NISM Series XXI-B Formula Sheet
Ten formulas, what each one is for, and a worked example.
Series XXI-B is for the people who run the portfolios, so it goes deeper into portfolio theory and the measures used to judge active management.
These ten formulas cover construction, risk and performance attribution, with a worked information-ratio example.
The 10 Formulas
1
Formula
Portfolio return
Expression
R_p = Σ wᵢ × Rᵢ
Use it for
Weighted return of the holdings.
2
Formula
Two-asset portfolio variance
Expression
σ²_p = w₁²σ₁² + w₂²σ₂² + 2w₁w₂ρσ₁σ₂
Use it for
Measuring diversification.
3
Formula
Beta
Expression
β = Cov(Rᵢ, R_m) ÷ σ²_m
Use it for
Market sensitivity of a stock or portfolio.
4
Formula
CAPM
Expression
E(R) = R_f + β × (R_m − R_f)
Use it for
Required return for a given beta.
5
Formula
Jensen's alpha
Expression
α = R_p − [R_f + β_p × (R_m − R_f)]
Use it for
Value added above CAPM.
6
Formula
Treynor ratio
Expression
(R_p − R_f) ÷ β_p
Use it for
Return per unit of market risk.
7
Formula
Tracking error
Expression
Standard deviation of (R_p − R_b)
Use it for
How closely the portfolio follows its benchmark.
8
Formula
Information ratio
Expression
(R_p − R_b) ÷ Tracking error
Use it for
Active return per unit of active risk.
9
Formula
Value at Risk (parametric)
Expression
VaR = Portfolio value × z × σ × √t
Use it for
Loss not expected to be exceeded at a confidence level.
10
Formula
Time-weighted return
Expression
(1 + R₁) × (1 + R₂) × … × (1 + Rₙ) − 1
Use it for
Reporting performance independent of client cash flows.
| # | Formula | Expression | Use it for |
|---|---|---|---|
| 1 | Portfolio return | R_p = Σ wᵢ × Rᵢ | Weighted return of the holdings. |
| 2 | Two-asset portfolio variance | σ²_p = w₁²σ₁² + w₂²σ₂² + 2w₁w₂ρσ₁σ₂ | Measuring diversification. |
| 3 | Beta | β = Cov(Rᵢ, R_m) ÷ σ²_m | Market sensitivity of a stock or portfolio. |
| 4 | CAPM | E(R) = R_f + β × (R_m − R_f) | Required return for a given beta. |
| 5 | Jensen's alpha | α = R_p − [R_f + β_p × (R_m − R_f)] | Value added above CAPM. |
| 6 | Treynor ratio | (R_p − R_f) ÷ β_p | Return per unit of market risk. |
| 7 | Tracking error | Standard deviation of (R_p − R_b) | How closely the portfolio follows its benchmark. |
| 8 | Information ratio | (R_p − R_b) ÷ Tracking error | Active return per unit of active risk. |
| 9 | Value at Risk (parametric) | VaR = Portfolio value × z × σ × √t | Loss not expected to be exceeded at a confidence level. |
| 10 | Time-weighted return | (1 + R₁) × (1 + R₂) × … × (1 + Rₙ) − 1 | Reporting performance independent of client cash flows. |
What the Symbols Mean
- R_b
- Benchmark return.
- z
- 1.65 for 95% and 2.33 for 99%, one-tailed.
Free account, this exam preselected.
Worked Example: Information Ratio
A portfolio returned 15% against a benchmark's 12%, with a tracking error of 4%.
- 1
Active return
15% − 12% = 3%.
- 2
Divide by active risk
3 ÷ 4 = 0.75.
- 3
Read it
The manager earned 0.75% of excess return for each 1% of tracking error taken.
Alpha vs Active Return
Active return is simply portfolio minus benchmark. Alpha adjusts for beta. A high-beta portfolio can beat its benchmark in a rising market and still have zero alpha.
FAQs
Is a high tracking error bad?expand_more
Not on its own. It means the manager takes active bets. What matters is whether those bets pay, which is what the information ratio measures.
When does diversification fail?expand_more
When assets are highly correlated. With correlation of +1, combining them doesn't reduce risk.
How is VaR scaled to more days?expand_more
By multiplying one-day VaR by the square root of the number of days, assuming returns are independent.
Next steps
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