# Introduction to Fixed-Income Valuation: Bond Prices and Time Value of Money | CFA Level I Fixed Income

Let’s dive into the world of fixed income valuation. In this study note, we’ll cover the application of time value of money principles in pricing bonds. Get ready to discover how bond prices are influenced by factors like interest rates, credit quality, and time to maturity.

## Using Time Value of Money Principles in Bond Pricing

Bond pricing is an application of **discounted cash flow analysis**. We’ll explore how the present value of a bond’s future cash flows can be calculated using the market discount rate. But first, let’s take a look at an example:

**EXAMPLE**

**Consider a 5-year, $1,000 par, 10% coupon, annual-pay bond. At issuance, the market discount rate for the bond is 10%. Calculate the price of the bond using the time value of money (TVM) principles.**

To calculate the bond’s price, we need to find the present value of all its promised cash flows using the market discount rate.

In the example, the price turns out to be $1,000. Here’s a quick summary of the relationships:

- When the market discount rate is equal to the coupon rate, the bond is priced at par value.
- When the market discount rate is less than the coupon rate, the bond is priced at a premium above par value.
- When the market discount rate is greater than the coupon rate, the bond is priced at a discount below par value.

**This means that there is an inverse relationship between the market discount rate and the price of the bond.**

## Adjusting Bond Pricing Calculations for Semi-Annual Pay Bonds

Most bonds in the market pay coupons semi-annually. To adjust our calculations for such bonds, we simply need to modify the number of periods and the discount rate per period. Here’s how:

**EXAMPLE**

**Calculate the price of a 5-year, $1,000 par, 10% coupon bond that pays out coupons semi-annually, with a market discount rate of 12%.**

With semi-annual payments, the bond pays coupons of $50 every half a year. The discount rate per half a year is 6%, half the market discount rate of 12%. The bond’s price turns out to be $951.

## Calculating Yield-to-Maturity (YTM) Using Market Price

In a liquid market where a bond is publicly traded with a known market price, we can use the price to calculate the precise yield-to-maturity of the bond based on this price.

**EXAMPLE**

**Calculate the yield-to-maturity of a bond with a market price of $932 and expected future cash flows.**

Using the TVM calculator, enter N = 4 (four half-year periods left to maturity), PMT = 50, PV = -$932 (an outflow of cash from the investor), and FV = $1,000. Solving for IY gives us 7% per half-year period. Multiply by 2 to get the annualized yield-to-maturity, which is 14%.

## Important Assumptions for Yield-to-Maturity

While calculating yield-to-maturity, it’s crucial to keep the following assumptions in mind:

- The bond is held to maturity.
- The issuer does not default on any of the scheduled payments.
- The investor can reinvest coupon payments at the same yield.

## Bond Price-Yield Relationship: Inverse and Convex

The bond price and its yield-to-maturity (market discount rate) share an **inverse relationship**. When the bond price goes up, the yield goes down, and vice versa. Furthermore, the relationship is **convex**, meaning the percentage decrease in value when the yield increases is smaller than the increase in value when the yield decreases by the same amount.

## Other Factors Affecting Bond Price Sensitivity

Besides the inverse and convex effects, there are two more factors affecting bond price sensitivity:

**Coupon effect**: For the same time-to-maturity, a lower-coupon bond has a greater percentage price change than a higher-coupon bond when their market discount rates change by the same amount.**Maturity effect**: For the same coupon rate, a bond that is further from maturity has a greater price sensitivity to a change in market discount rate than a bond that is closer to maturity.

## Constant-Yield Price Trajectory

The **constant-yield price trajectory** is the convergence of the bond price to its par value at maturity, regardless of whether the market discount rate is above or below the bond’s coupon rate. This is because the par value is repaid to the investor at maturity.

## Using Spot Rates for Bond Pricing

Using the discounted cash flow method, we can use a sequence of market discount rates corresponding to the cash flow dates, called **spot rates**, to determine the price of a bond more precisely. These rates can be viewed as the yield of zero-coupon bonds maturing at the date of the cash flow.

**EXAMPLE**

## Key Takeaways

- The bond price and yield-to-maturity have an inverse and convex relationship.
- Other factors affecting bond price sensitivity include the coupon effect and the maturity effect.
- Bond prices follow a constant-yield price trajectory, converging to the par value at maturity.
- Using spot rates for bond pricing provides a more precise valuation method.

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