We have seen that a bond can be valued using spot rates by discounting each cash flow by the spot rate for the maturity. We also saw that forward rates can be derived from spot rates. If so, we can also value a bond using forward rates instead of spot rates. Let’s take a specific

# Fixed Income Securities

## 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

## Option-adjusted Spreads (OAS)

The Z-spread handles one problem present in nominal spread effectively, i.e., that is, it measures the spread over the entire spot rate curve instead of only at one point in the curve. However, there is another problem that comes because of the embedded options. Due to the embedded options in the bonds, there is uncertainty

## Z-Spread: Definition and Calculation

The problem with nominal spread is that it measures the spread at just one point on the yield curve. The z-spread solves this problem by considering the spot yield curve instead of the standard yield curve. The z-spread, also known as the zero-volatility spread or the static spread, measures the spread that the investor will

## Nominal Spread

Nominal Spread, or the nominal yield spread, is the most simple yield spread for non-Treasury bonds. Nominal spread measures the difference between the yield of a bond and the yield to maturity of a similar maturity Treasury bond. Consider the following two 10-year bonds: A Treasury bond having a YTM of 6.5% A non-Treasury bond

## How to Price a Bond Using Spot Rates (Zero Curve)

The simplest way to calculate the value of a bond is to take the cash flows of the bond till its maturity and then discount them by a single discount rate. The method is quick but not very accurate because the yield curve is not flat and the interest rates are different for different maturities.

## Bootstrapping Spot Rate Curve (Zero Curve)

A spot rate curve, also known as a zero curve refers to the yield curve constructed using the spot rates such as Treasury spot rates instead of the yields. A spot rate Treasury curve is more suitable to price bonds because most bonds provide multiple cash flows (coupons) to the bond holders at different points

## Cash Flow Yield

For some debt products such as mortgage-backed securities, we calculate cash flow yield instead of yield to maturity. This is because investors in mortgage-backed securities receive their cash flow from the cash flow of the underlying collateral pool and this cash flow is not fixed. In the underlying collateral pool, the home owners make the

## How to Calculate Yield to Call of a Bond

For callable bonds that are likely to be called before their maturity, it is more useful to calculate yield to call instead of yield to maturity. The formula and steps to calculate yield to call are exactly the same as how we calculate yield to maturity, i.e., you calculate the discount rate that makes the

## Calculate Bond-Equivalent Yield of Annual-Pay Bonds

In the US, most bonds are generally semi-annual coupon paying bonds, so we calculate the semi-annual yield and then calculate the bond-equivalent yield (annualized) by simply doubling the semi-annual yield. This is done when the bonds have semi-annual coupon payments. However, not all bonds pay semi-annual coupon, especially there are many non-US bonds that pay