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FRM Level 1 Formulas – Financial Markets and Products

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

Sis 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:

F is the Forward Price

Sis 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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