Introduction to Financial Risk Management (FRM)

Discount Rates calculated using the bootstrapping methodology (Forward Substitution). The first step is to calculate discount factors for all short-term points (MM term points generally < 1 year). Then follows the forward substitution methodology to bootstrap all the discount factors for each term point.

Where,

cfn = is the coupon of the n-year bond

dfi = is the discount factor for that time period

dfn = is the discount factor for the entire period, from which we derive the zero-rate.

The system uses linear interpolation methodology for interpolating yields between term points. The closed ended formula for linear interpolation used is:

Where,

Rn = Interpolated yield

R1 = Previous term point Yield

R2 = Next term point yield

T1 = Previous term point days

T2 = Next term point days

Tn = interpolated yield term point days

In the system, parametric VaR is calculated for portfolio using the following matrix algebra:

Where,

VaRP = Portfolio VaR

V = Vector matrix of position VaR

C = Correlation Matrix of the assets in the portfolio

VT = Transpose of the Vector matrix

Historical simulation takes the portfolio of assets at a particular point in time and then revalues the portfolio using the historical prices of the assets in the portfolio. The portfolio revaluation produces a distribution of P&L which is then used to determine the portfolio VaR at a given confidence factor. This process can get computationally intensive if there is large number of assets in the portfolio and price history over a significant time period.

Where,

Pn = Price of asset i

Hn = Holding of asset i

Monte Carlo simulation involves artificially generating large set of events (correlated price changes) from which VaR is derived. This involves two steps: Generate random numbers and then converting them into set of normally distributed price changes. And for multi asset portfolio this is done using the Jacobi’s method of generating EignValues and EignVectors for the correlation matrix.

Where,

Xk = correlated random price change for asset k with a normal distribution and the volatility of the asset k

= square root of the eigenvalue for the asset k

Xnorm = random price change from a normally distributed series

= the kth element of the eigenvector for the ith asset

Volatility of the kth asset

Where,

Xk = Price of asset calculated

Hn = Holding of asset

For pricing illiquid debt securities over a benchmark issue, system uses the interpolated yield of the benchmark issue for the maturity date of the security. Then using the closed ended formula, price is calculated based on the interpolated yield and the spread.

Where,

Redemption = redemption value

Yield = Interpolated Benchmark Yield + spread

Frequency = frequency of coupon payment

DSC = number of days from settlement to next coupon date

E = number of days in coupon period in which the settlement date falls

N = number of coupons payable between settlement date and redemption date

A = number of days from beginning of coupon period to settlement date

Yield is calculated from the pricing function using an iterative process by keeping the yield variable to match the user defined dirty / clean price with the price computed with yield. The yield is changed until the estimated price given the yield is close to price.

Bond prices tend to revert to par as it nears the maturity date. To price a floating rate bond, discount margin is used. An investor can make the additional yield if the bond is priced at a discount and vice a versa, if the security is priced at a premium, then the DM will equal reference rate less the DM.

Where,

DM = Discount Margin

Df = Discount Rate

IR = Current Indicator Rate

QM = Quoted Margin

n = Period till maturity

Cash flow allocation:

Cash flow allocation in the system is completed using the Weighted Duration methodology. The below flowchart gives the complete flow of the calculation process.

Below are the procedure for calculating the price of a fixed rate bond:

1. Calendar date wise cash flows have to be arrived.
2. Present Value Factor has to be determined based on YTM and Frequency.
3. Calculate the present value of future cash flows.
4. Sum them up to get full value of a bond, also called Dirty Price.

Calculate interest accrued and subtract its value from Dirty Price to get Clean Price of a bond.

System Illustration:

Following are the procedure to calculate price for a fixed rate bond via Market Risk module screens.

Select Home > Master Data Configuration > Portfolio Configuration > Instruments to access the Instruments screen.

Enter or select required values in the respective fields. Click Generate Cashflow.

Bond Pricing:

Select Home > Portfolio Analytics > Bond Pricing > Fixed Rate Bonds to access the Fixed Rate Bonds screen.

Enter or select values in the fields given below:

After entering the fixed rate bond details, click Calculate Price. The Fixed Rate Bond screen is displayed with the price details. The user can expand the Issuance field to view the price details.

