding returns.

Thus the discounted present value (for one cash flow in one future period) is expressed as:

where

• DPV is the discounted present value of the future cash flow (FV), or FV adjusted for the delay in receipt;
• FV is the nominal value of a cash flow amount in a future period;
• i is the interest rate, which reflects the cost of tying up capital and may also allow for the risk that the payment may not be received in full;
• d is the discount rate, which is i/(1+i), ie the interest rate expressed as a deduction at the beginning of the year instead of an addition at the end of the year;
• n is the time in years before the future cash flow occurs.

Where multiple cash flows in multiple time periods are discounted, it is necessary to sum them as follows:

for each future cash flow (FV) at any time period (t) in years from the present time, summed over all time periods. The sum can then be used as a net present value figure. If the amount to be paid at time 0 (now) for all the future cash flows is known, then that amount can be substituted for DPV and the equation can be solved for i, that is the internal rate of return.
All the above assumes that the interest rate remains constant throughout the whole period.
(1+i)^(-t) can of course also be expressed as exp(-it).