Observing Duality

This guide explains and demonstrates a fundamental bookkeeping principle: Observing Duality.

This principle is essential to accounting practice, yet very simple to master. We will first use a contrived example from a day-to-day economic exchange to understand the context and the problem, and to derive the principle. Then, we will discuss how to observe the same principle in DECAF context.

Let's Go Shopping

Scenarios which we will describe here should be familiar to all of us. We will go shopping to buy a pair of socks that costs US$ 10 for a pair.

Simple Economic Exchange with Duality

By the beginning of the month, we have US$ 100 in our pocket ($t_{0}$), ie. our total assets (our contrived net asset value or NAV) is US$ 100.

Time	NAV (US$)
$t_{0}$	100

Our holdings at $t_{0}$ are as follows:

Asset	Quantity	Price	Value (US$)
US$	100	1	100
Total			100

Next day ($t_{1}$), we go to our favourite shop and ask the shopkeeper for a pair of socks. She shows us one pair, we like it and take it to the cashier. Then, we pay for it and receive our nicely packed pair of socks. Transactions due to this economic exchange are:

Transaction	Time	Description	Value (US$)
1	$t_{1}$	Give cash	-10
2	$t_{1}$	Take socks	+10

Therefore, our holdings as of next day ($t_{1}$) are:

Asset	Quantity	Price	Value (US$)
US$	90	1	90
Pair of Socks	1	10	10
Total			100

... and NAV evolution is:

Time	NAV (US$)
$t_{0}$	100
$t_{1}$	100

The two transactions (1 and 2) observe a duality, ie. The decrement event (Transaction 1) and the increment event (Transaction 2) have same monetary value (despite that one is negative and the other is positive) and they have the same timestamp ($t_{1}$).

Breaking Duality

If the monetary values would differ OR the timestamps would not be the same, we would break the duality.

A typical situation is that the shop has no stock of the pair of socks we want to buy:

You: I want to buy a pair of socks as displayed here.

Shopkeeper: We do not have them on the stock at the moment. If you pay now, you can come and collect tomorrow.

If you accept the offer, pay the monies now ($t_{1}$) and collect the pair of socks tomorrow ($t_{2}$), the naive picture would be as follows:

Transactions due to this economic exchange

Transaction	Time	Description	Value (US$)
1	$t_{1}$	Give cash	-10
2	$t_{2}$	Take socks	+10

Our holdings as of today ($t_{1}$)

Asset	Quantity	Price	Value (US$)
US$	90	1	90
Total			90

Our holdings as of tomorrow ($t_{2}$)

Asset	Quantity	Price	Value (US$)
US$	90	1	90
Pair of Socks	1	10	10
Total			100

NAV evolution

Time	NAV (US$)
$t_{0}$	100
$t_{1}$	90
$t_{2}$	100

As you see, breaking the duality results in a sudden decrease in our net wealth, followed by a sudden increase.

Indeed, we did not lose money today ($t_{1}$). It is just that we are missing something in our bookkeeping.

Gaining Duality Back

Following the example in the previous section, we will gain the duality back with 2 additional transactions of a different nature.

When we pay the monies to the shopkeeper, we are actually entering into an agreement whereby the the shopkeeper has promised us to supply the goods tomorrow. This is called a Materialized Claim and it can be reflected in two different ways. Let's see each:

In the form of an token (asset)

If the shopkeeper gives us a gift certificate with a value of US$ 10, our new picture would look like this:

Transactions due to this economic exchange

Transaction	Time	Description	Value (US$)
1	$t_{1}$	Give cash	-10
2	$t_{1}$	Take Gift Certificate	+10
3	$t_{2}$	Redeem Gift Certificate	-10
4	$t_{2}$	Take socks	+10

Our holdings as of today ($t_{1}$)

Asset	Quantity	Price	Value (US$)
US$	90	1	90
Gift Certificate	1	10	10
Total			100

Our holdings as of tomorrow ($t_{2}$)

Asset	Quantity	Price	Value (US$)
US$	90	1	90
Pair of Socks	1	10	10
Total			100

NAV evolution

Time	NAV (US$)
$t_{0}$	100
$t_{1}$	100
$t_{2}$	100

In the form of a promise

If the shopkeeper's promise is legally binding (in any way), we can say that we have some receivable although it is not in asset form (Can not be materially custodied like a token or gift certificate in our wallet). In this case:

Transactions due to this economic exchange

Transaction	Time	Description	Value (US$)
1	$t_{1}$	Give cash	-10
2	$t_{1}$	Increase in Receivables	+10
3	$t_{2}$	Decrease in Receivables	-10
4	$t_{2}$	Take socks	+10

Our holdings as of today ($t_{1}$)

Asset	Quantity	Price	Value (US$)
US$	90	1	90
Receivables	10	1	10
Total			100

Our holdings as of tomorrow ($t_{2}$)

Asset	Quantity	Price	Value (US$)
US$	90	1	90
Pair of Socks	1	10	10
Total			100

NAV evolution

Time	NAV (US$)
$t_{0}$	100
$t_{1}$	100
$t_{2}$	100

Summary

The principle of Duality is essential to any economic exchange whether the goods and/or services are physically exchanged at the same time or not. If there is a delay in the delivery of the goods or services, a materialized claim arises: Either the receiving party obtains a physical token or the giving party's promise is legally binding. But in either case, both parties can book such materialized claims in their books.

