# Present Value Documentation

## Introduction

A very common idea in finance is that of a Present Value, which is used to calculate the present-day equivalent of a future amount of money. This is necessary because the future nominal amount is not equivalent to the same nominal amount of money today.

For example, $1000 today is not equivalent to $1000 in 10 years for two main reasons: (1) We can invest the $1000 we have today so even with a modest annual return, the amount would be considerably higher after 10 years. (2) Inflation is going to erode the purchasing power of the $1000 so it cannot buy the same amount of goods and services 10 years into the future.

If the investment return and inflation were to cancel each other out perfectly, then there would not be a real difference between the current and future amounts of money, but this is rarely the case so we need the concept of present value to adjust future amounts into present-day equivalents.

## Models

When comparing the value of an investment to its purchase price, we personally prefer using the IRR models, because their output is the actual rate of return on the investment, so it avoids having to choose a discount rate.

Fortunately it is very easy to copy-paste your input data from one simulation model to another, so you can easily try your input data with all the models, to find the one you prefer to work with.

## Discount Rate

To calculate the present value we first need to define the so-called Discount Rate, which is used to convert future amounts into their present-day equivalents. You may think of the discount rate as the annualized rate of return that you would expect to make on alternative investments with similar risk-profiles.

Apart from zero-risk government bonds, all future investment returns are uncertain, because bonds may default and the stock-markets will fluctuate. Instead of using the average expected returns for the discount rate, as is commonly done in financial analysis, the simulation models allow you to use probability distributions, so the present value calculations can be done with thousands of different discount rates that are chosen randomly from the probability distribution you provide.

We provide historical data so you can create your own probability distribution for the discount rate:

- Inflation Rate as measured with a Consumer Price Index (CPI) can be used as a discount rate, to estimate the inflation-adjusted (aka. real) value of future amounts.
- Bond Yields for low-risk government bonds can be used as a discount rate, to estimate the present value of future amounts that are invested in such bonds. This may be appropriate when the future amounts have very low risk of being different than expected.
- Stock Index returns can be used if the future amounts have about the same risk as stocks in general. We provide different stock indexes e.g. for large-cap, mid-cap and small-cap stocks. We also provide average annualized returns for different periods such as 1-3 years, 3-5 years, etc. that you can match to your investment period.

The historical bond-yields and stock-index returns are given as both nominal and real values (i.e. adjusted for inflation). It is important to match these with either nominal or real future values, as explained below.

You can also add a Risk Premium to the probability distributions for the discount rate, e.g. if you want to earn 5% more per year than a stock-index, so as to compensate for the extra risk of a particular stock.

You can also combine several probability distributions for the discount rate, e.g. by generating weighted averages using the model DIST-SUM-PCT.

## Present Value (PV)

The following are the basic formulas for calculating the present value. Let \( PV \) denote the present value, let \( d \) be the discount rate, and let \( x_t \) be the future amount in year \( t \). The present value of \( x_t \) is then defined as: \[ PV = \frac{x_t}{(1+d)^t} \] For example, let us say the number of years is \( t=10 \), the discount rate is \( d = 9\% \), and the future amount is \( x_{10} = $1000 \), then the present value is: \[ PV = \frac{$1000}{1.09^{10}} \simeq $422.41 \] One way of interpreting this, is that if you have $422.41 today and you invest it for 10 years in e.g. the S&P 500 which you expect to return 9% per year over those 10 years, then the Future Value (FV) is: \[ FV = $422.41 \cdot 1.09 ^ {10} \simeq $1000 \] We can also calculate the present value of multiple future values by summing them: \[ PV = \sum_{t=1}^N \frac{x_t}{(1+d)^t} \] For example, let us say the future amount is $500 in year \( t=1 \), $600 in year \( t=2 \), and $700 in year \( t=3 \), and we again use a discount rate of \( d=9\% \), then the present value is: \[ PV = \frac{$500}{1.09^{1}} + \frac{$600}{1.09^{2}} + \frac{$700}{1.09^{3}} \simeq $1504.25 \] Whereas if we had just summed the amounts we would have gotten \( $500 + $600 + $700 = $1800 \).

In the simulation models, both the future values \( x_t \) and the discount rates \( d \) are generated randomly from the probability distributions you provide, so that thousands of different values are tried, to give you a better idea of the range of possible outcomes.

