HP 12C RPN calculator for FRM risk math

2. Build a square root with the yth-root key

The square-root-of-time rule needs √N. Put N on the stack, ENTER, key 2, then g x√y to take the 2nd root. Everything stays on the display for the next multiply.

3. Exponentiate for continuous compounding

Continuous discounting runs through e^x. Assemble the exponent on the stack — here −r·t — then press g e^x to get the discount factor, ready to multiply by an exposure.

4. Chain a whole formula on the stack

RPN shines when results feed the next step. Raise a factor with yˣ, divide by another stack value, and subtract a constant — the forward-rate formula runs end to end without re-keying intermediates.

Net two desk positions on the stack

A book carries a +$4.2m long and a −$1.7m short. Net them the RPN way: put both operands on the stack first, then apply the operator — no equals key. Key 4.2, ENTER, 1.7, minus.

RPN pushes operands onto a four-level stack and applies the operator to the top two. Once the stack feels natural, chained risk arithmetic runs with fewer keystrokes and no parentheses.

Scale a 1-day VaR to 10 days with the root key

A desk reports a 1-day VaR of $2,000,000. Under the square-root-of-time rule, VaR_N = VaR_1 · √N. Build √10 on the stack with the yth-root key, then multiply by the 1-day figure.

The square-root-of-time rule assumes independent, identically distributed returns and a zero drift. The 10-day VaR is about $6.32m — roughly 3.16× the 1-day number.

Discount an expected exposure with e^x

An expected exposure of $5,000,000 falls due in 2 years. At a 3.5% continuously compounded rate, discount it with e^(−rt): build −0.07 on the stack, take g e^x, then multiply by the exposure.

e^(−rt) is the continuous discount factor. Reversing it with the LN key recovers the continuously compounded rate from a discount factor. The present value is about $4.66m.

Forward rate from two spot rates

The 1-year spot is 3% and the 2-year spot is 4%. The 1y-into-2y forward solves (1 + z2)² / (1 + z1) − 1. Raise 1.04 to the power 2 with yˣ, divide by 1.03, then subtract 1.

The implied one-year forward beginning in year one is about 5.01%, above both spot rates because the curve is upward sloping. yˣ raising and stack division keep the whole chain on the display.

Convexity-adjusted price move for a 100bp yield rise

A bond has modified duration 6.5 and convexity 80. For a +100bp yield move (Δy = 0.01) the second-order estimate is ΔP/P ≈ −D·Δy + ½·C·Δy². Build the positive convexity term ½·C·Δy² on the stack first, then key the duration term D·Δy and subtract it — the running total never leaves the display.

Duration alone predicts −6.50%; the +0.40% convexity term pulls the estimate to about −6.10%. Convexity always softens the loss on a yield rise and adds to the gain on a fall, which is why long-convexity positions are prized in rate selloffs.

Aggregate two desks into one correlated VaR

Desk A carries $3m VaR and desk B $4m VaR with a 0.5 return correlation. Combine them with VaR_p = √(VaR_A² + VaR_B² + 2ρ·VaR_A·VaR_B). Working in $m, square each leg, park the sum in the stack, build the cross term 2ρ·VaR_A·VaR_B, add it, then take the root.

The undiversified sum of $7m collapses to about $6.08m once the 0.5 correlation is credited — the $0.92m gap is the diversification benefit. The four-level stack parks the 25 (sum of squares) in Y while the cross term is assembled in X.

Net present value of an uneven exposure profile

A hedging trade costs $5m today and returns $2m in each of years 1 and 2, then $3m in year 3, discounted at 8%. Load it into the HP 12C cash-flow registers: CFo as a negative, the repeated $2m with Nj = 2, then the $3m, set i, and press f NPV.

g Nj repeats an identical cash flow without re-keying it, which is where the RPN cash-flow registers beat a plain TVM worksheet on uneven streams. The positive $948k NPV means the trade clears an 8% hurdle rate.