Financial Balances and Abstract Assets#

Introduction#

The development of processes or products does not occur inside a mythical universe that is free from market forces. Instead, processes and products must be, at a minimum, financially neutral to be viable. Processes and products that are not financially viable, while perhaps being technically innovative or otherwise advantageous, will not survive in the marketplace without external support e.g., local, state or federal government subsities.

In this lecture we will:

Abstract Assets#

../_images/Fig-Asset-CashFlowDiagram.pdf

Fig. 13 Abstract asset diagram. During each time period \(t=1,2,\dots,T\) an asset has cash flow(s) \(CF_{t}\). The value of the asset is some function \(\mathcal{V}\) of these cash flows; the function \(\mathcal{V}\) is the called the valuation operator.#

In abstract sense, an asset is simply a sequence of current and future cash flows demarcated in some currency, for example, Euros, Dollars, Yuan, or cryptocurrencies such as Bitcoin. Thus, the value of an asset can be determined using a valuation operator \(\mathcal{V}\) applied to these current and future cash flows:

(38)#\[\mathcal{V}\left(Asset_{t}\right) = \mathcal{V}\left(CF_{t},CF_{t+1},\dots,CF_{t+T-1}\right)\]

where \(CF_{t}\) denotes the cash flow in time period \(t\) up to \(T\) time periods in the future. The most common (and intuitive) valuation operator \(\mathcal{V}\) is the sum of net cash flows in each time period, i.e., we compute the net cash in each period (cash inflow - cash outflow in period \(i\)) and then sum these net cash flows over all \(T-1\) periods. However, there is a critical and fundamental challenge to this intuition.

Remark 3 (Time value of Money)

The value of money is not conserved over time. One dollar today is not worth the same as one dollar tomorrow. The change in the value of money over time is called the time value of money.

Time value of money#

The value of money is not conserved; one dollar T years from now is worth less than one dollar today. The change in the value of money over time is called the time value of money. The time value of money is an empirical observation that has been seen over hundreds of years. But why is this the case?

The short answer: money given to us today has a greater utility than the same amount tomorrow; because we have an extra day to invest that money.

Observation 15 (Why does that value of money change over time?)

Hypothetically, suppose a risk-free investment was guaranteed to return \(i>0\) in some period. If we invested \(P\) dollars today in this risk-free investment, then at the end of the investment period, we are guaranteed to get \(F\) dollars back (because the investment is risk-free) where \(F\) is given by:

\[F = (1+i)\cdot{P}\]

Thus, if given a choice between \(P\) dollars today, or the same \(P\) dollars one investment period in the future, you would always choose (assuming you were a rational investor) to take the \(P\) dollars now.

Why?

  • By taking \(P\) dollars today, you could invest those \(P\) dollars in a risk-free product and get \(F = (1+i)\cdot{P}\) back, where \(F>P\). Thus, the only case where it makes sense to take the future money is if \(i=0\), i.e., \(F = P\), which is forbidden by the condition \(i>0\).

  • This hypothetical scenario assumes that a risk-free investment exists that always returns \(i>0\). Does such an investment exist? Unfortunately, the risk-free investment is only a theoretical concept. However, the risk-free rate of return \(i\) is often approximated by the yield on 10-year US Treasury bonds or some other US Treasury debt security.

Unfortunately, our intuitive valuation operator (net cash flows) does not consider (yet) the time of value of money. Let’s develop some tools to address this issue.

Exchange rate model#

A useful mental model for the time value of money is to think of money from different time periods as being in different currencies (e.g., \(\$\) and \(\def\euro{\unicode{x20AC}} \euro\)) that must be exchanged. Thus, to compare money or cash flows from different time periods we formulate the equivalent of an exchange rate.

Discrete one-period conversion#

For example, suppose we wanted to convert the value of a cash flow from one time period in the future into today’s dollars (or vice-versa, we want to compute the value of a future cash flow to a current value). To do this, we use a one-period discrete conversion:

Definition 16 (One Period Discrete Conversion)

For any one-period (\(t\rightarrow{t+1}\)), the present value of cash flow, denoted as \(CF_{t}\), and the future value of cash-flow, denoted by \(CF_{t+1}\), there exists a one time-step discrete conversion:

(39)#\[CF_{t+1} = \left(1+r_{t+1,t}\right)\cdot{CF_{t}}\]

where \(r_{t+1,t}>{0}\) is called the rate of return between periods \(t\) and \(t+1\). The term \((1+r_{t+1,t})\), which denotes the hypothetical exchange rate between time period \(t\) and \(t+1\), is called the discrete discount factor. Because \(r_{t+1,t}>{0}\), the discrete discount factor is always greater than one, thus \(CF_{t+1}>{CF_{t}}\) for any \(t\).

