Estimated reading time: 4 minutes
Introduction
Below are FRM formulas for the level 1 Financial Markets and Products segment.
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Price of a Bond
= [ Present Value of Coupon Payments ] + [ Principal Payment at End of Life ]
= [ SUM [ Ce –rt ] ] + P e -rT
Where:
C is the Coupon Payment
P is our Principal
t is the time to Maturity
r is the Interest Rate
Forward Pricing
Forward Pricing (for a continuous compounding rate, r) is given by:
Fo = So e rt
Where:
Fo is the Forward Price
So is the Spot Price
t is the Time of Contract
Forward Pricing with an annual rate
Forward Pricing (for an annual rate, r) is given by:
Fo = So (1+r)t
Where:
Fo is the Forward Price
So is the Spot Price
t is the Time of Contract
Value of a Long Forward
Value of a Long Forward, f (for a continuous dividend yield):
f = So e –qt – K e -rt
Where:
So is the Spot Price
K is the Price of Delivery
t is the time of Payoff
q = Continuous return % / Total Asset Price
Value of a Long Forward, f (for a discrete dividend):
f = So – I – K e -rt
Where:
So is the Spot Price
I is the asset’s Present Value
K is the Price of Delivery
t is the time of Payoff
Maximum Bond Values
Maximum Value of American Call = St
Maximum Value of American Put = X
Maximum Value of European Call = St
Maximum Value of European Put = X / ( 1 + Rf)t
Minimum Bond Values
Minimum Value of American Call = Ct ≥ Max ( 0 , St – ( X / 1 + Rf )t )
Minimum Value of American Put = Pt ≥ Max ( 0 , ( X – St ) )
Minimum Value of European Call = Ct ≥ Max ( 0 , St – ( X / 1 + Rf )t )
Minimum Value of European Put = Pt ≥ Max ( 0 , X / ( 1 + Rf )t ) – St
Special Note: For American options, the following relationship must hold:
S0 – X ≤ C – P ≤ S0 – X * e-rt
Value of a Swap
V = (Present Value of Payments) – [ (Present Value of Par Values) + (Accrued Interest) ] * e -rt
Basis
(Asset’s Spot Price) – (Futures Price) = ‘Basis’
Also, when Futures Price = Spot Price, then Basis = 0
Forward Rate Agreement
Forward Rate Agreement Payment to Long:
= Principal X [ ( Settlement Rate – Forward Rate ) X ( # of Days / 360 ) ] / [ 1 + ( Settlement Rate ) X ( # of Days / 360 ) ]
Dollar Default Rate
Dollar Default Rate during a particular year may be considered as:
= (Par Value of defaulted bonds) / (Total Par Value of all outstanding bonds)
Required Rate of Return
Required Rate of Return will be given as:
= Risk Free Rate + [ (Beta) * (Market Risk Premium) ]
Cheapest to Deliver
Cheapest to Deliver bond is the bond with the lowest cost of delivering.
Cost of delivering = Quoted price – (Current Futures Price x Conversion Factor)
Use: “Current Futures Price” or “Settlement Price”
Riskless Pure Discount
Riskless Pure Discount Position through the formula:
X / (1 + Rf)T = P0 + S0 – C0
KMV Model
The KMV model ( which measures a normalized distance ), is given:
( Expected Assets – Weighted Debt ) / ( Assets’ Volatility )
Convexity
C = (1 / B) X ( d2B / dy2 )
Interest Rate Parity
Ft = St * e (rf −q) * (T−t)
Where:
rf is the risk-free-rate
q is our dividend yield
(T − t) is the time until contract maturity
Ft is the theoretical contract price
St is the underlying security’s spot price
Interest Rate Parity for Currencies
Forward = Spot [ (1 + Local Currency Rate) / (1 + Foreign Currency Rate) ] T
Value of a European Call
The Value of a given European Call assuming that there will be no income payment on the security, we have:
c = [ S * N(d1) ] − [ K * e ( −rτ ) ] X [ N(d2) ]
Interest Earned between Dates
= [ ( # of Days between Dates ) X ( Interest Earned for Period ) ] / ( # of Days in Ref Period )
Interest Rate Parity for Currencies
On Currencies, The interest rate parity theory contends that:
Ft = So * e ( rbc− rfc ) T
Fair Value of a Futures Contract
F = S * e ( −r *T ) / e ( −r*T )
No-Arbitrage Forward Price
F ( 0 , T ) = S0 ( 1 + r )T
Short Position’s Value at Expiration:
VT ( 0, T ) = ST – F (0,T)
Put-Call Parity Formula
Call Premium + PV of Strike = Put Premium + Asset Price
or
C + X e −rT = P + S
Where:
C is the Call premium
X e−rT is the PV of the strike
P is the put premium
S is the underlying asset’s current price
Re‐arranging, we see that: S = C − P + X * erT
Amount of Contracts to Sell
N = ( Beta X Position Size ) / ( Size of single futures contract )
Beta
β = Cov ( Spot futures ) / Var ( futures )
Cov = σ spot * σ futures * correlation
Optimal Hedge Ratio for a Fund
h = ρ * ( σ fund / σ hedge )
Where:
Ρ is the Correlation Coefficient between σ fund and σ hedge
σ is the Standard Deviation (the change during the hedging period)
Value of the Fixed Payment, i.e. Bfix
Bfix = (fixed rate coupon)e(-1Yr LIBOR) + (nominal amount + fixed rate coupon)e(-2Yr LIBOR*2)
Fixed Rate Coupon
Fixed Rate Coupon = Given notional value X Annual fixed rate %
Cost of Carry
Cost of Carry = Interest Cost + Storage Cost – Income Earned
Convenience Yield for a Consumption Asset
Fo = So e (c-y)T
Issuer Default Rate
= ( # of Issuers Defaulting ) / ( Total # of Issuers at Start )
Dollar Default Rate
= ( Dollar Sum of All Default Bonds ) / ( Dollar Sum of all Issues X Wt. Avg. of Amount of Years Outstanding )
Key Rate Duration
= ( -1 / Bond Value ) X ( Bond Value Change / Key Rate Change )
Summary
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