## What is the 'Zero-Volatility Spread - Z-spread'

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 a 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.

## BREAKING DOWN 'Zero-Volatility Spread - Z-spread'

The Z-spread helps analysts discover if there is a discrepancy in a bond's price. Because the Z-spread measures the spread that an investor will receive over the entirety of the Treasury yield curve, it gives analysts a more realistic valuation of a security instead of a single-point metric, such as a bond's maturity date.

## Z-Spread Calculation

A Z-spread calculation is different than a nominal spread calculation. A nominal spread calculation uses one point on the Treasury yield curve (not the spot-rate Treasury yield curve) to determine the spread at a single point that will equal the present value of the security's cash flows to its price.

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 components that go into a Z-spread calculation are as follows:

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

C(x) = bond coupon payment

r(x) = the spot rate at each maturity

Z = the Z-spread

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

n = the relevant time period

The generalized formula 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)}

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.