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FRM Level 1 Formulas – Valuation and Risk Models

Estimated reading time: 3 minutes

 

Introduction

We present the formulas for the Valuation and Risk Models segment.

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Put Option’s Value

To find a put option’s value, we may utilize the following formula:

Put Value  = { [  Strike Price  X  e( – RFR * T ) ]  X  [ 1 – N ( d2 ) ] }   –   [  Stock Price X ( 1 – N ( d1 ) ]

 

Call Option’s Value

To find the value of the call option, we may utilize the following formula:

Call Value  =  [ Stock Price X  N ( d1 ) ]  –  [  Strike Price  X e ( – r  X  days/365)  ]  X [  N ( d2 ) ]

 

Putup

Putup  =  Max ( 0, Strike Price – StockUp )

 

Putdown

Putdown  =  Max ( 0, Strike Price – StockDown

 

The Put-Call Parity relationship

Call Price  –  Put Price  =  Stock Price  –  [ Strike Price  / ( 1 + RFR )]

Rearranging:

Call Price  =  Stock Price  – [  Strike Price  /  ( 1 + RFR )T ] + Put Price

or

Put-call parity relationship:

P0  +  S0  =  C0  +  X  / ( 1 + Rf )T

Where:

C = Call Premium

P = Put Premium

X = Strike Price of Call & Put

r = Annual Interest Rate

t = Time in Years

So = Initial Price of Underlying Security

 

Value at Risk

To calculate the Value at Risk, we may utilize the following formula:

VaR  =  [ Expected Change in Value ] –  [ Z-Score X  Price Change Standard Deviation ]

 

Annual Standard Deviation

Annual Standard Deviation =  Daily Standard Deviation  X ( 365 ) ½

 

Convexity Adjustment

Convexity Adjustment  =  Convexity  X  100 X  ( Δy )2

Where: Δy is the change in interest rates in decimal form

 

Synthetic Call Price

Formula to determine the synthetic call price:

c0  =  ( p0   +  S0  )  −  [  X   /   (1 + r)T  ]

 

Covered Call

Consider the following as a quick reminder:

‘ Covered Call = Stock + Short Call ‘

 

The Unexpected Loss formula

Unexpected Loss, UL  =  EA X (  PD X σ2 LR   +   LR2  X σ2 PD   ) ½ 

Where:

LR = loss rate = ( 1 – recovery rate)

EA = exposure amount

PD = probability of default

 

Daily Standard Deviation

Please note: If information is based on a sample, ‘ N – 1 ‘ will be used in the denominator

Daily Standard Deviation =  [ ( Sum of Squared Deviations from Mean  / ( N – 1 ) )  ] ½ 

 

Implied 1‐Yr Forward Rate

From the Pure Expectations Theory: The expected 1-Yr spot rate, one (1) year from now will be equivalent to finding the forward rate for the second year, thus:

The Implied 1‐Yr Forward Rate is given:

=  [  ( 1 + Two-Yr Spot Rate )2  /  ( 1 + One-Yr Spot Rate )  ]  –  1

 

Risk Neutral Probability

We may calculate the risk neutral probability from the given formula:

Π  =  ( 1 + RFR ) – ( 1 – % Down ) )  /  ( ( 1 + % Up ) –  ( 1 – % Down ) )

 

Risk Neutral Probability (single step stock)

The risk neutral probability of a stock going up in a single step may be calculated as follows:

pup       =   ( e rΔt  –  d )  /  ( u – d ) 

 

Dirty Price

Dirty Price = Quoted Price + Interest Accrued from last Coupon Date

 

Daily Delta Normal VaR

Daily Delta Normal VaR formula is as given:

=  [ R –  ( z )( sigma ) ]  X  ( Portfolio Value )

Where:

R is our portfolio’s expected one‐day return

z is our z‐value with respect to the significance level

Sigma is the standard deviation ( one‐day returns )

 

Mean Loss Rate

Mean Loss Rate  = Probability of Default X ( 1 – Recovery Rate )

 

Portfolio’s Beta

Beta, β  =  Correlation Coefficient X ( Portfolio Standard Dev / Instrument’s Standard Dev )

 

Sample-Mean

Sample-Mean may be calculated as:

= ( Summation of change in each day’s yield ) / ( The Number of Observations – 1 )

 

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