CII J10 Discretionary Investment Management · Lesson 6 of 14

Two assets, one portfolio

Two investments that do not move together are safer together. This lesson shows why, with sliders, a worked example and the code that does the sum.

An example lesson built by Chalkwork Labs from public sources, to show the shapes. Your course is drafted from your material and checked by you.

Syllabus: J10 learning outcome 8, Understand the principles and limitations of modern portfolio theory (8.1). It also touches outcome 7 (returns and risk) and 14.2 (apply portfolio theory to construct portfolios).

1. Risk means the return wobbles

An investment's return is what it earns in a year, as a percentage of what you paid. Returns are not fixed. A good year and a bad year give different numbers.

The expected return is the average return you would see over many years.
The volatility is how far a typical year lands from that average. It is the standard deviation of the yearly returns.

In this unit, risk means volatility. A high-volatility asset has wide swings. A low-volatility asset stays close to its average. This lesson uses two illustrative asset classes:

Asset	Expected return (% a year)	Volatility (% a year)
(asset A)
(asset B)

These numbers are illustrative. They are plausible teaching values, not a forecast. The maths works the same for any two assets.

2. Why two assets that move differently are safer together

Think of two shops on one street: an ice-cream stall and an umbrella stall. In a hot summer one does well and the other does badly. Together, their combined takings are steadier than either alone.

The correlation measures how much two assets move together. It runs from +1 (always move the same way) through 0 (no link) to −1 (always move in opposite ways). Each dot below is one simulated year.

Forty simulated years for each correlation setting. Each dot is one year: the return of equities across, the return of gilts up. Illustrative, generated from the table above.

At +0.9 the dots form a tight diagonal: a bad year for one is a bad year for both. At −0.7 the diagonal runs the other way: one asset's bad year tends to be the other's good year. At +0.2, the setting this lesson uses, there is only a weak link.

When two assets do not move together, their swings partly cancel. The mix wobbles less than you would expect from the two volatilities alone. That cancelling is diversification.

3. Be the portfolio manager

A portfolio is a mix of investments. The weight of an asset is its share of the money. With two assets, the weight in A fixes the weight in B: they add to 1.

The chart plots every mix of A and B. Each point is one weight. Across is volatility, up is expected return. Move the sliders and watch the curve bend.

Weight in equities (A):

Correlation:

Volatility of equities: %

–

expected return (% a year)

–

volatility (% a year)

–

weighted-average volatility

–

minimum-variance weight in A

The curve is every mix from 100% gilts to 100% equities. The large dot is your mix. The hollow dot is the mix with the lowest volatility.

Three things to try:

Correlation to +1. The curve becomes a straight line. Volatility equals the weighted average of the two volatilities. Diversification does nothing.
Correlation to −1. The curve bends all the way to the axis. One mix has zero volatility.
Correlation back to 0.2, then slide the weight from 0 to 1. Watch the volatility readout. It falls as you add the first few percent of equities, then rises. Adding a riskier asset made the mix safer.

The mix with the lowest volatility is the minimum-variance portfolio. The part of the curve above it is the efficient frontier: for each level of risk, the highest return you can get. Mixes below it are never worth holding, because another mix gives more return for the same risk.

4. Add a risk-free asset: the capital allocation line

Now add a third choice: cash on deposit, or a short-dated government bill. Call its return the risk-free rate. Its volatility is zero, so it sits on the vertical axis.

Mix the risk-free asset with any risky portfolio and you move along a straight line from the axis to that portfolio. The steepest line you can draw from the risk-free point to the curve is the capital allocation line. It touches the curve at one mix, the tangency portfolio.

The slope of that line is the Sharpe ratio: the extra return above the risk-free rate, per unit of volatility. Higher is better. Move the sliders and watch the slope.

Risk-free rate: %

Correlation:

–

tangency weight in equities

–

tangency return (% a year)

–

tangency volatility (%)

–

Sharpe ratio of the tangency mix

–

Sharpe ratio, equities alone

Volatility of equities is fixed at % here. The dashed line starts at the risk-free rate and touches the curve at the tangency mix.

Two things to notice:

Lower correlation, steeper line. At correlation 0.2 the tangency Sharpe ratio is . At −0.5 it is . At +1 it is , the same as equities alone, because diversification adds nothing.
A higher risk-free rate moves the tangency mix towards equities. Gilts earn less above cash, so they contribute less to the Sharpe ratio.

Every investor who accepts this model should hold the same tangency mix and vary only how much cash sits beside it. That is the idea behind the one-fund result in portfolio theory. Its limits are real: the inputs are estimates, correlations change in a crisis, and the model treats upside and downside swings the same.

5. Worked example

Take % in equities and % in gilts, with correlation . The formulas, from the portfolio-theory sources below:

E(R_p) = w_A × μ_A + w_B × μ_B
σ_p² = w_A² σ_A² + w_B² σ_B² + 2 w_A w_B ρ σ_A σ_B

Expected return. = %. A plain weighted average.
Variance, term by term. Equities term . Gilts term . Cross term . Total .
Volatility. The square root of is %.
Compare. The weighted average of the two volatilities is %. Diversification saved points of volatility, for the same expected return.
Minimum-variance mix. w* = (σ_B² − ρ σ_A σ_B) ÷ (σ_A² + σ_B² − 2 ρ σ_A σ_B) = . About % in equities gives volatility %, below gilts alone at %.

The same 60/40 mix under four correlations:

Correlation	Expected return (%)	Volatility (%)	Minimum-variance weight in A

The return column never moves. Correlation changes only the risk. That is the whole lesson in one table.