## Inflation

Most goods and services tend to become more expensive over time. This is known as inflation and is measured with a Consumer Price Index (CPI). It basically means that $1000 today cannot purchase the same amount of goods and services as $1000 in the future — with few exceptions such as computers that get much cheaper and much better over time.

The term *nominal* means the future value has not been adjusted for
inflation. The term *real* means the future value has been adjusted
for inflation so it is comparable to present-day values.

We can adjust for inflation using present value calculations where the discount rate is the annualized inflation rate. But we have to be careful when using other discount rates such as bond yields or stock-index returns, because we must not mix nominal and real future values and discount rates.

For example, let us say the nominal discount rate is \( d_{nom} = 9\% \),
which is the annualized rate of return you expect to make on a stock-market
index in the future, without adjusting for inflation. If the annualized
inflation rate is going to be \( Infl.Rate = 4\% \), then we might think
that the real discount rate would simply be the difference:
\[
d_{real} = d_{nom} - Infl.Rate = 9\% - 4\% = 5\%
\]
but that is slightly incorrect as it should actually be calculated as follows:
\[
d_{real} = \frac{1 + d_{nom}}{1 + Infl.Rate} - 1
= \frac{1.09}{1.04} - 1 \simeq 4.8\%
\]
Now assume that the real future value is \( FV_{real} = $1000 \). This is
the present-day equivalent of the nominal future value \( FV_{nom} \),
which can be calculated by compounding the annualized inflation rate for
e.g. \( t = 10 \) years:
\[
FV_{nom} = FV_{real} \cdot (1 + Infl.Rate) ^ t
= $1000 \cdot 1.04 ^ {10} \simeq $1480.24
\]
The present value can be calculated using the *nominal* future value
and discount rate as follows:
\[
PV_{nom} = \frac{FV_{nom}}{\left(1 + d_{nom}\right) ^ t}
= \frac{$1480.24}{1.09 ^ {10}} \simeq $625.27
\]
Or the present value can be calculated using the *real* future value
and discount rate as follows:
\[
PV_{real} = \frac{FV_{real}}{\left(1 + d_{real}\right) ^ t}
= \frac{\$ 1000}{1.048 ^ {10}} \simeq $625.73
\]
Note that there is a small difference between these two present values,
but that is entirely due to rounding errors. If we had used the unrounded
numbers for \( d_{real} \) and \( FV_{nom} \) then the two present values
would be identical.

Now consider what happens if we mix the *nominal* future value and the
*real* discount rate, which severely over-estimates the present value:
\[
PV_{mix 1} = \frac{FV_{nom}}{\left(1 + d_{real}\right) ^ t}
= \frac{$1480.24}{1.048 ^ {10}} \simeq $926.23
\]
In the opposite case where we mix the *real* future value and the
*nominal* discount rate, the present value is severely under-estimated:
\[
PV_{mix 2} = \frac{FV_{real}}{\left(1 + d_{nom}\right) ^ t}
= \frac{$1000}{1.09 ^ {10}} \simeq $422.41
\]
So you have to be careful not to mix nominal and real future values and
discount rates. This basically means that if you grow future values
according to the expected inflation, then you should use a nominal discount
rate that also includes the expected inflation. Conversely, if you don't
grow future values according to the expected inflation, then you should
use a real discount rate that also excludes the expected inflation.

## Terminal Value (TV)

When we need to calculate the present value of future amounts that continue for eternity, we can split the calculation into two parts: First we forecast the future values for a limited period such as \( N = 10 \) years, and for the remaining period we calculate the so-called Terminal Value which assumes a constant growth-rate for eternity.

Let \( TV \) denote the terminal value, let \( g \) be the growth-rate that continues forever, and let \( x_N \) be the starting amount that must also be divided by \( (1+d)^N \) so as to calculate its present value, because it occurs \( N \) years into the future. Then the terminal value is calculated using the following formula, which is derived from a so-called Geometric Series in mathematics: \[ \label{eq_tv} TV = \frac{x_N}{(1+d)^N} \cdot \sum_{t=1}^\infty \left( \frac{1+g}{1+d} \right)^t = \frac{x_N}{(1+d)^N} \cdot \frac{1+g}{d-g} \] Note that the formula for the terminal value is not defined when the discount rate \( d \) is smaller than the growth-rate \( g \), because then the future values would continue to grow indefinitely so the Geometric Series would not converge. If this occurs in a simulation then it is marked as invalid and excluded from the final results.

