Infinite divisibility

Capital Market Theory draws smooth curves. The efficient frontier is a curve. The Capital Market Line is a straight line running through every possible mix of a risk-free asset and the market portfolio.

Smooth means every point on the line is available. If a model says the right answer is 63.4% of your money in the market portfolio, you must be able to buy exactly that.

In real markets you cannot. Shares come in whole numbers. Fund units come in whole numbers. You can buy 63 units or 64, not 63.4.

Infinite divisibility is the assumption that papers over this. All investments are assumed infinitely divisible, meaning it is possible to buy or sell fractional shares of any asset or portfolio.

The workbook is relaxed about it. This assumption is what lets the theory discuss investment alternatives as continuous curves, and changing it would have little impact on the theoretical models.

The exact assumption (Chapter 3, section 3.11). All investments are infinitely divisible, which means it is possible to buy or sell fractional shares of any asset or portfolio. This assumption allows investment alternatives to be discussed as continuous curves. Changing it would have little impact on the theoretical models.

Why the theory needs it. Capital Market Theory adds a risk-free asset to Markowitz's efficient portfolios of risky assets. The expected return of the resulting portfolio is a weighted average:

E(Rport) = WRF x RFR + (1 - WRF) x E(Ri)

With the risk-free rate at 5%, the risky portfolio at 12% and 70% of wealth in the risk-free asset, the workbook's own arithmetic gives 7.10%, and with the risky asset's standard deviation at 10% the portfolio risk is 3%. That calculation only means something if WRF can take any value between 0 and 1. Restrict it to whole units and the Capital Market Line stops being a line and becomes a scatter of dots.

Where it sits among the other assumptions. Section 3.11 states each assumption and then says what happens if it is relaxed:

So infinite divisibility is at the harmless end of the list. Equilibrium is at the other end.

Where the assumption meets this paper's own reality. The same workbook's Fund SER illustration issues Category III AIF units at Rs 1,000 each — 2,00,000 units for a Rs 20 crore commitment — and computes NAV per unit to four decimal places. The unit price is a whole number and the unit is indivisible; only the value is carried to four decimals. An AIF investor does not get fractional units; the theory assumes they do.

The investor and the amounts are illustrative; the risk-free rate, the risky return, the weights, the resulting 7.10% and 3% and the Rs 1,000 unit price are the workbook's.

Mr Prakash Menon has Rs 10,00,000 and follows the workbook's own illustration: 70% in the risk-free asset at 5%, 30% in a risky portfolio expected to return 12% with a standard deviation of 10%.

E(Rport) = (70% x 5%) + (30% x 12%) = 3.50% + 3.60% = 7.10%
Portfolio risk = 30% x 10% = 3%

So Rs 7,00,000 risk-free and Rs 3,00,000 in the risky portfolio. The theory treats that Rs 3,00,000 as a clean 30% slice.

Now buy it. The risky portfolio is accessed through a fund whose units cost Rs 1,000 each, as the workbook's Fund SER units do.

Rs 3,00,000 / Rs 1,000 = 300 units exactly

No problem. Divisibility never bites.

Change one number. Mr Menon has Rs 10,45,500 and wants the same 30%:

30% x Rs 10,45,500 = Rs 3,13,650
Rs 3,13,650 / Rs 1,000 = 313.65 units

He cannot buy 313.65 units. He buys 313 or 314.

The gap is Rs 650 on Rs 10,45,500, and 0.0062 percentage points of portfolio risk. Expected return moves by about 0.0044 percentage points — roughly Rs 46 a year.

That is the workbook's point made in rupees. The assumption is false, and being false costs Rs 46. Little impact on the theoretical models.

Where it would matter. Scale the same rounding error to a Rs 500 crore fund rebalancing weekly and the cost stops being Rs 46 — which is why the assumption is convenient rather than true.

Chapter 3, section 3.11 (Assumptions of Capital Market Theory and the implication of relaxing these assumptions), lists infinite divisibility as one of eight assumptions, and section 3.12 immediately uses it to draw the Capital Market Line. Chapter 3 is where the paper's theory marks are.

The reliable question is not the definition but the comparison: which assumption can be relaxed and which cannot. Infinite divisibility, homogeneous expectations, risk-free borrowing and the absence of taxes and transaction costs are all described as relaxable with limited consequences. Capital markets in equilibrium cannot be relaxed as such — that is the answer the question is usually hunting.

• It is a Capital Market Theory assumption. All of MPT's assumptions carry over into CMT, and CMT adds its own. Infinite divisibility appears in the CMT list in section 3.11.
• Relaxing it has little impact. Do not answer that the theory collapses without it. The assumption that cannot be relaxed is market equilibrium.
• It means fractional units, not liquidity. Infinite divisibility is about buying part of a unit. It says nothing about whether you can find a buyer.
• Divisibility is a different idea. That page, written from another paper, is about whether an asset can be liquidated in part — real estate cannot, mutual fund units can. It is an attribute of an asset class. Infinite divisibility is a modelling assumption about fractions.
• The purpose is continuity. The workbook's stated reason is that the assumption lets investment alternatives be discussed as continuous curves — which is what makes the efficient frontier and the Capital Market Line drawable.
• Do not confuse it with the no-transaction-cost assumption. They sit in consecutive bullets and are separately relaxable, and the workbook grades them differently: divisibility has little impact, while removing taxes and transaction costs modifies the results without changing the basic theory.