Min, max, floors and caps · RiskAtoms Academy

By the end you will be able to

1Find the min and max of two or more numbers, and write a floor as max(x, F), a cap as min(x, C) and a corridor as min(max(x, a), b).
2Read the kink that a floor or cap puts in a graph, and say which constraint binds.
3Use max(a, b) = −min(−a, −b), max(a, b) + min(a, b) = a + b and the positive part x⁺ = max(x, 0).
4Nest floors and caps in the right order, and spot when the order changes the answer.

Why you need it

RiskAtoms Bank plc's own models put its risk-weighted assets at £21,558.1m. The rules say: use that, or 72.5% of the figure the standardised approaches give, whichever is higher. That is £22,807.7m. “Whichever is higher” is a max, and the question that matters is the one every floor and cap asks: which one wins?

The rulebook is full of them. These Core Path lessons use this atom:

L4.1.04: Risk-weighted assets
RWA is the higher of the models' figure and the output floor (RBC20.4).
L4.1.03: Additional Tier 1 and Tier 2
Excess provisions count in Tier 2, but only up to a cap of 0.6% of IRB credit RWA (CAP10.19).
L4.1.05: Capital buffers
Inside the buffer, payouts are capped at a share of earnings that steps down as CET1 falls (RBC30.4).
L4.1.07: TLAC and MREL
Requirements with more than one leg: the one that needs more capital is the one that binds.
L4.4.01: SA-CCR: replacement cost and the multiplier
SA-CCR: replacement cost floored at zero, and a multiplier capped at 1 and floored at 5% (CRE52).
L4.6.08: Own funds and the MCR
Solvency II's MCR sits in a corridor between 25% and 45% of the SCR, above an absolute floor.

You need percentages from M-A01: Percentages and graphs of functions from M-A09: Functions and graphs. The kinks link to slopes in M-A10: Slopes and rates of change.

The idea

The higher of, the lower of

Rule writers rarely trust a single number. So they write “the higher of” to put a safety net under a figure, and “the lower of” to put a ceiling over it. A minimum wage is a floor: your pay is the higher of what the job offers and the legal minimum. A limit on how much of something you may count is a cap: you get the lower of what you have and the limit.

The bank's RWA works the same way. Its own models give one figure; the standardised approaches, scaled by the floor rate, give another. Move the rate and see which one the bank must use.

The higher of two numbers

RiskAtoms Bank plc, £m

Floor rate72.5%

RWA = 22,807.7

The floor binds: it adds 1,249.6 to the models' figure. The two are equal at a floor rate of 68.5%.

At 72.5% the floor binds: it decides the answer. Below 68.5% it does not, and the models' figure stands. Nothing in between: the answer is always one of the two inputs. Knowing which one is most of the work.

The rules

Min and max. Two new pieces of notation, max and min.

Floors, caps and corridors. A floor is a max, a cap is a min, and a corridor is one inside the other. Each puts a kink in the graph where the limit starts to bind.

Nesting order. A floor and a cap can be applied in either order. Usually that makes no difference. Sometimes it decides everything.

x	min(max(x, 60), 40)	max(min(x, 40), 60)
20	40	60
50	40	60
80	40	60

Three identities. They turn a max into a min, take the cases out of a max, and name the most common floor in finance, the positive part.

The rules, for reference

\begin{gathered} \text{floor: } \max(x, F) \qquad \text{cap: } \min(x, C) \\ \text{corridor: } \min(\max(x, a), b),\ a \le b \end{gathered}

\begin{gathered} \max(a, b) = -\min(-a, -b) \\ \max(a, b) + \min(a, b) = a + b \\ x^{+} = \max(x, 0) \end{gathered}

For example, with a = 7 and b = −3: max = 7, min = −3, −min(−7, 3) = 7, and max + min = 4 = 7 + (−3). In the calculator, max(a, b) and min(a, b) take two or more numbers.

Try it

Pick a shape, then move the floor and the cap: the kinks move with them. Drag along the chart, or use the x slider, and watch which branch gives the answer. On a flat stretch a limit binds and the slope is 0; on the diagonal nothing binds and the slope is 1. Then choose the corridor, push the floor above the cap, and switch the nesting order.

Try itMove the floor and the cap

Shape

Nesting order

Floor a30

Cap b70

Input x15

Output

min(max(15, 30), 70) = 30

Which bindsfloor

Adjustment, output − x15

Slope here0

The floor binds: the output is held up at 30, and the graph is flat here (slope 0).

output y = x floor capDrag on the chart to move x.

Worked examples

Figures are from RiskAtoms Bank plc's pro forma Basel 3.1 book at 31 December 2025 (£ millions) and RiskAtoms Insurance SE (€ millions). Both are illustrative.