For example, if the eternal growth-rate is \( g = 5\% \) then the discount rate must be greater such as \( d = 9\% \). If the growth-rate is negative such as \( g = -5\% \) then the discount rate must again be greater such as \( d = -3\% \) or \( d = 9\% \).

We can now calculate the present value for all future amounts into eternity, by summing the projection period for the first \( N \) years, and the terminal value for the period that continues forever: \[ PV = \sum_{t=1}^N \frac{x_t}{(1+d)^t} + \frac{x_N}{(1+d)^N} \cdot \frac{1+g}{d-g} \] In the simulation models, you can set the terminal value to zero, either by setting the last value to zero \( x_N = 0 \), or by setting the terminal growth-rate to \( g = -100\% \).

Let us continue the example from above where the future values are \( x_1 = $500 \), \( x_2 = $600 \), \( x_3 = $700 \), the discount rate is \( d=9\% \), and let us say the eternal growth-rate is \( g=2\% \). Then the terminal value is calculated as follows: \[ TV = \frac{x_3}{(1+d)^3} \cdot \frac{1+g}{d-g} = \frac{$700}{(1.09)^3} \cdot \frac{1.02}{0.09-0.02} \simeq $7876.27 \] So the present value for the first three years and the terminal value is: \[ PV \simeq $1504.25 + $7876.27 \simeq $9380.25 \]

## Net Present Value (NPV)

We often want to compare the present value to a current price by calculating their difference, which is called the Net Present Value (NPV): \[ NPV = PV - Price \] We can also consider the ratio between the NPV and the current price: \[ NPV\ Ratio = \frac{NPV}{Price} = \frac{PV}{Price} - 1 \] We do not call this ratio for the Return on Investment (ROI), because that would be confusing. To understand why, consider an investment whose present value equals the purchase price when the discount rate is 20%. So both the NPV and the NPV ratio are zero. But the annualized rate of return on the investment is actually 20%. So it would be confusing if we were to say the ROI is 0%. That is why we just refer to it as the NPV Ratio. A much better measure for the ROI is the Internal Rate of Return (IRR) described below, or the Total Return.

## Internal Rate of Return (IRR)

If you know the future values \( x_t \) and the current price, then we can find the discount rate that makes the present value of all the future values \( x_t \) equal to the current price. This is called the Internal Rate of Return (IRR). That is, we want to find the number \( IRR \) that satisfies the following equation: \[ Price = \sum_{t=1}^N \frac{x_t}{(1+IRR)^t} \] The IRR is used to measure the annualized rate of return that an investor will earn when buying an asset for a given price. For example, if we buy an asset for \( Price = $1200 \) that will pay us \( x_1 = $500 \) in the first year, \( x_2 = $600 \) in the second year, and \( x_3 = $700 \) in the third year, after which it will no longer pay us anything, then the IRR can be found to be around 21.92%, because that is the number satisfying this equation: \[ $1200 \simeq \frac{$500}{1.2192^{1}} + \frac{$600}{1.2192^{2}} + \frac{$700}{1.2192^{3}} \] So we will earn around 21.92% in annualized return on this asset, which can then be compared to the IRR of other investment opportunities to determine which ones are the most attractive.

### Caveats

- There is no general formula for calculating the IRR, so the simulation models use an optimization method to try and find the IRR. This can be rather finicky and may not always be able to find the IRR. Sometimes the IRR simply does not exist for your input. Or your input may be so irregular that it is too hard to find the IRR. If this happens then the simulation trial is marked as invalid and removed from the final results.
- You will get an error message if too many simulation trials were invalid. Unfortunately it is not possible to show the exact reason. So it can be a bit frustrating to work with these models, if your input is too difficult for the simulation models to handle, and you get a lot of error messages.
- For simulation models that use terminal values, the IRR must be greater than the terminal growth-rate, otherwise Eq.\eqref{eq_tv} is not defined. And the IRR often cannot be found if it is too close to the terminal growth-rate. In such cases the IRR is invalid.
- The price must be greater than zero otherwise the IRR is invalid. In some simulation models it is possible to inadvertently use a negative price. For example, in the simulation models for the IRR of stocks it is possible to effectively have negative prices, because the excess cash of the company is assumed to be paid out as dividends immediately, which just subtracts the excess cash (minus dividend-tax) from the market-cap of the company, thus making it possible for the price to become negative, if the excess cash is larger than the company's market-cap.

