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:

F_{o} = S_{o} e ^{rt }

Where:

F_{o} is the Forward Price

S_{o }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:

F_{o} = S_{o} (1+r)^{t }

Where:

F_{o } is the Forward Price

S_{o }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 = S_{o} e ^{–qt} – K e ^{-rt }

Where:

S_{o }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 = S_{o} – I – K e ^{-rt }

Where:

S_{o }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 = S^{t}

Maximum Value of American Put = X

Maximum Value of European Call = S^{t}

Maximum Value of European Put = X / ( 1 + R_{f})^{t}

## Minimum Bond Values

Minimum Value of American Call = C_{t} ≥ Max ( 0 , S_{t} – ( X / 1 + R_{f} )^{t} )

Minimum Value of American Put = P_{t} ≥ Max ( 0 , ( X – S_{t} ) )

Minimum Value of European Call = C_{t} ≥ Max ( 0 , S_{t} – ( X / 1 + R_{f} )^{t} )

Minimum Value of European Put = P_{t} ≥ Max ( 0 , X / ( 1 + R_{f} )^{t} ) – S_{t}

*Special Note: For American options, the following relationship must hold:*

S_{0} – X ≤ C – P ≤ S_{0} – 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 + R_{f})^{T} = P_{0} + S_{0} – C_{0}

## KMV Model

The KMV model ( which measures a normalized distance ), is given:

( Expected Assets – Weighted Debt ) / ( Assets’ Volatility )

## Convexity

C = (1 / B) X ( d^{2}B / dy^{2} )

## Interest Rate Parity

F_{t} = S_{t} * e ^{(rf −q) * (T−t)}

Where:

r_{f} is the risk-free-rate

q is our dividend yield

(T − t) is the time until contract maturity

F_{t} is the theoretical contract price

S_{t } 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:

F_{t} = S_{o} * e ^{( r}_{bc}^{− r}_{fc}^{ ) T }

## Fair Value of a Futures Contract

F = S * e ^{( −r *T )} / e ^{( −r*T )}

## No-Arbitrage Forward Price

F ( 0 , T ) = S_{0} ( 1 + r )^{T}

## Short Position’s Value at Expiration:

V_{T} ( 0, T ) = S_{T} – 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 * e^{rT}

## 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. B_{fix}

B_{fix } = (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

F_{o} = S_{o} 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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