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My question regards the rationale behind the Present Value computation of a single cash flow received after $n$ periods, $CF_{n}$. Most textbooks make the following statement:

$$PV =\frac{CF_{n}}{(1+x)^{n}}$$

While I totally get the formula, I am unsure about what information is actually considered in $x$. Some textbooks call it interest rate while others discount rate, and I am not sure if all authors mean the same thing. According to reviewed resources, some quantities that could affect the time value of money are:

  • the interest rate ($i$): how much money would be earned on a bank deposit
  • inflation ($if$): rising prices
  • time risk ($r$): analogous to the probability of not getting the cash flow

At this point, my first question is the following: even if the interest rates, inflation and time risk are all exactly zero ( $i$=$if$=$r$=0), is it still true that "a dollar today is worth more than a dollar tomorrow"? In case the answer is yes, I suppose there is a fourth quantity.

  • impatiance ($im$): Analogous to the degree of impatience of the cash flow receivers. It makes sense because who would care about a zero risk cash flow received 1 million years from now even under $i=if=r=0$, i.e. with the same purchasing power as today and unchanged prices.

The second question is, which of these factors are typically accounted for in $x$? For text books that name $x$ as the interest rate [1] I guess it is only $x=i$, such that

$$PV =\frac{CF_{n}}{(1+x)^{n}} = \frac{CF_{n}}{(1+i)^{n}}$$

For zero interest rate the above definition claims that $PV = CF_{n}$. How come this definition ignores factors such as inflation, risk and impatience? On the other hand, it is unclear if textbooks that call it simply "discount rate" with giving any more detail [2] include all these factors in the denominator $x$. For instance, if we consider all factors it is $x= (i+if+r+im)$:

$$PV =\frac{CF_{n}}{(1+x)^{n}} = \frac{CF_{n}}{(1+(i+if+r+im))^{n}}$$

Disclaimers:

  • I think multiple cash flows are not essential for understanding concept so I sticked to simple one-time cash flows.
  • As similar information content may be accounted for multiple times in the variables $i,if,r,im$, if addition is not the right way to go ($x=(i+if+r+im)$), consider more generally that $x=f(i,if,r,im)$ in the above formula.

[1] Mishkin, Money, Banking and Financial Markets.

[2] Damodaran, Investment Valuation

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First, of all you can use/calculate present value of money using different factors in different context, I will come back to this point at the end.

Next, generally speaking it is often in economics appropriate to just use prevailing (nominal) interest rate (see discussion in Mankiw Principles of Economics pp 564). This is simply because interest rates already include most of what you mention. Nominal interest rates are, by Fisher equation given as:

$$i \approx \pi +r$$

where $\pi$ is inflation and $r$ is real interest rate.

So first nominal interest rates already include inflation within them and putting it there twice would be double counting. Second, real interest rates critically depend on all the other parameters you mention such as how risky the loan is or how impatient people generally are.

Now on more fundamental level, it is slightly more complex because prevailing interest rates are based on supply demand interactions between large number of heterogenous people as a consequence prevailing interest rate might not represent very well your own personal impatience etc. even if it captures aggregate level of impatience and other factors.

Consequently, in more advanced models in economics you will find that economists use not just interest rate (although note some models use real interest rate as many economic models simplify by assuming no inflation) but also second term called discount rate (usually denoted $\delta$), which supposed to be a catch all term for all other personal factors. Discount rate is based on your utility, essentially you are asking yourself how much extra utility you need to get to be willing to not consume your money today but wait to the next (or more) future time periods (given the prevailing interest rate). So generally, it is enough to have these two terms (in fact I seen some authors to even subsume everything under discount rate, but most will split it at least into these two).

Of course, if you want you can always subdivide the terms into more granular sub categories. You can always split nominal interest $i$ into sum of real interest and inflation $r+\pi$ and then proceed to split real interest into risk free rate and compensation for risk etc. That is really up to you, but you need to be extremely careful not to double count. If you use nominal interest rate do not add inflation, that is double counting.

Lastly, going back full circle, in many business, legal etc settings or real world applications in general it might be difficult to know what's your client or your firms discount rate so you will just use interest rate and make calculations just with it. Or in some contract with your suppliers you might agree that if need will arise to compensate the other party for some past damages for time value of money you will use only inflation or only some predetermined interest rate and so on.

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  • $\begingroup$ Great response. Regarding the formula i≈π+r: I guess I had it backwards that π≈i-r such that π is the dependent variable and not $i$. Ultimately, what I had backwards is that $r$ carries more information than $i$, when in fact $i$ already includes the inflation. Some parts still seem counter-intuitive, so I will continue working on this to get the intuition: E.g. Doing the algebra to get future value for current cash $X$ gives $ FV = X(1+(r+π))$. Higher real return leading to higher FV is straightforward. But why would inflation also increase FV. $X(1+(i-π))$ makes more sense. $\endgroup$
    – Enk9456
    Jun 5 at 18:54
  • $\begingroup$ @Enk9456 1. in $i \approx \pi+r$ all variables are dependent - it is endogenous system (although r can be considered exogenous in long run). Also no matter how you reorder it, it carries the same information. Eg x+3=y describes completely the same function as y-3=x. 2. Inflation lowers value of your money if you have 100 today and there is 10% inflation then next year your money is only worth 90.9 so no you should not be subtracting it (from real return) <- you would subtract it if there would be deflation $\endgroup$
    – 1muflon1
    Jun 5 at 19:02

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