CISI IRT Return and Risk Calculations: A Worked Practice Set

Table of Contents

• • Use the six-step calculation routine
• • 1. Holding-period return
• • 2. Exact real return
• • 3. Arithmetic versus geometric return
• • 4. Time-weighted return around a contribution
• • 5. Select the correct risk-adjusted ratio
• • 6. CAPM required return and Jensen alpha
• • Formula-selection map
• • How to review a wrong calculation

The difficult part of many CISI Investment, Risk and Taxation calculations is not arithmetic. It is identifying what the question asks you to measure, selecting the correct denominator and keeping cash flows, periods and percentages consistent.

This workbench uses six connected examples from the current IRT calculation set. Do each example without the answer first, then explain what the result means. A number without an interpretation is fragile knowledge.

Use the six-step calculation routine

Before touching the onscreen calculator:

1. Name the measure. Is the question asking for wealth growth, purchasing-power growth, manager performance or risk-adjusted performance?
2. Mark external cash flows. Contributions and withdrawals can change the required method.
3. Align periods and units. Annual return must be compared with annual risk; percentages should be converted consistently.
4. Write the formula. This exposes denominator errors before they become calculator errors.
5. Estimate the direction. A positive nominal return above inflation should normally produce a positive but smaller real return.
6. Calculate and interpret. State the answer with its unit and one sentence explaining its meaning.

1. Holding-period return

An investment starts at £5,000, ends at £5,250 and pays £150 of income separately during the period.

Holding-period return = (ending value − starting value + income) ÷ starting value

The total gain is:

£5,250 − £5,000 + £150 = £400

The return is:

£400 ÷ £5,000 = 8%

The key phrase is “paid separately.” If the ending value already included reinvested income, adding it again would double count the same benefit.

If a comparable benchmark returned 6%, the arithmetic excess return is 2 percentage points. That does not automatically make it alpha; Jensen alpha requires a risk-adjusted CAPM comparison.

2. Exact real return

Suppose the nominal return is 8% and inflation is 5% over the same year.

Real return = (1 + nominal return) ÷ (1 + inflation) − 1

1.08 ÷ 1.05 − 1 = 2.86%

Subtracting 5% from 8% gives 3%, which is a useful mental approximation but not the exact answer. The exact method compares growth factors because both changes apply to the same starting purchasing power.

3. Arithmetic versus geometric return

An investment returns +20% in year one and −10% in year two.

The arithmetic average is:

(20% − 10%) ÷ 2 = 5%

The two-year wealth factor is:

1.20 × 0.90 = 1.08

The geometric annual return is:

√1.08 − 1 = 3.92% per year

Why are the answers different? The arithmetic mean averages observations. The geometric mean finds the constant annual return that would turn the opening wealth into the actual closing wealth. After volatile returns, the geometric rate is normally lower because losses reduce the capital base on which recovery occurs.

4. Time-weighted return around a contribution

A portfolio starts at £10,000 and grows to £11,000. The investor then contributes £2,000, after which the portfolio finishes at £14,300.

Split the calculation at the external cash flow.

First subperiod:

£11,000 ÷ £10,000 − 1 = 10%

Second subperiod opening value after the contribution:

£11,000 + £2,000 = £13,000

Second subperiod:

£14,300 ÷ £13,000 − 1 = 10%

Link the subperiod growth factors:

(1.10 × 1.10) − 1 = 21% TWRR

Do not treat the £2,000 contribution as investment profit. Do not add the two 10% returns to get 20%; time-weighted returns compound.

TWRR is designed to reduce the effect of investor-controlled cash-flow timing. A money-weighted measure answers a different question because it reflects the timing and size of the investor’s actual cash flows.

5. Select the correct risk-adjusted ratio

Assume these annual figures:

• portfolio return: 10%;
• risk-free return: 2%;
• standard deviation: 16%;
• beta: 0.8;
• target return: 4%;
• supplied target-downside deviation: 6%;
• benchmark return: 7%; and
• tracking error: 5%.

Measure	Formula	Result	What the denominator represents
Sharpe	(10% − 2%) ÷ 16%	0.50	Total volatility
Treynor	(10% − 2%) ÷ 0.8	0.10	Systematic risk measured by beta
Sortino	(10% − 4%) ÷ 6%	1.00	Supplied downside-risk measure
Information ratio	(10% − 7%) ÷ 5%	0.60	Tracking error relative to benchmark

The numerator and denominator must belong together. Standard deviation cannot be silently substituted for tracking error, and beta cannot be used merely because it is present in the question.

Also preserve units. Treynor’s 0.10 can be described as 10% excess return per unit of beta under the stated inputs; it is not directly comparable with a Sharpe ratio whose denominator is return volatility without understanding the conventions used.

6. CAPM required return and Jensen alpha

The risk-free return is 3%, the market return is 9%, beta is 1.2 and the portfolio actually returned 12%.

First calculate the market risk premium:

9% − 3% = 6%

Then calculate the CAPM-required return:

3% + 1.2 × 6% = 10.2%

Jensen alpha is:

12% − 10.2% = 1.8 percentage points

The most common structural mistake is multiplying beta by the market return rather than by the market risk premium. CAPM first removes the risk-free component from the market return, scales the remaining premium by beta and then adds the risk-free return back.

Formula-selection map

If the question asks…	Start with…	Main trap
How much the holding gained, including income	Holding-period return	Double-counting income
How purchasing power changed	Exact real return	Using subtraction when an exact answer is required
The constant compound rate over several periods	Geometric return	Reporting the arithmetic mean
Manager performance despite investor cash flows	TWRR	Treating contributions as gains
Return per unit of total volatility	Sharpe ratio	Using beta instead of standard deviation
Return per unit of systematic risk	Treynor ratio	Using standard deviation instead of beta
Return above a target per unit of downside risk	Sortino ratio	Using the wrong downside definition
Active return per unit of relative risk	Information ratio	Using portfolio volatility instead of tracking error
Performance above CAPM requirement	Jensen alpha	Confusing benchmark excess return with alpha

How to review a wrong calculation

Label the failure before repeating the question:

• measure error: you answered a different question;
• formula error: the wrong relationship was selected;
• cash-flow error: a contribution, withdrawal or income payment was mishandled;
• unit error: periods, pounds, decimals or percentages were inconsistent;
• input error: the correct formula received the wrong number;
• interpretation error: the arithmetic was correct but the conclusion was not.

That diagnosis makes revision efficient. Ten more calculations will not fix a recurring formula-selection error unless you practise identifying the measure first.

The companion CISI Investment, Risk and Taxation study plan covers all eight syllabus elements and the current Candidate Update. Use the official CISI Investment Advice Diploma page to confirm the qualification route and check the live Candidate Update page for corrections applying to your sitting.

The reliable exam habit is: name the measure, control the cash flows, match the denominator, calculate and interpret.