The capital market line

Every portfolio so far has been built from risky assets. Adding one riskless asset changes the answer more than any other single step in the subject.

What a riskless asset does

A riskless asset has a known return $r_f$ and, by definition, zero standard deviation — and therefore zero covariance with everything.

Put weight $a$ in some risky portfolio $P$ and $1-a$ in the riskless asset. From chapter 2's formulas, with $\sigma_f = 0$ and $\sigma_{fP} = 0$:

$E(R) = a\,\mu_P + (1-a)\,r_f = r_f + a\left(\mu_P - r_f\right)$

$\sigma = a\,\sigma_P$

Both are linear in $a$. Eliminate it:

$E(R) = r_f + \underbrace{\frac{\mu_P - r_f}{\sigma_P}}_{\text{slope}} \times \sigma$

A straight line, starting at $r_f$ on the vertical axis and running through the point $P$. Every mixture of cash and $P$ lies on it.

That is the whole mechanism. Combining anything with a riskless asset produces a line rather than a curve, because the thing that bends the curve — covariance — is zero.

Which line to pick

Every risky portfolio generates its own line from $r_f$. The investor wants the highest one, since a steeper line gives more return at every level of risk.

Rotate the line upward from $r_f$ until it just touches the efficient frontier. Beyond that it leaves the achievable region. The portfolio at the point of contact is the tangency portfolio, and the line is the capital market line.

Its slope has a name you already know:

$\text{slope} = \frac{\mu_P - r_f}{\sigma_P}$

That is the Sharpe ratio. So maximising the Sharpe ratio and finding the tangency portfolio are the same operation — which is why chapter 9 is a continuation of this one rather than a separate topic.

Two-fund separation

Now the consequence, and it is the strongest result in the subject.

Every investor holds some point on the capital market line. Every point on it is a mix of the riskless asset and the same tangency portfolio. So:

All investors hold the identical portfolio of risky assets. The cautious and the aggressive hold the same proportions of the same things. They differ only in how much cash they hold alongside it.

Sharpe states it directly, crediting Tobin:

every investor will hold a combination of the riskless asset and the market portfolio — and hence all investors will hold risky assets in the same proportions.

This is called two-fund separation, and it splits the investment problem cleanly in two:

Question	Answer depends on
Which risky portfolio?	the assets — same for everybody
How much of it?	the investor — different for everybody

Why it matters practically. The common advice to "hold safer assets as you get older" is often implemented by changing which equities you hold — moving towards defensive sectors, large caps, dividend payers. Two-fund separation says that is the wrong lever. Hold the same risky portfolio and hold less of it.

And whether that is right in practice is a real question, because it depends on assumptions chapter 12 takes apart. But it is a sharper claim than the advice it contradicts, and it is worth knowing which one you are following.

The arithmetic

Equity at 11% and 14%, debt at 7% and 4%, correlation 0.2, and cash at 5.5% — the RBI's current policy repo rate, used here as a stand-in for a riskless rate.

The tangency portfolio turns out to be 23.45% equity, 76.55% debt, returning 7.938% with 4.917% risk, for a Sharpe ratio of 0.496.

Now mix it with cash:

Cash	Tangency portfolio	Expected return	Risk	Sharpe
75%	25%	6.110%	1.229%	0.496
50%	50%	6.719%	2.458%	0.496
25%	75%	7.329%	3.688%	0.496
0%	100%	7.938%	4.917%	0.496
−25%	125%	8.548%	6.146%	0.496
−50%	150%	9.157%	7.375%	0.496

The Sharpe ratio is identical in every row. That is what being on one straight line through $r_f$ means — and it is why chapter 9 can treat the ratio as a property of a strategy rather than of a position size.

The negative-cash rows are borrowing. Weight above 100% in the risky portfolio means leverage, which is chapter 1's negative weight.

How much better is the line than the curve?

Compare the capital market line against chapter 4's curve, at each level of risk:

Equity weight on the curve	Curve return	Risk	Line return at that risk	Line better by
0.0	7.000%	4.000%	7.483%	+0.483
0.2	7.800%	4.654%	7.808%	+0.008
0.4	8.600%	6.519%	8.732%	+0.132
0.6	9.400%	8.860%	9.893%	+0.493
0.8	10.200%	11.387%	11.146%	+0.946
1.0	11.000%	14.000%	12.442%	+1.442

The line is above the curve everywhere except at one point, where they touch — the tangency portfolio, at about 23% equity. That single point is the only place the curve is as good as the line.

