The Tangency Portfolio and the Capital Market Line

Suppose you have done the hard work and drawn the The Efficient Frontier for a set of risky assets. You now face an awkward question: which point on that curve is yours? A cautious saver wants the left end, an aggressive one wants the right end, and there seems to be no way to say either is wrong. Every point is defensible, so the frontier gives you a menu rather than an answer.

Then someone hands you a savings account paying a known rate with no risk at all. Something surprising happens: the menu collapses. There is now exactly one mix of risky assets worth owning, and the only remaining decision is how much of your money to put into it versus leave in the savings account. Cautious and aggressive investors hold the same basket, just in different amounts.

An analogy before any symbols

Think of a juice bar that sells one concentrate and unlimited water. The barista's job is to get the concentrate blend exactly right, the strongest, best-tasting syrup possible. Your job is to decide how much water to add. A child gets a weak dilution, an adult gets it neat, and someone reckless asks for it double-strength. Nobody argues about the recipe, only about the dilution.

The tangency portfolio is the concentrate: the single best-tasting blend of risky assets. Cash is the water. Borrowing to invest more than 100% of your money is asking for it double-strength. And "best-tasting" has a precise meaning here, it is the blend with the highest reward per unit of risk.

What "best" means

Reward per unit of risk is the Sharpe Ratio. For a portfolio with weights $w$ (a list saying what fraction of your money sits in each asset), expected returns $\mu$, covariance matrix $\Sigma$ (the table of how the assets move together), and a risk-free rate $r_f$, it is

$\text{SR}(w) = \frac{w^\top \mu - r_f}{\sqrt{w^\top \Sigma w}}.$

In words: the top is how much more than cash you expect to earn, and the bottom is how much the portfolio bounces around. Divide one by the other and you get "extra return earned per unit of wobble endured". Higher is better.

Maximising that ratio over all fully-invested weight vectors gives the tangency portfolio:

$w_{\text{tan}} = \frac{\Sigma^{-1}(\mu - r_f\mathbf{1})}{\mathbf{1}^\top \Sigma^{-1}(\mu - r_f\mathbf{1})}.$

Read it in two halves. The top, $\Sigma^{-1}(\mu - r_f\mathbf{1})$, is "each asset's edge over cash, discounted for how much risk it brings and how much it duplicates what the other assets already give you". The bottom is just the sum of those numbers, dividing by it rescales the weights so they add to one. The whole expression says: tilt toward assets with a big edge, low volatility, and low overlap with everything else.

The capital market line

Now mix the tangency portfolio with cash. Put a fraction $a$ of your money into the tangency portfolio and leave $1-a$ in the savings account. Because cash has zero risk, the combination's volatility is simply $a$ times the tangency portfolio's volatility, and its expected return is $r_f$ plus $a$ times the tangency portfolio's excess return. Both scale with the same $a$, so as you slide $a$ from 0 upward you trace a perfectly straight line:

$\mathbb{E}[R_p] = r_f + \underbrace{\frac{\mathbb{E}[R_{\text{tan}}] - r_f}{\sigma_{\text{tan}}}}_{\text{slope}} \cdot \; \sigma_p.$

In words: start at the risk-free rate, and every extra unit of volatility you take on buys you a fixed amount of extra expected return. That fixed amount, the slope, is the tangency portfolio's Sharpe ratio, and it is the best exchange rate available anywhere in the market. This line is the capital market line (CML).

Notice why the line beats the curve. Any point on the curve is achievable, but the line lies above it at every risk level except one. So for any amount of risk you are willing to take, mixing cash with the tangency portfolio beats the best pure-risky-asset portfolio at that same risk. The only exception is the touch point itself, where the two coincide.

Here is the underlying curve for two assets, so you can feel where a tangency point would sit. Drag the correlation slider down and watch the frontier bow further left, a lower correlation means the whole curve, and therefore the best line you can draw from the risk-free rate to it, gets steeper:

Worked example: finding the tangency weights by hand

Two assets. A bond fund with expected return 6% and volatility 10%. An equity fund with expected return 11% and volatility 20%. Their correlation is 0.25. Cash pays 3%.

Step 1, excess returns. Subtract the cash rate: the bond fund earns 3% over cash, the equity fund 8%.

