Bonds consist of fixed coupons and the pricing of the bond involves the use of combinations of infinite series. A typical example would be the sum of terms that increase at an infinite rate. This formula can be proved by multiplying both sides of the equation by (1-a) and then cancelling common terms on both […]

# Fixed Income Securities

## Price Yield Relationship for Bonds

For a bond with a cash flow pattern the present value can be calculated using the following formula: Where the numerator denotes the coupon or principal in that period, t is the number of periods, T is the number of periods to final maturity and y is the discount factor per period. For a bond […]

## Discounting of Cash Flows

A zero coupon bond is a fixed income instrument with only one cash flow i.e. the face value of the bond which is paid to the investor at the maturity date. It is a more appropriate instrument for exemplifying the discounting of cash flows. Present Value The price of the zero coupon bond is determined […]

## Measuring Interest Rate Risk: Full Valuation Approach

We know that bond prices are sensitive to interest rate changes. A portfolio of bonds will suffer a loss if the interest rates rise and vice versa. Similarly, a short bond position will make losses when interest rates fall. What a portfolio manager is interested in is to know the exact losses his portfolio will […]

## How to Value a Bond Using Forward Rates

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 […]

## 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. […]