- Sources of Return from Investing in a Bond
- How to Calculate Current Yield
- How to Calculate Yield to Maturity
- Bond Equivalent Yield Convention
- Yield to Maturity (YTM) Approximation Formula
- YTM and Reinvestment Risk
- Factors Affecting Reinvestment Risk
- Calculate Bond-Equivalent Yield of Annual-Pay Bonds
- How to Calculate Yield to Call of a Bond
- Cash Flow Yield
- Bootstrapping Spot Rate Curve (Zero Curve)
- How to Price a Bond Using Spot Rates (Zero Curve)
- Nominal Spread
- Z-Spread: Definition and Calculation
- Option-adjusted Spreads (OAS)
- What are Forward Rates?
- How to Calculate Forward Rates from Spot Rates?
- How to Value a Bond Using Forward Rates

# How to Calculate Forward Rates from Spot Rates?

Once we have the **spot rate curve**, we can easily use it to derive the forward rates. The key idea is to satisfy the no arbitrage condition – no two investors should be able to earn a return from arbitraging between different interest periods. Let’s take an example of how this works. Let’s say an investor wants to invests his funds for two years. He is faced with two choices:

- Directly invest in a 2-year bond
- Invest in a one-year bond, and again invest the proceeds after one year in a one year bond.

Assuming the same nature of investments, the returns from both choices should be the same.

Let’s say *s _{1}* is the one-year spot rate,

*s*is the two-year spot rate and 1f1 is the one year forward rate one year from now.

_{2}Assuming $1 as the initial investment, the value of investment in first choice after two years:

= (1+s_{2})^{2}

The value of investment in second choice after two years:

= (1+s_{1}) (1+_{1}f_{1})

If there are no arbitrage opportunities, both these values should be the same.

**(1+s _{2})^{2 }= (1+s_{1}) (1+_{1}f_{1})**

If we have the spot rates, we can rearrange the above equation to calculate the one-year forward rate one year from now.

_{1}f_{1} = (1+s_{2})^{2}/(1+s_{1}) – 1

Let’s say s_{1} is 6% and s_{2} is 6.5%. The forward rate will be:

_{1}f

_{1}= (1.065^2)/(1.06) – 1

_{1}f

_{1}= 7%

Similarly we can calculate a forward rate for any period.