Below are the procedure for calculating the price of a floating rate bond:

1. Calendar date wise cash flows have to be arrived. For Floating Rate Bonds, cash flows is based on Reset Rate and Initial Margin put together, which can be called as “Effective Coupon Rate”.
2. Present Value Factor is based on YTM and Frequency of payments. This is arrived at by adding Reset Rate and Quoted Margin.
3. Calculate the present value of future cash flows.
4. Sum them up to get full value of a bond, also called Dirty Price.
5. Calculate interest accrued and subtract its value from Dirty Price to get Clean Price of a bond.

System Illustration:

Following are the procedure to calculate price for a floating rate bond via Market Risk module screens.

Select Home > Portfolio Analytics > Bond Pricing > Floating Rate Bonds to access the Floating Rate Bonds screen.

Enter or select values in the fields given below:

After entering the floating rate bond details, click Calculate Price. The Floating Rate Bond screen is displayed with the price details. The user can expand the Issuance field to view the price details.

Below are the procedure for calculating the price of a Treasury Bill:

• Arrive at the residual maturity of the bond in actual days.
• Discount the face value by (1 + YTM adjusted for the no. of days).
• The assumption is that year consists of 360 days as it is the normal convention.

System Illustrations:

Following are the procedure to calculate price for a Treasury bill via Market Risk module screens.

Select Home > Master Data Configuration > Instruments to access the Instruments screen.

Enter or select required values in the respective fields. Click Generate Cashflow.

Select Home > Portfolio Analytics > Bond Pricing > Fixed Rate Bonds to access the Fixed Rate Bonds screen.

Enter or select values in the fields given below:

After entering the fixed rate bond details, click Calculate Price. The Fixed Rate Bond screen is displayed with the price details. The user can expand the Issuance field to view the price details.

An FX forward contract is an agreement to purchase or sell a set amount of a foreign currency at a specified price for settlement at a predetermined time in the future.

System uses following model to value an FX Forward:

Where,

NPV - FX trade mark-to-market in valuation currency,

C - Cash flow in foreign currency,

d - Number of days from calculation date to settlement date,

r - Continuous zero coupon rate on settlement date in domestic currency,

r_f - Continuous zero coupon rate on settlement date in foreign currency.

B - Interest basis (360 or 365)

Below table is an example describing the FX forward calculation:

An interest rate swap (IRS) is an agreement between two counterparties to exchange fixed for floating cash flows. Although the notional principal is not exchanged in a swap, the assumption is, without changing the value of the swap that the two counterparties pay each other the same notional amount. Another possibility is, a portfolio consisting of one fixed leg, which is equivalent to a fixed-coupon bond, and one floating leg, which is equivalent to an FRN.

IRS is calculated on the below model:

Where,

• fi is the estimated forward rate on the i^th period of the floating side
• ri is the continuously compounded zero coupon rate of the i^th receiver cash flow
• BR is the receiver interest basis (360 or 365)
• Mi is the number of days in period i (using the appropriate day count convention) of the floating side
• di is the time to maturity from the As Of Date to the i^th cash flow date of the floating side
• NR is the receiver notional amount
• Ml is the number of periods on the floating side
• k is the payer’s fixed rate
• rj is the continuously compounded zero coupon rate of the j^th payer cash flow
• BP is the payer interest basis (360 or 365)
• Mj is the number of days in period j (using the appropriate day count convention) of the fixed side
• dj is the time to maturity from the As Of Date to the j^th cash flow date of the fixed side
• Mx is the number of periods on the fixed side
• NP is the payer notional amount

Currency swaps are swaps that consists element of IR risk as well as FX risk. The valuation model is same as IRS. Refer below tables for valuation examples:

INR equivalent PV of Leg 1 = 936,506.03 * 65 = 60,872,89.00

PV of Leg 2 = 47,588,257.09

Value of Swap (L-S) = 13,284,634.91

Options valuation using Black-Scholes model

Options valuation using Black-Scholes model would use the standard formula:

Where,

Below tables are working examples:

The binomial tree valuation model is performed by means of a binomial lattice (tree), for a number of time steps between the valuation and expiration dates. Each node in the lattice represents a possible price of the underlying at a given point in time.

Working example for a 1-step binomial tree

Below table is a working example for 1-step binomial tree:

Where,

The same principle is used for modelling for puts and multi-step binomial tree model.