**Forward Rate Formula (Table of Contents)**

## What is the Forward Rate Formula?

Forward Interest Rate is the interest rate which is decided initially at the today price for a certain future period. It is the only rate that is decided on the basis of mutual concern and agrees upon it to borrow or lend a sum of money at some future date. Forward rates can also be derived from spot-interest rates that are the yields that we are obtaining on zero-coupon bonds through a process called bootstrapping.

A forward rate arises due to the forward contract. Even though the commitment between two parties leads to the successful execution of a forward contract. And it has been split into two legs; the first commitment is to deliver, sell, or take a short position on the asset and on another leg, to take delivery, buy, or take a long position on the asset.

It states that it produces a rate in such a way that two consecutive one-year maturities offer the same return as two -year maturity offers. It is the interest rate an investor has a guaranteed to get the same returns between the first investment maturity and second maturity while picking between the shorter or longer-term.it is represented mathematically as below:

**Forward Rate f(**

**t-1, 1)**

**= [(1 + s(**

**t**

**))**

^{t}**/ (1 + s(**

**t-1**

**))**

^{t-1}**] – 1**

where,

**st:**t-period Spot Rate**st-1:**t-1-period Spot Rate**ft-1, 1:**forward Rate applicable for the period (t-1,1)

**Examples of Forward Rate Formula (With Excel Template)**

Let’s take an example to understand the calculation of the Forward Rate in a better manner.

#### Forward Rate Formula – Example #1

**Suppose an investor tries to determine what the yield will he obtain on a two-year investment made from three years from now. While the spot rate of interest for three years is 8.2% p.a and spot yield for five years is 10.4% p.a on zero-coupon bonds.**

**Solution:**

Forward Rate is calculated using the formula given below

**Forward Rate f(****t-1, 1) ****= [(1 + s(****t****))**^{t}** / (1 + s(****t-1****)**^{t-1}** ] – 1**

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- (1+f(3,2))^2 = (1+s(5))^5 / (1+s(3))^3
- f(3,2) =[{(1+s(5))^5/(1+s(3))^3)^(1/2)}] -1
- f(3,2) =
**0.1378 = 13.78%p.a**

It shows 2-year yields after 3 years from now would be 13.784%p.a

#### Forward Rate Formula – Example #2

**If we take the same as discussed above, the three-year zero-coupon bond price is $0.70 per unit nominal, and the five-year bond cost is $0.5323 per unit nominal.**

**Solution:**

From the Forward rate equation, we can assume (1+s(t))^t = P(t)^-1 and (1+s(t-1))^(t-1) = P(t-1)^-1, where P(t-1) is a unit zero-coupon price with term(t-1) and P(t)is the unit zero-coupon price bond with term t year. So we can write an equation in such a way of these prices as follows:

Forward Rate is calculated using the formula given below

**Forward Rate f(****t-1, 1) ****= [(1 + s(****t****))**^{t}** / (1 + s(****t-1****)**^{t-1}** ] – 1 = P(t-1) / P(t)**

- (1+f(t,1))^1 = [(1 + s(t))
^{t}/ (1 + s(t-1)^{t-1}] - (1+f(3,2))^2 = P(3) / P(5)
- f (3,2) =[P(3) / P(5)^(1/2)}] -1
- f(3,2) = [(0.07 / 0.5323 )^(1/2)}] -1
- f(3,2) =
**0.14676 = 14.68%p.a**

### Explanation

Let’s elaborate on this concept with the help of an example; an oil trader is expecting a delivery of oil barrels in four months. Due to price fluctuation, its price keep on varying. So the trader has decided to enter into a forward contract to buy oil from the seller. Through this contract, both parties enter into a contract and agree to fix a rate of interest in advance at a certain time ‘t’.

**How to Calculate Forward Interest Rates?**

To determine the forward rates, we need to satisfy the no-arbitrage condition, where no two investors should earn the arbitraging between interest periods. Suppose an investor is interested in invest his funds’ for two years. He has two choices:

- Invest directly in a 2-year bond or,
- Invest for a 1-year bond and then end of the year again invest for the next one-more year in a one year bond.

**Step1**. Determine the spot rate s1 of the on-year, s2 spot rate of the two years and one -year forward rate 1f1 for one-year from now.

**Step2**. If the initial value of an investment for the 2-year bond is $1, then the final outcome after 2-years would be =(1+s2)^2

**Step3**: To avoid the arbitrage between the two methods of investment, the value of an investment in the second choice for 2 years would be = (1+s1)(1+1f1)

**(1+s****2****)****2 ****= (1+s****1****) (1+****1****f****1****)**

Step4: When we two spot rates, we can rearrange the above equation and can obtain the one-year forward rate for one year from now.

**1f1 = (1+s2)^2/(1+s1) – 1**

The standard representation of the formula in a mathematical term as such:

**Forward Rate f****t-1, 1****=(1+s****t****)****t**** ÷ (1+s****t-1****)****t-1**** -1**

where,

**st:**t-period Spot Rate**st-1:**t-1-period Spot Rate**ft-1, 1:**forward Rate applicable for the period (t-1,1)

### Relevance and Use of Forward Rate Formula

Normally, the forward rates are used by the investors, who believe that they have a good understanding of market trends from immediate past to current market scenario relative to prices of specific item changes with respect to time. Merely, it is the belief of the potential investors that the real future rates will always be higher or lower than the present stated market rate. It could be a signal for potential investors and crack the opportunity.

If we try to identify the situation on the basis of an economic indicator, we can analyse the spot rate and the forward rate changes with respect to time, where a spot rate is used by the buyers and the sellers, who believe in immediate buy and sale and act as a starting point to any financial transaction. While a forward rate is merely a market’s expectations for future prices. It serves as an economic indicator, how may the market expect to perform in the future.

In order to make the best decision- it is more likely to return the highest possible yield on investment. It draws a positive impact on the forward rate. Above that, it is merely an estimate of where interest rates more likely to be in the next six months from the time of the investor’s initial investment.

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