6. What this means for a discretionary manager

A discretionary manager buys and sells without asking the client each time. The client's mandate sets the limits: the objective, the assets allowed, and how much risk the portfolio may carry. The client's risk profile is the level of volatility they can accept and afford.

The mandate is a point on the frontier. A cautious profile sits near the minimum-variance mix. An adventurous profile sits further up the curve. The manager's job is to stay on the frontier, not below it.
Volatility is a limit, not a target. If the modelled volatility breaches the profile, the manager cuts the risky weight. Adding more of a risky asset to chase return is a breach, not a strategy.
Correlations are inputs, and they drift. A 60/40 mix at correlation 0.2 has volatility %. At correlation 1 it has %. In a crisis correlations often rise. A prudent mandate review re-checks them.
Report in the client's words. "Your portfolio should move about 10% in a typical year" means more to a client than a Sharpe ratio. Both belong in the review, the plain one first.

Quick check

CII style: one correct answer in four.

1. A manager holds 60% equities and 40% gilts. The correlation between them falls from +1 to 0. The weights do not change. What happens to the portfolio?

2. Two assets have a correlation of exactly +1. The portfolio's volatility is:

3. A client's risk profile caps portfolio volatility at 9% a year. The manager's 60/40 mix has a modelled volatility of %. Which action is most consistent with the mandate?

4. The risk-free rate is %. A portfolio has an expected return of % and volatility %. Its Sharpe ratio is closest to:

7. Read the code

Show the working in code

Optional. The same calculation as Python you can step through.

The block below is real Python that runs in this page. You will not write or edit it. Read the steps on the left, then press Next step to run one step at a time. The panel shows the variables each step made.

Three bits of Python you will see:

x = ... stores a result under the name x, so later lines can use it.
w_a**2 means w_A squared. variance ** 0.5 is the square root.
print(...) shows a result in the output box.

Block 1: Portfolio variance, one term at a time

Press Next step five times. Check each printed number against the worked example in section 5.

Set the illustrative inputs: weights, expected returns, volatilities, correlation.
Expected return: weight times return, added up.
The three variance terms: equities, gilts, and the cross term.
Add the terms, then take the square root.
Compare with the plain weighted average of the volatilities.

# Illustrative inputs, in percent per year
w_a = 0.6          # weight in UK equities (A)
w_b = 1 - w_a      # weight in UK gilts (B)
mu_a, mu_b = 7.0, 4.0
sigma_a, sigma_b = 16.0, 7.0
rho = 0.2

# Expected return: the weighted average
exp_return = w_a * mu_a + w_b * mu_b
print("expected return:", round(exp_return, 2), "%")

# Variance, one term at a time
term_a = w_a**2 * sigma_a**2
term_b = w_b**2 * sigma_b**2
term_ab = 2 * w_a * w_b * rho * sigma_a * sigma_b
terms = round(term_a, 3), round(term_b, 3), round(term_ab, 3)
print("terms:", *terms)

# Add the terms, then take the square root
variance = term_a + term_b + term_ab
volatility = variance ** 0.5
print("variance:", round(variance, 3))
print("volatility:", round(volatility, 2), "%")

# Compare with the plain weighted average
naive = w_a * sigma_a + w_b * sigma_b
saved = round(naive - volatility, 2)
print("weighted average:", round(naive, 2), "%")
print("saved by diversification:", saved, "points")

The block prints variance 110.752 and volatility 10.52. Why take the square root?

What to remember

Risk in this unit means volatility: the standard deviation of yearly returns.
Portfolio expected return is a weighted average. Correlation does not change it.
Portfolio variance has three terms. Only the cross term holds the correlation.
Correlation below +1 makes volatility lower than the weighted average. That is diversification.
The minimum-variance mix can be safer than the safest single asset.
Add a risk-free asset and the best risky mix is the tangency portfolio, with the highest Sharpe ratio.
For a discretionary manager, the risk profile is a cap on volatility. Stay on the frontier, under the cap.
The model's limits: inputs are estimates, correlations drift, and up and down swings count the same.

Sources

Chartered Insurance Institute, J10 Discretionary investment management examination syllabus, published June 2023 (examined 1 September 2023 to 31 August 2024). Learning outcome 8: "Understand the principles and limitations of modern portfolio theory"; 8.1: "Explain the fundamental principles and limitations of modern portfolio theory." Also outcome 7, "Understand how investment returns are related to investment risk", and 14.2, "Apply the principles of portfolio theory in order to construct and/or evaluate portfolios." cii.co.uk, J10 unit page. Check the current edition's wording before an exam sitting.
Markowitz, H., "Portfolio Selection", The Journal of Finance, vol. 7, no. 1 (March 1952), pp. 77–91. doi.org/10.1111/j.1540-6261.1952.tb01525.x. The original paper for portfolio expected return and variance, diversification and the efficient frontier.
Sharpe, W. F., "Mutual Fund Performance", The Journal of Business, vol. 39 (January 1966). doi.org/10.1086/294846. The original paper for the reward-to-variability ratio, now called the Sharpe ratio.
Tobin, J., "Liquidity Preference as Behavior Towards Risk", The Review of Economic Studies, vol. 25, no. 2 (February 1958). doi.org/10.2307/2296205. The original paper for mixing a risk-free asset with one risky portfolio.
The two asset classes, their expected returns, volatilities and correlation are illustrative values chosen for teaching. They are not a forecast and not drawn from market data. Every number on this page is computed in export_09.py.

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