Asset-stock

An asset stock, or simply “Stock”, is a collection, group, population, mass or volume of materials or entities that accumulates or depletes over time. Examples include the volume of liquid in a tank, the amount of cash in a bank account (or debt on a credit card), the population of a country (or fish in a fishery, bears in a forest ...), the inventory of goods in a store (often called the store's stock), and the quantity of CO₂ in the atmosphere. Asset stocks fill and drain over time, driven by the flow-rates of material or items entering or leaving the Stock, so the most common example used to explain how Stocks work is the “bath-tub metaphor”. In this metaphor, changes over time to the level of water in a bath-tub reflect both the rate of water flowing in through the faucet and the rate of water flowing out through the drain.

How the rate water flows into and out of a bathtub changes the quantity or "Stock" of water each minute.

Mathematical properties

Asset stocks accumulate and deplete (fill and drain) over time. The quantity of a Stock at any time is equal to the quantity at a previous point in time, plus any amount that was added, minus any amount that was lost in the intervening period. Extend that previous point to the time when the Stock first came into existence and that quantity is also equal to the sum of everything that was ever added, minus everything that was ever lost - for example, a bank account contains the sum of every dollar ever paid in, minus every dollar ever paid out since the day the account was opened.

Units of measurement.

An asset stock’s units are measured at points in time, as number-of-items for discrete factors (people, fish, bears, inventory units), or weight, volume or value for continuous items (tons of CO₂, litres of liquid, dollars of cash). However, average Stock-quantities during a period may also be calculated, such as the average bank balance during a month, or the average population during a year.

In contrast, the in- and out-flow rates are measured for some period of time, often the period between the two times at which the stock’s quantity is measured. The units for each flow-rate are therefore “items per period” (people or fish per year, inventory units delivered or sold per week), or “weight/volume/value per period” (tons/year, litres/minute, dollars/month). These relationships are captured at each point in time by a differential calculus equation:

$d(Stock)/dt=Inflow(t)-Outflow(t)$

… where ‘t’ is the period over which the change in the Stock is measured and ‘d(Stock)/dt’ is the math notation for the net change in the Stock during a very small period. To operationalise the principle that Flows drive changes to a Stock's quantity requires the formulation to be reversed with an integral equation:

$Stock(t)=Stock(t_{0})+\int \limits _{t_{0}}^{t}[Inflow(s)-Outflow(s)]ds$

… where ‘Inflow(s)’ is the value of the inflow-rate at any time ‘s’ between the initial time ‘t₀‘ and the current time ‘t’.

Example of an asset stock.

The following table shows the relationships between the balance (Stock) in a bank account and the payments into and out of the account each month (the Flows). The balance at the end of the month is equal to the balance at the start of the month plus or minus the net amount paid in or out. The end-of-month balance then becomes the starting balance for the next month, as highlighted for the end of January and start of February. Making the calculations with a different frequency would give the changing balance each year, week, day or hour.

How amounts are paid in and out of a bank account each month change the account balance.

The impacts of asset stocks on system behaviour

Asset stocks are fundamental to the behaviour of many systems, due to four roles that they fulfill .

• Stocks characterise the state of the system and its performance on any measure at each point in time. For example, interest charges on a credit card are driven by the debt on that card (the Stock), as is the interest-cost of Government debt. The demand for food and the rate of waste-generation are driven by the population Stock, the supply of food is driven by the Stock of agricultural land, the workload and cost of health-care, schooling and crime-prevention are driven by the Stocks of patients, children and criminals respectively.
• Stocks provide the system with inertia and memory. Today’s credit card debt reflects all historic purchases made on that card.
• Stocks are the source of delays - if a customer buys an item today, it enters a Stock of outstanding orders and is only delivered to that customer after production delays and/or delivery delays. All delays involve asset stocks, since the only mechanism for a factor to influence another at a future point in time is for something (material or information) to be stored in a Stock for the intervening period.
• Stocks separate rates-of-flow, causing disequilibrium dynamics. If food could not be stored, for example, then the food consumption rate would have to match the food production rate.

Managing complex systems thus relies on managing the relationship between Flows and Stocks - slow the rate of credit-card spending or accelerate the rate that debt is paid off, reduce the birth-rate, bring new land into agricultural production, deter young people from becoming criminals, and so on.

Historical development of asset-stock applications

The role of asset stocks was first recorded in ancient Mesopotamia where settled cities started to store and distribute grain and other goods. Checking the additions to those stocks at harvest time, the distribution of those stocks during the rest of the year, and the changing level of the remaining stocks required stock-and-flow calculations, a practice that formed the basis of modern-day accounting. Stock-flow relationships were also exploited for engineering purposes, especially irrigation and flood control, for example in the Nile valley.

With the Industrial Revolution, more complex movements of raw materials and goods required stock-flow principles to be replicated for all cases of inventory or stock management and at each stage of complex supply chains. That period also saw the development of modern engineering disciplines, which depend on mathematical manipulation of stock-flow relationships. Chemical engineering, in particular, relies on controlling the movement of liquids between vessels, driven by pumps and controlled by valves. Information on the levels of such vessel liquids, and on the rates at which those liquids are moving is used to adjust the pumps and valves that control those flow-rates. This exploitation of information feedback is the central feature of control engineering (also traceable back to ancient times).

Non-engineering fields also exploit analysis of stock-flow principles, such as demography, where population dynamics is the analysis of population stocks and flows (births, deaths, migration and ageing). The role of accumulating asset stocks has also been recognised and exploited in environmental policy, security and defence modelling, public-health policy, and the control of infectious diseases based on stocks and flows of susceptible, infected and resistant individuals (the SIR model).

Recognition and exploitation of asset-stock analysis and modelling have been somewhat slower in other fields. Enterprises - including for-profit corporations, non-profit organisations and public-services - also consist of interdependent stocks and flows (customers, staff, production-capacity, products and services). Although the role of those entities in such cases has long been noted, that role features little in the study of enterprise management, with the exception of the strategy dynamics method. Similarly, while a national economy consists of interdependent stocks and flows (jobs and net job-creation, capacity and investment-driven capacity-changes, quantities and flows of money, trade flows, populations and demographic changes), the study of macroeconomics offers little formal representation of those stock-flow relationships.

Time-based simulation.

Accumulating asset-stocks form the basis of the system dynamics simulation method. The following figure illustrates a simulation of the bank account example, matching the table above. (Several system dynamics software tools use this same visual metaphor to display simulated changes to asset-stock quantities). Simulation is valuable because research has established that estimating the time-path behaviour of the Stock that is caused by any given changes to the in-flow and out-flow rates is difficult. In this case, the time-path of the bank account balance bears no obvious relationship to the changing in- and out-flow rates.

A working quantified model of changes to a bank account over 12 months

System dynamics models work by capturing the interdependencies between the multiple stocks in a system, in order to simulate the system's overall behaviour. It is one of a class of dynamic simulation methods, the others being discrete event simulation (DES) and agent-based modelling (ABM) System dynamics models deal with aggregate quantities of material or entities in each asset stock, while DES and ABM both model the movements of individual entities into, out of, and between different stocks.

References

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