## Calculating Forward Prices, Forward Rates and Forward Rate Agreements (FRA) – Calculation reference

*in*Derivatives

Calculation and formula reference for Forward Price, Spot Rates & Forward Rates, Yield to Maturity, Forward Rate Agreement (FRA), Forward Contract and Forward Exchange Rates.

## Short and sweet Lessons in Forward Pricing

- Valuing a forward contract in Excel – Lesson Zero
- Calculating Forward Prices in Excel – Lesson 1
- Calculating Forward Prices and YTMS in Excel – Lesson 2
- Calculating Forward Foreign Exchange Rates in Excel – Lesson 3
- Defining the Par Interest Rate Term Structure
- Calculating the Zero interest rate curve
- Calculating the Forward interest rate curve
- Calculating the MTM of the Interest Rate Swaps – IRS
- Pricing Basis swap
- Projecting Foreign Exchange Currency Rates for Cross Currency Swap Valuation (from CCS)
- CCS – Cross Currency Swap Valuation & Pricing

- Forward price of a security with no income
- Forward price of a security with known cash income
- Forward price of a security with known dividend yield

- Relationship between spot rates and forward rates-1
- Relationship between spot rates and forward rates-2

- Value of a long forward contract (continuous)
- Value of a long forward contract (discrete)
- Value of a long forward contract (continuous) which provides a known income
- Value of a long forward contract (continuous) which provides a known yield
- Value of a forward foreign current contract (continuous)

# 1. Forward Price

## a. Forward price of a security with no income

Where S_{0} is the spot price of the asset today

T is the time to maturity (in years)

r is the annual risk free rateof interest

## b. Forward price of a security with known cash income

(Securities such as stocks paying known dividends or coupon bearing bonds)

Where I is the present value of the cash income during the tenor of the contract discounted at the risk free rate.

## c. Forward price of a security with known dividend yield:

(Securities such as currencies and stock indices)

Where q is the dividend yield rate. For a foreign currency q will be the foreign risk free rate.

# 2. Spot Rates and Forward Rates

## Relationship between spot rates and forward rates-1

## Relationship between spot rates and forward rates-1

Where s_{t} is the t-period spot rate and

f_{t-1,t} is the forward rate applicable for the period (t-1,t)

# 3. Yield to Maturity (YTM)

To solve for YTM we are solving for the interest rate (r) in the bond valuation formula:

Where CP_{t} is the coupon payment at time t and MV is the maturity value at time n (i.e. at maturity).

# 4. Forward Rate Agreement (FRA)

The value of the FRA at time 0, V_{FRA}, for someone receiving fixed and paying floating will be

if R_{2} (the zero coupon rate for a maturity of T_{2}) is calculated on a discrete basis or

if R_{2} is calculated on a continuous basis.

Where, L is the principal amount

R_{K} is the fixed interest rate

R_{F} is the forward interest rate assuming that it will equal the realized benchmark or floating rate for the period between times T_{1} and T_{2}

# 5. Forward Contract

## Value of a long forward contract (continuous)

Where S_{0}is the spot price

T is the remaining time to maturity

r is the risk free rate

K is the delivery price which is set in the contract

## Value of a long forward contract (discrete)

## Value of a long forward contract (continuous) which provides a known income

I is the present value at time 0 of the known income on the investment assets

## Value of a long forward contract (continuous) which provides a known yield

q is the know yield rate provided by the investment asset

## Value of a forward foreign current contract (continuous)

Where r_{f} is the value of the foreign risk free interest rate when the money is invested for time T.

# Forward exchange rates

Where r and r_{f} are compounded continuously

or

if the interest rates were compounded on a discrete basis.

r is the risk free rate of the domestic currency

r_{f} is the risk free rate of the foreign currency