FRM Part I · FRM Exam Part I

Pricing Conventions, Discounting, and Arbitrage: formula sheet

Full chapter guide

Key formulas

Future value, discrete compounding: FV = PV × (1 + R/m)^(m×T); R is the annual quoted rate, m is compounding periods per year, T is years.
Present value, discrete compounding: PV = FV ÷ (1 + R/m)^(m×T); Same growth factor, used in reverse.
Future value, continuous compounding: FV = PV × e^(R×T); R is the continuously compounded annual rate.
Present value, continuous compounding: PV = FV × e^(−R×T); The discount factor is e^(−RT).
Effective annual rate: EAR = (1 + R/m)^m − 1; For continuous compounding, EAR = e^R − 1.
Continuous rate from discrete rate: Rc = m × ln(1 + Rm/m); Rm is the rate compounded m times a year.
Discrete rate from continuous rate: Rm = m × (e^(Rc/m) − 1); Inverse of the line above.
Equivalent rates: (1 + R1/m1)^m1 = (1 + R2/m2)^m2; Equal annual growth factors mean equivalent rates.
Discount factor from spot rate: d(t) = 1 ÷ (1 + z(t)/m)^(m×t); m is compounding periods per year. Use m = 2 for semiannual, as most bond questions do. Annual: d(t) = 1 ÷ (1 + z)^t.
Continuous compounding: d(t) = e^(−z×t); Use only when the question states continuous compounding.
Spot rate from discount factor: z(t) = m × [d(t)^(−1/(m×t)) − 1]; Annual compounding: z = d^(−1/t) − 1. Continuous: z = −ln d(t) ÷ t.
Bond price using spot rates: P = Σ C(t) × d(t) + F × d(T); C is each coupon, F is face value, T is maturity. Each cash flow uses its own discount factor.
Par rate from discount factors: c = m × [1 − d(T)] ÷ Σ d(t_i); Sum the discount factors at every coupon date up to T. Gives the annual coupon rate for a par bond.
Bootstrapping the final discount factor: d(T) = [P − Σ C × d(t) for earlier dates] ÷ (C + F); C + F is the final coupon plus principal. All earlier discount factors must already be known.
Forward rate, annual compounding: (1 + z₂)^T₂ = (1 + z₁)^T₁ × (1 + f)^(T₂ − T₁), so f = [(1 + z₂)^T₂ ÷ (1 + z₁)^T₁]^(1 ÷ (T₂ − T₁)) − 1; z₁ and z₂ are spot rates for maturities T₁ and T₂. f is the forward rate between T₁ and T₂.
Forward rate, continuous compounding: f = (z₂ × T₂ − z₁ × T₁) ÷ (T₂ − T₁); Valid only when all rates are continuously compounded. Here the forward is a weighted difference of the spot rates.
Forward rate from discount factors: (1 + f) = [d(T₁) ÷ d(T₂)]^(1 ÷ (T₂ − T₁)); d(T) = 1 ÷ (1 + z)^T is the discount factor. For a one-year forward period, 1 + f = d(T₁) ÷ d(T₂).
Par yield, annual coupons: c = (1 − d(N)) ÷ Σ d(t), for t = 1 to N; Per 1 of face value. Gives the annual coupon rate that prices the bond at par.
Par yield, m coupons per year: c = m × (1 − d(N)) ÷ Σ d(tᵢ); The sum runs over all coupon dates, spaced 1/m years apart. d(N) is the discount factor at final maturity.
Yield to maturity: Price = Σ C ÷ (1 + y)^t + Face ÷ (1 + y)^N; Solve for y by trial or with a financial calculator (N, PV, PMT, FV, then CPT I/Y).
Dirty price: Dirty price = Clean price + Accrued interest; Dirty price is the cash paid at settlement. Clean price is the quote.
Accrued interest: AI = Coupon per period × (Days from last coupon to settlement ÷ Days in coupon period); Coupon per period = annual coupon ÷ payments per year. Use the day count of the bond for both day numbers.
30/360 day count: Days = 360 × (Y2 − Y1) + 30 × (M2 − M1) + (D2 − D1); Before computing: if D1 = 31, set D1 = 30. If D2 = 31 and D1 is 30 or 31, set D2 = 30. This is the standard US rule.
Accrued interest, 30/360 shortcut: AI = Annual coupon × Days (30/360) ÷ 360; Equivalent to the period formula when the period is 180 days (semiannual) or 360 days (annual).
T-bill price from discount rate: P = 100 − d × (n ÷ 360) × 100, with d as a decimal; n is actual days to maturity. P is per 100 face value.
T-bill bond equivalent yield: BEY = ((100 − P) ÷ P) × (365 ÷ n); Uses price as the base and 365 days. For bills of up to six months this is the usual simple-interest form. BEY is higher than the discount rate.
Law of one price: Same cash flows ⇒ same price; If prices differ, an arbitrage exists in frictionless markets.
Replication price: Price of target = cost of replicating portfolio; Match cash flows at every date, then add up the cost.
Coupon bond as sum of zeros: P = Σ CFₜ × d(t); d(t) is the price of ₹1 or $1 received at time t, i.e. the discount factor.
Discount factor from a spot rate: d(t) = 1 ÷ (1 + z/m)^(m×t); z is the annual spot rate, m is compounding per year. For continuous compounding, d(t) = e^(−z×t).
Arbitrage profit: Profit = |Market price − Replication price| × quantity; Buy the cheaper side, sell the dearer side.
Arbitrage definition: Cost ≤ 0 today, payoff ≥ 0 later in all states, and > 0 in some state; A trade with risk of loss is not an arbitrage.
Forward price, no income (continuous compounding): F = S0 × e^(rT); With annual compounding use F = S0 × (1 + r)^T. Use the compounding the question gives.
Forward price with known cash income: F = (S0 − I) × e^(rT), where I = PV of income; Discount each income payment at the risk-free rate to today before subtracting.
Forward price with known yield q: F = S0 × e^((r − q)T); q is the continuously compounded yield. For a commodity with storage cost u, use e^((r + u)T).
Value of a long forward at time t: f = S_t − PV(income) − K × DF(t, T); Equivalent to (F_t − K) × DF(t, T). A forward has zero value at inception when K = F.
Discount factor from a spot rate: DF(T) = 1 ÷ (1 + z)^T, or e^(−zT) if continuous; z is the spot (zero) rate for maturity T.
Par swap rate: c = (1 − DF_n) ÷ Σ(τ_i × DF_i); τ_i is the accrual fraction of each period (1 for annual, 0.5 for semi-annual). Sum runs over all fixed payment dates.
Value of a swap to the fixed payer: V = N × [(1 − DF_n) − K × Σ(τ_i × DF_i)]; Valid on a reset date. The fixed receiver's value is the negative of this.
Forward rate from discount factors: f(t1, t2) = (DF(t1) ÷ DF(t2) − 1) ÷ (t2 − t1); Simple-compounded forward rate over the period. The par swap rate is a weighted average of these forward rates.

