FRM resources
Market, credit, liquidity and operational risk
- Worked examples
Funding the discounted invoice amount
Compare cash costs over the actual gap between payment dates under a stated simple-rate model.
- Worked examples
Reconciling all five DuPont factors
Use the full precision of each factor and check the product against direct ROE.
- Worked examples
Estimating a price change with duration and convexity
Follow the signs, decimal yield change and squared curvature term.
- Worked examples
Expected credit loss from a two-outcome table
Derive PD × LGD × EAD from explicit default and non-default outcomes.
- Worked examples
Checking nearby order quantities around EOQ
Compare the objective value rather than assuming a rounded quantity is automatically optimal.
- Worked examples
A finite annuity compared with the perpetuity limit
Compare a bounded payment stream with an infinite stream under the same positive rate.
- Worked examples
A level-payment amortization model
Derive a payment from an annuity factor and separate interest from principal.
- Worked examples
Comparing current, quick and cash ratio components
Use one liability denominator and three explicitly defined numerators.
- Worked examples
Interpreting debt ratios when equity is negative
Identify why a negative debt-to-equity quotient is not a low-leverage signal.
- Worked examples
Expected revenue from a two-stage scenario model
Calculate conditional expectations, then apply the outer scenario probabilities.
- Worked examples
NPV of unequal annual cash-flow amounts
Discount an initial outflow and three unequal annual receipts.
- Worked examples
Converting odds against default into a default probability
Work through the event label, quotation direction and normalization.
- Worked examples
Comparing ordinary and due annuity values
Shift each payment one period earlier while preserving the valuation date.
- Worked examples
Counting an investable universe with overlapping exclusion lists
Avoid subtracting the same issuer more than once when screening a universe.
- Worked examples
Purchases versus COGS in payable days
Show why an inventory movement changes the flow used for a payables comparison.
- Worked examples
Quarterly and annual discounting of the same payment
Check that an effective annual conversion preserves the value of a dated single sum.
- Worked examples
Calculating the probability of at least one event
Subtract the overlap once when adding two event probabilities.
- Worked examples
Future value of end-quarter deposits
Use a quarterly rate, quarterly payment count and a defined final date.
- Worked examples
Reconciling assets, liabilities and retained earnings
Use the accounting equation and explicit equity-component assumptions.
- Worked examples
The same ROE from different factors
Separate an equal headline ratio from its different arithmetic components.
- Worked examples
Scenario variance and squared deviation from a target
Calculate the implied mean before distinguishing variance from a target-based squared deviation.
- Worked examples
Calculating a score-linked management fee under a stated contract
Apply a fictional score-improvement condition before converting basis points to an annual fee.
- Worked examples
Why two ESG scoring methods can produce different results
Recalculate a fictional issuer score using two explicit sets of weights.
- Worked examples
Combining conditional probabilities with scenario weights
Calculate an unconditional probability from a complete weighted scenario table.
- Worked examples
Annualizing monthly tracking error: why the square root appears
Derive the square-root-of-12 calculation and identify its covariance assumption.