And the gap widens at both ends. At the top it reaches 1.44 percentage points: an investor willing to bear 14% volatility can have 12.44% expected return on the line against 11.00% from holding equities outright. Same risk, 1.44 points more return, which over a long horizon is the difference between outcomes.

Why you probably cannot have the top of the line

The honest part, and it is not a detail.

The 12.442% row requires borrowing at 5.5%. To reach 14% risk on the line you hold 2.85 times the tangency portfolio, funded by borrowing 1.85 times your capital at the riskless rate.

Nobody lends to individuals at the policy repo rate. The RBI's rate corridor runs from a standing deposit facility rate of 5.25% to a marginal standing facility rate of 5.75%, and those are rates for banks at the central bank, not for households at banks. An individual borrows at well above the rate they can lend at, which means the real picture is two different lines — a steeper one for lending, a flatter one for borrowing — joined by a segment of the original curve between them.

So the part of the capital market line most investors can actually use is the lower half, from cash up to the tangency portfolio. That part is genuinely available, and it is where the theory's advice is soundest: to take less risk, hold cash alongside the same risky portfolio rather than changing the risky portfolio.

Above the tangency point, the theory's prescription quietly stops being implementable, and an investor who wants more risk than the tangency portfolio offers has to move along the curve after all — accepting a lower Sharpe ratio, because that is the only route available. That is not a flaw in the mathematics; it is the borrowing assumption failing, and chapter 12 lists it among the assumptions that do.

Working the problem

Someone wants 14% risk.

On the capital market line. The tangency portfolio has 4.917% risk, so reaching 14% needs

$a = \frac{14.000}{4.917} = 2.847$

That is 284.7% of capital in the tangency portfolio, financed by borrowing 184.7% at the riskless rate. The expected return:

$E(R) = 5.5 + 2.847 \times (7.938 - 5.5) = 5.5 + 6.942 = 12.442\%$

Against holding equities outright, which gives 11.000% at exactly the same 14% risk. The line wins by 1.442 percentage points.

Why the leveraged diversified portfolio beats the concentrated one. The tangency portfolio earns 0.496 of excess return per unit of risk; all-equity earns $(11 - 5.5)/14 = 0.393$. Leverage scales risk and excess return together, so it preserves the ratio — and a position built from the better ratio wins at every risk level above zero. Leverage does not improve a portfolio; it lets a better portfolio be scaled up to meet a risk target, which is a different and much more defensible claim than the one leverage is usually sold with.

Why they probably cannot do it — three reasons, in order of severity.

The borrowing rate. At a realistic personal borrowing rate the arithmetic changes sign. If borrowing costs 9% rather than 5.5%, the levered position returns $7.938 \times 2.847 - 9 \times 1.847 = 5.98\%$ for the same 14% risk — worse than simply holding equities. The entire advantage lived in the spread between 5.5% and the actual cost of money, and for a household that spread is negative.

The margin mechanics. A levered position can be closed out against you before the expected return arrives. The Option pricing subject's margin discussion and the Financial institutions subject's forced-selling mechanism both apply: being right on average is worth nothing if the position does not survive to average out.

And the inputs. The 1.442-point advantage is computed from estimates of three numbers. Chapter 5's warning applies in full — the tangency portfolio is the single most estimation-sensitive object in the subject, since it depends on expected returns, which are the least reliably estimated inputs of all.

What I would actually take from this chapter, stripped of the leverage: the lower half of the line is real, free and usable. Holding cash alongside a well-diversified risky portfolio dominates holding a de-risked version of that portfolio, and it requires no borrowing, no margin and no leverage at all.

The point

A riskless asset combines with any risky portfolio along a straight line from $r_f$, because the covariance term that bends the frontier is zero. The highest such line touches the frontier at the tangency portfolio, and its slope is the Sharpe ratio — so maximising the ratio and finding that portfolio are one operation. The consequence is two-fund separation: every investor holds the same risky portfolio and differs only in how much cash sits beside it, which means de-risking should change the amount rather than the contents. The line beats the curve everywhere but the tangency point, by as much as 1.44 points at 14% risk — but the upper half needs borrowing at the riskless rate, which no individual can do, so the usable part runs from cash up to the tangency portfolio.