## Stocks

The present value of a stock that is bought now and sold \( N \) years into the future, is split into three parts:

- Excess cash that could be paid out as dividends now, so it should not be discounted.
- Present value of the earnings for the intermediate years, assumed to be paid out as dividends.
- Present value of the future market-value of all the shares, which is denoted \( MarketCap_N \).

In the simulation models, the user inputs the excess cash and earnings for future years. The market-value of all the company's shares is then calculated from e.g. the simulated earnings multiplied with the simulated P/E (Price-To-Earnings) ratio, as follows: \[ MarketCap_N = P/E_N \cdot Earnings_N \] The P/E ratio is undefined when the earnings are zero or negative, in which case you should use a simulation model that uses the P/S (Price-To-Sales) ratio instead.

We can use different tax-rates for dividends and capital gains. The capital gains tax depends on each investor's purchase price for the shares, so it may be different for all investors and we therefore have to make a reasonable compromise and calculate the capital gains between the current \( MarketCap \) and \( MarketCap_N \) for future year \( N \): \[ Capital\ Gains_N = MarketCap_N - MarketCap \] If the capital gains are negative then there is no tax, which is calculated as follows: \[ TaxCapGains_N = (1 - TaxRateCapGains) \cdot \max(Capital\ Gains_N, 0) \] The future market-cap net of the capital gains tax is then: \[ Net\ MarketCap_N = MarketCap_N - TaxCapGains_N \] We can now calculate the after-tax present value as follows: \[ \displaylines{ PV_N = (1 - TaxRateDividend) \cdot \left( Excess\ Cash + \sum_{t=1}^{N} \frac{Earnings_t}{(1+d)^t} \right) \\ + \frac{Net\ MarketCap_N}{(1+d)^N} } \]

### Caveats

There is a problem when the earnings can be negative. In the real world this would be a deficit that the company must fund somehow, either from cash holdings, or borrowings, or making a new share issuance. The loss can typically also be carried forward as a future tax-credit for the company. All of this would impact the present value of the stock.

Furthermore, when assuming the earnings are being paid out as dividends, it is not possible to pay out negative dividends to shareholders. So there is also the question of how to account for negative dividends, as well as the taxes on negative dividends.

We do need to somehow account for negative earnings and dividends, because they do have a negative impact on the shareholder value. But to keep the valuation models simple, we typically ignore all these issues when simulating the present value of a stock, and simply include the negative earnings and dividends in the calculation of the present value without any adjustments. We don't even adjust the tax-rate for negative dividends, so if the dividend tax-rate is 30% and the loss is -100m, then we use the value -70m in the present value calculation.

This means the present value should be considered a somewhat more rough estimate of the shareholder value, when there are negative earnings and dividends in the future years.

## Drift

You may sometimes notice a peculiar phenomenon in the simulation results, where the present value seems to drift over time. The plot below shows an example of the present value of a stock, where there is no tax, no excess cash, constant earnings of USD 1b per year, a constant discount rate of 20%, and the only source of variation is the P/E ratio which is normal-distributed with mean 15 and std.dev. 3. Notice how the simulated present values drift lower for the years that are further into the future.

This may seem bizarre at first and you may wonder if there are errors in the simulations. But it is just a natural result of comparing present values for successive years. If you try running this example yourself, you will see that the valuation drift gets worse for larger discount rates. It is also possible for the present values to drift upwards instead of downwards. It all depends on the choice of discount rate, earnings, and market-cap.

We can demonstrate this with a small mathematical example by inserting \( Excess\ Cash = $0 \), \( Earnings_t = $1b \), \( d = 20\% \), and \( MarketCap_N = $15b \) into Eq.\eqref{eq_pv_stock}, so we get this simple formula for the present value of the stock in future year \( N \): \[ PV_N = \sum_{t=1}^{N} \frac{$1b}{1.2^t} + \frac{$15b}{1.2^N} \] Using this formula with different number of years \( N \) we get the present values \( PV_1 = $13.33b \), \( PV_2 = $11.94b \), \( PV_3 = $10.79b \), etc. These are drifting downwards because the present value of the market-cap is discounted with 20% more for each year into the future, but one more year of discounted earnings is not sufficient to make up for the decrease in the present value of the market-cap, and hence the total present value drifts downwards.

What this means in practice, is that you should always consider entire distributions for the present value, as there is rarely just a single present value of a stock. Even if the future earnings, market-cap, and discount rate are all held constant as in the example above, the present value of a stock may vary simply from changing the holding period.