The Zero-volatility spread (Z-spread) is the constant spread that makes the price of a security equal to the present value of its cash flows when added to the yield at each point on the spot rate Treasury curve where cash flow is received. In other words, each cash flow is discounted at the appropriate Treasury spot rate plus the Z-spread. The Z-spread is also known as a static spread.

## Formula and Calculation for the Zero-Volatility Spread

To calculate a Z-spread, an investor must take the Treasury spot rate at each relevant maturity, add the Z-spread to this rate, and then use this combined rate as the discount rate to calculate the price of the bond. The formula to calculate a Z-spread is:

P = {C(1) / (1 + (r(1) + Z) / 2) ^ (2 x n)} + {C(2) / (1 + (r(2) + Z) / 2) ^ (2 x n)} + {C(n) / (1 + (r(n) + Z) / 2) ^ (2 x n)}

where:

P = the current price of the bond plus any accrued interest

C(x) = bond coupon payment

r(x) = the spot rate at each maturity

T = the total cash flow received at the bond's maturity

n = the relevant time period

The generalized formula is:

For example, assume a bond is currently priced at \$104.90. It has three future cash flows: a \$5 payment next year, a \$5 payment two years from now and a final total payment of \$105 in three years. The Treasury spot rate at the one-, two-, and three- year marks are 2.5%, 2.7% and 3%. The formula would be set up as follows:

\$104.90 = \$5 / (1 +(2.5% + Z) / 2) ^ (2 x 1) + \$5 / (1 +(2.7% + Z) / 2) ^ (2 x 2) + \$105 / (1 +(3% + Z) / 2) ^ (2 x 3)

With the correct Z-spread, this simplifies to:

\$104.90 = \$4.87 + \$4.72 + \$95.32

This implies that the Z-spread equals 0.5% in this example.

### Key Takeaways

• The zero-volatility spread of a bond tells the investor the bond's current value plus its cash flows at certain points on the Treasury curve where cash-flow is received.