Let’s do a few examples to illustrate one-period conversions.

Discrete multiple-period conversion#

Suppose, instead of a single period, we are interested in an asset that will be active over multiple periods, e.g., a multi-year project or some other transaction that occurs over many years. In this case, we develop a multi-year conversion that is easily constructed by sequentially applying many one-period calculations.

To see this idea, let’s start with period one to period two:

\[CF_{2} = \left(1+r_{21}\right)\cdot{CF_{1}}\]

Then, period two to period three is given by:

\[CF_{3} = \left(1+r_{32}\right)\cdot{CF_{2}}\]

but we can substiture \(CF_{2}\) into the expression above to give:

\[CF_{3} = \Bigl[\left(1+r_{32}\right)\left(1+r_{21}\right)\Bigr]\cdot{CF_{1}}\]

If we do this computation between 3 and 4, and then 4 to 5, etc, we develop a relationship between the current value of cash flow (\(t=1\)) and the future (\(t>1\)) value of cash flows:

Definition 17 (Multiple Period Discrete Conversion)

Let \(CF_{1}\) denote the cash-flow in period 1 (current value), and \(CF_{t}\) the cash-flow in period \(t>1\) (future value). Further, let \(r_{j+1,j}\) denote the rate of return between discrete time period \(j\) and \(j+1\). Then:

(40)#\[CF_{t} = \left[\prod_{j=1}^{t-1}\left(1+r_{j+1,j}\right)\right]\cdot{CF_{1}}\qquad{t=2,3,\dots,T}\]

The product term (the \(\prod\star\) in the brackets) in Eqn. (40) is called the multi-period discount factor and given the symbol \(\mathcal{D}\). Thus, Eqn. (40) can be written in a more compact form as:

\[ CF_{t} = \mathcal{D}_{t,1}\cdot{CF_{1}}\qquad{t=2,3,\dots,T}\]

In the particular case where the rates of return \(r_{\star}\) are equal in each period (let’s call this value \(\bar{r}\)), the multi-period discount factor is given by:

(41)#\[\mathcal{D}_{t,1} = (1+\bar{r})^{t-1}\qquad{t=2,3,\dots,T}\]

Let’s do some multi-period discounting examples to better understand Definition 17.

Continuous discounting#

When formulating Eqn. (39) we used a finite discrete-time difference, e.g., days, weeks, or maybe even years to compute the discount factor. However, suppose the time difference between \(t\rightarrow{t+1}\) were infinitesimally small, i.e., the time difference was continuous. In this case, we have continuous compounding where the discount factor \(\mathcal{D}\) is defined as:

\[\mathcal{D}(r,t) = \exp\left(rt\right)\]

A continuous discount factor gives a future value expression of the form:

(42)#\[CF(t) = \exp\left(rt\right)CF(t_{o})\]

where \(r\) denotes the instantaneous discount rate, \(t\) denotes the current time, \(t_{o}\) denotes the initial time.

Observation 16 (Discrete and continuous discount factor)

There is a relationship between the discrete and continuous discount factors; the discrete discount factor is the linear portion of the continuous discount factor. We can see this by looking at the series representation of the exp function:

\[\exp\left(x\right) = \sum_{k=0}^{\infty} \frac{x^{k}}{k!} = 1+x+\frac{x^{2}}{2}+\frac{x^{3}}{6}+\dots\]

The first two terms are the discrete discount factor. Thus, for small values of \(rt\) the discrete and continuous discount factors will be similar.

Let’s do a continuous discount factor example.

Net Present Value (NPV)#

../_images/Fig-FinanicalNode-Schematic.pdf

Fig. 16 Node cash-flow balance diagram. During time \(t=k\) monety flows enter (revenues) and exit (expenses) the time-node.#

Now that we have tools to account for the time value of money, we can return to the question of how to value an asset. The most intuitive aproach is to compute the net cash flow at every node. Imagine at node \(t=k\) we have \(\mathcal{S}^{t=k}\) cash streams, denoted as \(\dot{C}_{s}\), entering (or exiting) the asset node. Then, the net cash would at node \(t=k\) is given by:

\[CF_{k} = \sum_{s\in\mathcal{S}^{t=k}}\nu_{s}\dot{C}_{s}\]

where \(\nu_{s}\) denotes a direction parameter; \(\nu_{s}=+1\) if stream \(s\) enters node \(t=k\), \(\nu_{s}=-1\) if stream \(s\) exists node \(t=k\). Finally, to value an asset, we could imagine adding up all the current and future cash flows, converted to some shared basis, e.g., current dollars. This sum is called the Net Present Value (NPV); NPV is a widespread method for asset valuation. As well shall see, NPV is also a widely used tool for financial decision-making.