1. The output floor: a max

Step 1 of 5

1
The rule
RWA is the higher of the RWA from the bank's approved approaches and 72.5% of the RWA it would have with the standardised approaches throughout (RBC20.4).

2. Excess provisions in Tier 2: a positive part, then a cap

Step 1 of 4

1
The rule
Under the IRB approach, where provisions exceed expected loss, the excess may count in Tier 2 up to 0.6% of IRB credit RWA (CAP10.19).

3. The SA-CCR multiplier: a min with an exponential inside

Step 1 of 4

1
The rule
The potential future exposure of a netting set is its add-on times a multiplier that gives credit for excess collateral. The multiplier is capped at 1 and floored at 5% (CRE52.22 to CRE52.23):
$\begin{aligned} &\text{multiplier} \\ &= \min\left\{1,\ 5\% + 95\% \times \exp\left(\frac{V - C}{2 \times 95\% \times \text{AddOn}}\right)\right\} \end{aligned}$

4. The Solvency II MCR: a corridor and an absolute floor

Step 1 of 4

1
The rule
The combined MCR is the linear MCR kept between 25% and 45% of the SCR, and the MCR is the higher of that and an absolute floor, the AMCR (Delegated Regulation (EU) 2015/35, Article 248; Directive 2009/138/EC, Article 129(3)):
$\begin{aligned} \text{MCR}_{\text{combined}} &= \min\big(\max(\text{MCR}_{\text{linear}},\ 25\%\,\text{SCR}),\ 45\%\,\text{SCR}\big) \\ \text{MCR} &= \max(\text{MCR}_{\text{combined}},\ \text{AMCR}) \end{aligned}$

Which one binds? More floors and caps on the bank's books

Guess, then reveal

Payouts in the buffer (RBC30.4). Distributions may be at most (1 − conservation ratio) × earnings. The bank's CET1 left for buffers is 7.01% of RWA against a combined buffer of 4.5%. Does the cap bind?
Reverse repos in the leverage exposure (LEV30.37). Counterparty exposure E* = max(0, E − C), with £3,000m lent and £3,060m of collateral received. What is E*?
Derivatives in the leverage exposure (LEV30.16). The SA-CCR multiplier is fixed at one. What does that do to Saltney's min(1, 0.805)?
Level 2B assets in the LCR (LCR30.36). After haircuts the bank holds £11,500m of Level 1, £1,700m of Level 2A and £500m of Level 2B. Level 2B counts up to 15/85 of Level 1 plus 2A, and at most a quarter of Level 1. Which of the three binds?
Inflows in the LCR (LCR40.1). Inflows count up to 75% of outflows. Over the 30-day stress the bank has £10,080m of outflows and £1,375m of inflows. What counts?

Common mistakes

Each line below is a tempting slip. Spot what is wrong, then check.

Reaction

Check what stuck

Question 1 · objective 1

A bank's models give RWA of 900. With the standardised approaches throughout its RWA would be 1,400. The output floor is 72.5%. What RWA does it report?

Question 2 · objective 1

An IRB bank has provisions of 50 against expected loss of 30, and IRB credit RWA of 2,500. The excess counts in Tier 2 up to 0.6% of IRB credit RWA. How much counts?

Question 3 · objective 3

Two numbers add up to 10, and the smaller of them is 3. What is max(a, b)?

Question 4 · objective 4

Take a floor a = 5, a cap b = 3 and x = 4. Which is true?

Amin(max(4, 5), 3) = 3 but max(min(4, 3), 5) = 5BBoth orders give 4, because x lies between the two limitsCBoth orders give 3DNeither can be worked out when the floor is above the cap

Question 5 · objective 3

A netting set holds two trades worth +8 and −5 to the bank, with no collateral. What is its replacement cost, max(V − C, 0), where V is the net value?

Glossary

Maximum, max: max(a, b): the larger of a and b. With more inputs, the largest of them all.
Minimum, min: min(a, b): the smaller of a and b. With more inputs, the smallest of them all.
Floor: A lower limit: the answer may not fall below it. Written max(x, F).
Cap: An upper limit: the answer may not rise above it. Written min(x, C).
Corridor: A floor and a cap together, min(max(x, a), b): the answer always lands between a and b.
Binding constraint: A floor or cap binds when it changes the answer: the result is the limit, not the original number. One that does not bind leaves the number as it was.
Kink: A sharp corner in a graph, where its slope jumps. A floor or cap puts one where the limit starts to bind.
Positive part, x⁺: x⁺ = max(x, 0): x when x is positive, zero otherwise. Also called the positive part of x.