Step 2, the covariance matrix. The diagonal entries are the variances, $0.10^2 = 0.01$ and $0.20^2 = 0.04$. The off-diagonal is $0.25 \times 0.10 \times 0.20 = 0.005$:

$\Sigma = \begin{pmatrix} 0.010 & 0.005 \\ 0.005 & 0.040 \end{pmatrix}.$

Step 3, invert it. For a $2\times2$ matrix, swap the diagonal, flip the sign of the off-diagonal, divide by the determinant. The determinant is $0.010 \times 0.040 - 0.005^2 = 0.000375$, so

$\Sigma^{-1} = \frac{1}{0.000375}\begin{pmatrix} 0.040 & -0.005 \\ -0.005 & 0.010 \end{pmatrix} = \begin{pmatrix} 106.67 & -13.33 \\ -13.33 & 26.67 \end{pmatrix}.$

Step 4, multiply by the excess returns $(0.03,\ 0.08)$:

• bond row: $106.67 \times 0.03 - 13.33 \times 0.08 = 3.200 - 1.067 = 2.133$
• equity row: $-13.33 \times 0.03 + 26.67 \times 0.08 = -0.400 + 2.133 = 1.733$

Step 5, normalise. They sum to $3.867$, so the tangency weights are $2.133/3.867 = 0.552$ in bonds and $1.733/3.867 = 0.448$ in equities. Roughly 55% bonds, 45% equities.

Step 6, check it is actually better. That portfolio expects $0.552 \times 6\% + 0.448 \times 11\% = 8.24\%$, and its volatility works out to 11.6%. Its Sharpe ratio is $(8.24 - 3)/11.6 = 0.45$. Compare: the bond fund alone scores $(6-3)/10 = 0.30$, the equity fund alone $(11-3)/20 = 0.40$. The blend beats both, which is exactly what the tangency portfolio is supposed to do.

Worked example: turning the dial

You now hold the concentrate. How much water?

Case 1, you want 8% volatility. The tangency portfolio has 11.6% volatility, so put $8/11.6 = 0.69$ of your money in it and leave 31% in cash. Expected return: $3\% + 0.69 \times (8.24\% - 3\%) = 3\% + 3.6\% = 6.6\%$. Sharpe ratio: $(6.6-3)/8 = 0.45$. Identical to before, exactly as the straight line promised, diluting does not change the exchange rate.

Case 2, you want 18% volatility. Now $18/11.6 = 1.55$, so you invest 155% of your money in the tangency portfolio, borrowing the extra 55% at the cash rate. Expected return: $3\% + 1.55 \times 5.24\% = 11.1\%$.

Look at what that means. The equity fund on its own gives 11% at 20% volatility. The levered tangency portfolio gives 11.1% at 18% volatility, more return for less risk. That gap is the entire practical payoff of this theory: instead of reaching for a racier asset to get more return, you take a well-diversified blend and lever it.

What this means in practice

• Index-plus-cash is the retail version. "80/20 stocks and bonds, rebalanced" is a crude tangency portfolio with a fixed dilution. The theory says the split between the risky blend and cash is a personal choice; the blend itself is not.
• Risk parity and vol targeting are the same move. Both build a diversified portfolio with a modest natural volatility and then lever it to hit a return target, rather than concentrating into high-return assets. See Risk Parity and Vol Targeting.
• It is the bridge to asset pricing. If everyone holds the same risky portfolio, and markets clear, that portfolio must be the market itself. Follow that one step and you have the The Capital Asset Pricing Model (CAPM).
• The formula is a cousin of bet sizing. $\Sigma^{-1}(\mu - r_f\mathbf{1})$ is the same object that appears in the multi-asset The Kelly Criterion, just normalised differently. Both say: size by edge, divided by risk, adjusted for overlap.

Practice

1. Redo the worked example with correlation $0.75$ instead of $0.25$. The bond fund's diversification value falls, so its weight should drop. By how much?
2. Set both assets' Sharpe ratios equal (say the bond fund earns 7% instead of 6%). Does the tangency portfolio become 50/50? Explain why not, in one sentence, using the covariance term.
3. If cash paid 5% instead of 3%, which asset's excess return shrinks proportionally more, and which way does the tangency portfolio tilt?
4. An investor can lend at 3% but only borrow at 6%. Sketch what the capital market line looks like now. (Hint: two straight segments and a piece of the original curve between them.)

In interviews

Be able to state the formula, explain the two-fund separation result in a sentence, and answer the standard follow-up: "why does adding a risk-free asset turn the curved frontier into a straight line?" The answer is that you are now blending a point with zero risk against a single portfolio, and blending two things is linear in both risk and return, so the set of achievable combinations is a line segment. Then note that the line's slope is the maximum Sharpe ratio available, which is what makes the touch point special.