[kenrexford]

For purposes of this question, ignore any realities that might cause hedging.

Suppose you own a small business that generates on average $100K a year. Suppose, also, that you were to plan to sell the business in, say, exactly 5 years. What would be the theoretical value in today's dollars of this asset?

In other words, to perhaps make this simpler, if you were to sell the asset today, with an agreement that you would not transfer the asset until five years later, but with payment for that asset made today, what price would be fair theoretically for that future asset? And, what is the math going into the calculation (to be able to translate this into different annual averages or different point of sale)?

[FrancesHinden]

You need to define the riskiness of the $100K a year. For example, is it as certain as the coupons from US treasury stock? Or the interest on Greek debt? Or the interest on Argentinian debt?
And, is the assumption that it continues to generate $100K a year for ever? Increasing with inflation or constant?

This is theoretically a very simple question, but the main question you need to answer is the first. That will define the so-called 'discount rate'

[akwoo]

I need a presumptive interest rate to make the calculation.

Also, to make this clear, the business generates $100K a year in perpetuity? So 100,000 years from now, it will still be generating $100K (although $100K will be worthless by then due to inflation)?

[kenrexford]

I knew that I would get insane accounting caveats. Assume a return of 100K exactly for 5 years and then permanently forever at an expected increase each of years 6 and forward that's tied exactly to the inflation rate. The presumptive interest rate is whatever the market rate is on average for the past 100 years.

[hrothgar]

kenrexford, on 2014-August-25, 15:00, said:

For purposes of this question, ignore any realities that might cause hedging.

Suppose you own a small business that generates on average $100K a year. Suppose, also, that you were to plan to sell the business in, say, exactly 5 years. What would be the theoretical value in today's dollars of this asset?

In other words, to perhaps make this simpler, if you were to sell the asset today, with an agreement that you would not transfer the asset until five years later, but with payment for that asset made today, what price would be fair theoretically for that future asset? And, what is the math going into the calculation (to be able to translate this into different annual averages or different point of sale)?

What's your discount rate?

[hrothgar]

Hi Ken,

Rather than spending a bunch of time going back and forth regarding this calculation, I recommend that you familiarize yourself with formula's for the present value of a perpituity. (Alternatively, a perpetual annuity)

[steve2005]

the value in 5 years V(5) = 100,000/(interest rate-inflation)

then you would have to calculate the present value of V(5)

which would be V(5)/(1+(interest rate-inflation))^5

http://www.financefo...Perpetuity.html

[SteveMoe]

Ken, Excel spreadsheets can calculate this in a jiffy for you.
Simply set a cell with the NPV function (Finance) and input the discount rate (matching the time frame of the cash flows - annual flows get an annual rate, monthly flows get a monthly rate, etc).
The key is to have the discount rate reflect the risk of the asset. If the asset is completely risk free then the discount rate is the inflation rate.
Doing this gives the following results:
NPV..........Discount Rate
$457,970.72.....0.03
$445,182.23.....0.04
$432,947.67.....0.05
$421,236.38.....0.06
$410,019.74.....0.07
$399,271.00.....0.08
$388,965.13.....0.09
$299,061.21.....0.20

As the discount rate climbs the NPV drops.
For zero risk and zero inflation, the NPV would be $500,000.

[akwoo]

Ken,

Let me explain the accounting caveats. Then you can try to figure out the reasonable question.

The value of the business is exactly equal to the amount you would have to buy in US Treasury bonds (or your other preferred index asset), right now, to get exactly the same income over time.

Let's suppose, for a moment, that Treasury bonds pay exactly inflation rate in interest, and the profit from the business also increases by inflation starting in year 5. Let's suppose inflation over the next 5 years totals 11%.

In that case, the value of the business is infinite. You would need $90K to cover year 5 (since $90K * 111% is $100K), $90K to cover year 6 ($90K * 1.11 * (1+inflation in year 5) to cover $100K * (1+inflation in year 5)), $90K to cover year 7, and so on ad infinitum.

Fortunately for this calculation, Treasury bonds usually pay higher than inflation rate. (That's actually not true right not - TIPS are paying negative interest at the moment, reflecting a belief by investors that inflation will be going up soon, and also that lots of rich people have money but can't find any good investments because the economy isn't so good.) Let's suppose that Treasuries pay inflation + 1%.

In that case, you would need $90K to cover year 5, plus $90K/1.01 to cover year 6, plus $90K/(1.01)^2 to cover year 7, and so on.

This sum is indeed finite, because the amount to cover year 1005 is $90K/(1.01)^1000 which is less than 50 cents, and the amount keeps going down so quickly that the rest is pretty much negligible. The formula (from calculus) is that you would need $90K/(1-1.01) or $9000K to cover everything.

Well, actually, that previous calculation is more complicated than it needs to be for this simple situation. Think of it this way. How much money would you have to put in the bank, assuming it's paying 1% interest, to get $100K every year? Since you want to get $100K every year forever, you'd better never be dipping into the principal, so you need to be getting $100K a year in interest. So to get $100K every year at 1% interest, you need to put $10000K into the bank. You can get away with a little less since you will earn interest without taking out money for 5 years - that's where the difference between $9000K and $10000K comes in.

[kenrexford]

Thanks all. This helps.