We have to make a note here: Whether the shopkeeper will (or will be able to) hold her promise or not is not so much relevant in the booking process. The risk associated with the commitment and its fulfillment is rather a valuation issue (like the issuer risk in financial markets): What is the real value of my gift certificate considering the risk that I may not be able to redeem it due to bankruptcy or misconduct of business?

Financial Transactions and DECAF

Let's consider a common case in financial transactions: Subscribing to a fund before its subscription period kicks in.

For the sake of this example, assume that we start with US$ 1,000,000 on $t_{0}$, send US$ 500,000 to a fund on $t_{5}$ for subscription, but the fund subscription happens on $t_{10}$ at an NAV/share price of US$ 500.

Broken Duality

Transactions are:

Transaction	Time	Description	Value (US$)
1	$t_{5}$	Give Cash	-500,000
2	$t_{10}$	Take Shares in Fund	+500,000

Holdings at $t_{0}$ are:

Asset	Quantity	Price	Value (US$)
US$	1,000,000	1	1,000,000
Total			1,000,000

Holdings at $t_{5}$ are:

Asset	Quantity	Price	Value (US$)
US$	500,000	1	500,000
Total			500,000

Holdings at $t_{10}$ are:

Asset	Quantity	Price	Value (US$)
US$	500,000	1	500,000
Fund Shares	1,000	500	500,000
Total			1,000,000

Therefore, NAV evolution is:

Time	NAV (US$)
$t_{0}$	1,000,000
$t_{5}$	500,000
$t_{10}$	1,000,000

We know this picture: It is not correct! We are missing additional bookings, which in return creates an invalid fluctuation in our NAV figures.

Solution (1): A Dummy Instrument "Pending Subscription"

Create a dummy instrument of type "Other Asset" and name it as "Pending Subscription".

Once the money outflow happens, key-in an inflow of "Pending Subscription" with the same but positive quantity. Once the shares are in, key-in an outflow of "Pending Subscription". This will keep the gross asset value same if the pending subscription is booked under a custody account.

Transactions are:

Transaction	Time	Description	Value (US$)
1	$t_{5}$	Give Cash	-500,000
2	$t_{5}$	Increase in Pending Subscription	+500,000
3	$t_{10}$	Decrease in Pending Subscription	-500,000
4	$t_{10}$	Take Shares in Fund	+500,000

Holdings at $t_{0}$ are:

Asset	Quantity	Price	Value (US$)
US$	1,000,000	1	1,000,000
Total			1,000,000

Holdings at $t_{5}$ are:

Asset	Quantity	Price	Value (US$)
US$	500,000	1	500,000
Pending Subscription	500,000	1	500,000
Total			1,000,000

Holdings at $t_{10}$ are:

Asset	Quantity	Price	Value (US$)
US$	500,000	1	500,000
Fund Shares	1,000	500	500,000
Total			1,000,000

Therefore, NAV evolution is:

Time	NAV (US$)
$t_{0}$	1,000,000
$t_{5}$	1,000,000
$t_{10}$	1,000,000

Note that you can create a separate "Pending Subscription" instrument for each fund if you wish to.

Solution (2): Using Analytical Accounting (Accounts Receivable)

If analytical accounting is enabled on your system (it is enabled by default in fund management), you can create an analytical account for "Accounts Receivable".

Once the money outflow happens, key-in a partial journal entry with the same but positive quantity. Once the shares are in, key-in a partial journal entry with negative quantity. This will keep the net asset value same, but gross asset value will be different as gross asset value accounts for positions under custody accounts, not analytical accounts.

Transactions are:

Transaction	Time	Description	Value (US$)
1	$t_{5}$	Give Cash	-500,000
2	$t_{5}$	Partial Journal Entry	+500,000
3	$t_{10}$	Partial Journal Entry	-500,000
4	$t_{10}$	Take Shares in Fund	+500,000

Holdings at $t_{0}$ are:

Asset	Quantity	Price	Value (US$)
US$	1,000,000	1	1,000,000
Total			1,000,000

Holdings at $t_{5}$ are:

Asset	Quantity	Price	Value (US$)
US$	500,000	1	500,000
US$ (under Accounts Receivable)	500,000	1	500,000
Total			1,000,000

Holdings at $t_{10}$ are:

Asset	Quantity	Price	Value (US$)
US$	500,000	1	500,000
Fund Shares	1,000	500	500,000
Total			1,000,000

Therefore, NAV evolution is:

Time	NAV (US$)
$t_{0}$	1,000,000
$t_{5}$	1,000,000
$t_{10}$	1,000,000

Note that you can create a separate analytical account for each subscription if you wish to. This is not advised as it will unnecessarily increase the number of accounts under a given portfolio.