Definition 18 (Discrete Net Present Value)

The discrete Net Present Value (NPV) is the sum of the discrete current and future cash flows, corrected to current dollars:

\[\text{NPV} = \sum_{t=1}^{T}{\mathcal{D}_{t,1}^{-1}}\cdot{CF_{t}}\]

where \(CF_{t}\) denotes the cash flow in time period \(t\), and \(\mathcal{D}_{t,1}\) denotes the discrete multistep discount factor between the current time period, and time period \(t\). However, we know that \(r_{1,1} = 0\) because there is no possibility of an instantaneous return; thus, \(\mathcal{D}_{t,1} = 1\). Therefore NPV can be re-written as:

\[\text{NPV} = CF_{1} + \sum_{t=2}^{T}{\mathcal{D}_{t,1}^{-1}}\cdot{CF_{t}}\]

where the value of each cash flow at time \(\star\), denoted by \(CF_{\star}\), is the net cash flow:

\[CF_{\star} = \sum_{s\in\mathcal{S}^{t=\star}}\nu_{s}\dot{C}_{s}\]

NPV as a Decision Tool#

The net present value is a widely used tool for financial decision-making. However, to understand the basis of the approach, we must first answer a technical question: What discount rate should we use?

The discount rates \(r_{t+1,t}\), i.e., the rates of return between time periods \(t\rightarrow{t+1}\) appearing in the discount factors, have an exciting interpretation within the context of net present value calculations. These rates can be specified by the decision maker as the minimum acceptable rate of return that some hypothetical alternative investment could earn.

We can set up the following criteria to decide between a possible asset or project and a hypothetical alternative investment:

  • \(\text{NPV}<0\): A negative NPV indicates the proposed project will generate less income than the alternative investment, e.g., a zero-coupon bond at the same discount rate and time-to-maturity as the project.

  • \(\text{NPV}=0\): A zero NPV indicates the proposed project will generate the same income as the alternative investment, e.g., a zero-coupon bond at the same discount rate and time-to-maturity as the project.

  • \(\text{NPV}>0\): A positive NPV indicates the proposed project will generate more income than a hypothetical alternative investment, e.g., a zero-coupon bond at the same discount rate and time-to-maturity as the project.

Let’s do a net present value example:

The Internal Rate of Return (IRR)#

The NPV decision rule above relies on computing the sum of discounted future cash flows. However, in these calculations, what discount rate should we use? This is not an easy question; the correct discount rate varies between applications and industries. Thus, let’s think of the discount rate as a variable.

  • If the net present value of an investment or project is positive, the project earns more than a zero-coupon bond with a yield equal to the discount rate used to discount future cash flows.

  • If the net present value of a project is negative, the project earns less than a zero-coupon bond with a yield equal to the discount rate used to discount future cash flows.

  • If the net present value of a project or investment is precisely $0, the project earns the same as a zero-coupon bond with a yield equal to the discount rate used to discount future cash flows.

The special discount rate where the net present value is zero is called the Internal Rate of Return (IRR):

Definition 19 (Internal Rate of Return)

Assume the discount rate is constant between time periods, denote this rate as \(\bar{r}\). Then, the internal rate of rerturn (IRR) is the discount rate that satifies the following condition:

\[\sum_{t=1}^{T}{\mathcal{D}_{t,1}^{-1}}\cdot{CF_{t}} = 0\]

Discount rates greater than the IRR favor the alternative investment.

Thus, IRR can thought of as a decision boundary of sorts; the IRR is the discount rate where the project manager or investor is indifferent to the project.

Summary#

In this lecture:

  • We introduced the definition of a Abstract Assets. An asset is a sequence of current and future cash flows demarcated in some currency. Assets can be tangible things like cars or houses, but they can also be projects or ideas.

  • We introduced the Net Present Value (NPV), a tool to calculate the valuation of an asset. The net present value is the sum of net cash flows discounted to the current period.

  • Used NPV as a Decision Tool to evaluate the relative attractiveness of assets and projects. Assets with a positive net present value have a larger return than hypothetical alternative investments. On the other hand, investments with a negative net present value will have a smaller return than a hypothetical alternative investment.