Quick revision

Compounding more often at the same stated rate gives a higher future value; continuous compounding is the limit.
Convert rates between compounding bases by matching growth over the same period: e^rc = (1 + r ÷ m)^m.
A discount factor is the present value of 1 paid at time T; bond price = Σ cash flow × discount factor.
A spot rate is the yield on a zero-coupon investment to a given maturity.
With annual compounding, the forward rate between T1 and T2 satisfies (1 + z2)^T2 = (1 + z1)^T1 × (1 + f)^(T2 − T1).
With continuous compounding, the forward rate = (z2 × T2 − z1 × T1) ÷ (T2 − T1).
The par yield is the coupon rate that makes a bond price equal to its face value.
Accrued interest depends on the day count convention; clean price = dirty price − accrued interest.
The law of one price says identical cash flows must have the same price in a frictionless market.
An arbitrage requires no net investment (or yields a cash inflow today) and has no risk of loss, with a positive probability of profit.
A forward on an asset with no income, priced with continuous compounding: F = S × e^(r×T).
A plain vanilla swap has a fixed rate set so the swap's value is zero at initiation, which is the par swap rate.

Common mistakes

Using the quoted annual rate directly in (1 + R)^T when compounding is semiannual or monthly. Fix: Always divide R by m and multiply T by m.
Dividing the rate by m in the continuous formula. Fix: Continuous compounding uses e^(RT) with no division by m.
Using years instead of periods in the exponent under semiannual compounding. Fix: Use (1 + z/2)^(2t). For 1.5 years the exponent is 3.
Discounting all cash flows at one yield and calling it a spot-rate price. Fix: Use a separate discount factor for each date when spot rates are given.
Taking the simple difference of spot rates as the forward rate Fix: Use the growth-factor ratio for discrete compounding. The simple difference works exactly only with continuous compounding, and then only after weighting by maturity.
Using the wrong exponent when taking the root Fix: Write the exponent 1 ÷ (T₂ − T₁) before you compute. For a one-year forward period it is simply 1.
Using the wrong day count, for example real days on a corporate bond that uses 30/360. Fix: Read the instrument first. Treasury notes and bonds use Actual/Actual, corporates use 30/360, bills use Actual/360. Use the convention the question names.
Dividing by 365 or 360 for a Treasury bond's accrued interest instead of the days in the coupon period. Fix: For Actual/Actual, the denominator is the actual days between the two coupon dates, such as 181 to 184 for a semiannual bond.
Trading the wrong direction Fix: Always say it in words: buy whatever costs less for the same cash flows, sell whatever costs more.
Forgetting the final coupon plus principal at maturity Fix: List every date in a table. The final row is coupon + face value.

Exam tips

Questions often hide the compounding frequency in the wording. Underline it before calculating.
Practise ln and e^x on the approved calculator until the key sequence is automatic.
Expect conversion questions between continuous, semiannual and annual rates. Equate growth factors.
Sense-check: for the same growth, continuous rate < annual rate with fewer compounding periods, and EAR > quoted rate.
Later topics such as bond pricing, forwards and swaps all reuse these formulas, so build speed here.
Read the compounding convention first. Most wrong answers on this topic come from mismatched compounding.
Keep at least six decimals in discount factors. Rounding to three decimals can move you to a different option.
On a financial calculator, use the power function: store 1.022, raise to 2 with y^x, then take 1/x. Do not re-enter intermediate values by hand.