Wealth relative
The ending value of one unit of money over a period — 1 plus the period's return in decimals. Multiplying wealth relatives chain-links sub-period returns into TWRR or geometric return.
In plain language
A return of +8% means ₹1 became ₹1.08. A return of −5% means ₹1 became ₹0.95. Those ending values — 1.08 and 0.95 — are wealth relatives.
The workbook defines a wealth relative as the ending value of one unit of money. You create it by adding 1 to the period's return in decimal form.
Why bother? Because returns for successive periods cannot simply be added. They compound. Multiplying wealth relatives links sub-period returns correctly, giving the cumulative wealth relative — what one rupee has grown to over the whole period.
How it works
The chain-linking steps (Chapter 20, sections 20.2.2 and 20.2.7):
- Calculate each sub-period's return.
- Convert each into a wealth relative: 1 + return.
- Multiply them to get the cumulative wealth relative.
- Raise it to 1/n (n = number of years) and subtract 1 for the annualised rate.
Where it is used:
| Measure | Role of wealth relatives |
|---|---|
| Time-weighted rate of return (TWRR) | Links sub-period returns between external cash flows |
| Geometric mean return | Same process — the workbook says TWRR is the same as geometric return |
| Annualised return | Product of wealth relatives raised to 1/n |
The workbook's TWRR example (Table 20.7). Returns for 2015–2019 of −5%, −15.2%, 8.1%, 30.75% and 17.65% give wealth relatives of 0.95, 0.848, 1.081, 1.3075 and 1.1765.
Cumulative = 0.95 × 0.848 × 1.081 × 1.3075 × 1.1765 = 1.3396. One rupee became ₹1.3396 in five years.
TWRR = 1.3396^(1/5) − 1 = 6.021%.
The formula
Wealth relative (period t) = 1 + r_t (r_t in decimals)
Cumulative wealth relative = (1 + r_1) × (1 + r_2) × … × (1 + r_n)
Annualised (geometric) return = (Cumulative wealth relative)^(1/n) − 1
A worked example
Illustrative figures. Anita's PMS account returned the following in three years:
| Year | Return | Wealth relative |
|---|---|---|
| 1 | +20% | 1.20 |
| 2 | −10% | 0.90 |
| 3 | +12% | 1.12 |
Cumulative wealth relative = 1.20 × 0.90 × 1.12 = 1.08 × 1.12 = 1.2096
On ₹25 lakh invested at the start (with no further flows), the account is worth 25 × 1.2096 = ₹30.24 lakh.
Annualised return = 1.2096^(1/3) − 1 ≈ 1.0655 − 1 = 6.55%. Check: 1.0655³ ≈ 1.2096.
The wrong way: adding the returns (20 − 10 + 12 = 22%) suggests ₹30.5 lakh, and averaging them (22 ÷ 3 = 7.33%) overstates the annual rate. Only the product of wealth relatives gives the right ₹30.24 lakh.
Why NISM asks about it
Chapter 20 (Performance Measurement and Evaluation of Portfolio Managers) uses wealth relatives in the TWRR calculation (section 20.2.2, Table 20.7), the geometric mean return (section 20.2.3) and annualising returns (section 20.2.7). Performance questions on TWRR, GMR and annualised return all require this step.
Common exam traps
- Wealth relative = 1 + return in decimals. A −15.2% year has a wealth relative of 0.848, not −0.152 or 1.152.
- Multiply, do not add, to link periods.
- Raise to 1/n, then subtract 1 — forgetting the final "− 1" gives a wealth relative, not a return.
- The workbook's annualised-return example has a slip. It computes 1.1550 × 1.0950 × 0.9310 = 1.1775, then 1.1775^(1/3) = 1.0559, and writes the answer as "0.0559 or 5.56%". 0.0559 is 5.59%, and the exact figure is about 5.60%. Use the method; expect answer options close to 5.6%.
- Only annualise periods of at least a year. The workbook says it is inadvisable to annualise a measurement period shorter than a year.
Where this is taught
Free preparation for NISM Series XXI-BRelated terms
- Annualised returnA return scaled up or down to a one-year basis so that investments held for different lengths of time can be compared — a simple scaling that, unlike CAGR, ignores compounding.
- Holding period returnThe total of coupons, income earned on reinvesting them and any capital gain, expressed as a percentage of the purchase price — a crude return for the period actually held, with no compounding in it.
- Geometric Mean ReturnThe single constant yearly rate that would turn the starting value into the ending value — found by chain-linking period returns and taking the nth root.
- Money Weighted Rate of ReturnThe internal rate of return on an investor's actual cash flows into and out of a portfolio — so it rewards or punishes the timing of those flows, not just the manager's skill.
- Time Weighted Rate of ReturnA compounded return that neutralises client deposits and withdrawals by linking sub-period returns, so it measures the manager’s decisions. Mandatory for PMS investment approach performance.