# What is the Swap Rate?

3-30-2017

Lenders generally determine their interest rates by adding a risk-based spread over a risk-free index. In commercial real estate lending, two commonly used indices are the Treasury yield and the Swap rate. For example, life insurance lenders generally use the Treasury yield, and CMBS lenders generally use the Swap rate.

The Treasury yield is the current interest rate on US Treasury bonds. Since Treasury bonds are constantly traded, the Treasury yield can be constantly observed in the bond market. But what is the Swap rate?

An interest rate Swap is a contract in which one party agrees to pay a fixed interest rate to another party in exchange for receiving a variable rate. In effect, this contract converts a variable rate loan into a fixed rate loan.

At the time of the Swap contract, the present value of the fixed rate payment stream must be equal to the present value of the variable rate payment stream. Otherwise, it would not be a fair exchange. The Swap rate is defined as the interest rate that makes the present value of the fixed rate payment stream equal to the present value of the variable rate payment stream.

Let’s use a simplified example to show how the Swap rate is determined. One party agrees to pay the 10-year Swap rate to another party in exchange for receiving 10 years of variable interest payments based on 90-day LIBOR. (The principal amount does not matter.) In order to determine the Swap rate, we need to determine the present value of 40 expected variable rate interest payments: four payments per year for 10 years.

To do this, we need to know the expected LIBOR rates at 40 different times in the future. A graph of expected future rates is called the forward yield curve. We will use the forward curve to determine the dollar amounts of the 40 interest payments in the variable rate payment stream.

Where do we find the 10-year LIBOR forward curve? It is derived from the current LIBOR rates over the next 10-years. A graph of current rates with different maturities is called the spot yield curve. The forward curve is mathematically derived from the spot curve, based on the assumption that long term spot rates are an average of short term forward rates. This is an example of arbitrage pricing theory.

But we have a problem. LIBOR based loans only have maturities up to 12 months. So where do we find the spot curve from 12 months to 10 years, from which we can derive the forward curve?

The answer is that the 10-year LIBOR spot curve is mathematically constructed from 12 months to 10 years based on other observable interest rates: LIBOR forward contracts from 1 to 4 years, and US Treasuries from 4 to 10 years. After constructing the 10-year LIBOR spot curve in this manner, we can mathematically derive the forward curve from it, using the arbitrage theory mentioned above.

Now that we have the 10-year LIBOR forward curve, we can identify the expected LIBOR rates and the dollar amounts of the 40 monthly payments in the variable rate payment stream. We use the corresponding spot rate to discount each payment to the present. We add up the discounted payments to get the present value of the variable rate payment stream.

In the fixed rate payment stream, there are 120 monthly fixed payments during the 10-year term. These payments are discounted to the present using the corresponding spot rates. The sum of the 120 payments is the present value of the fixed rate payment stream. The Swap rate is the interest rate that makes the present value the fixed rate payment stream equal to the present value of the variable rate payment stream.

The Swap rate is called a derivative interest rate because it is derived from other interest rates. Notice that the Swap rate is derived from a derivative. First, the 10-year LIBOR spot curve is mathematically derived from other interest rates, and second, the forward curve is mathematically derived from the spot curve.

The Swap rate is used as an index on billions of dollars of CMBS loans every year, not to mention trillions of dollars of Swap contracts. I suspect that many real estate professionals do not realize that this widely used interest rate is not a market rate, but is mathematically derived from other interest rates in the financial markets.