# Bond - Yield

This is an introductory page in Fixed Income. If you are unfamiliar with any of the terms, you can refer to the Fixed Income Glossary.

In economics, the **yield** of an investment refers to the income return on an investment, expressed on an annual percentage. As such, the yield of a bond is the annualized percentage return that an investor will obtain from buying a bond.

In general, the yield of a bond is inversely proportional to its price. This means that as the yield increases, the price decreases (and vice versa).

Note that there are many different types of bond yields, so you have to be aware of the type. If no specific type is mentioned, then it generally refers to the Yield to Maturity.

## Current yield

The **current yield** is a straight forward calculation where we divide the annual coupon payments by the clean price of the bond. This represents the return that an investor would expect if they purchased the bond and held it for a year. Note that it is not an accurate reflection of annual return, because the market price is subject to change.

\[ \text{Current Yield } = \frac{ \text{ Annual interest payment} } { \text{ Clean price } } \]

For a bond that pays $100 in coupons annually, and is priced at 1000, the current yield is \( \frac{ $100} { $ 1000} = 10.0 \% \). However, your actual return would depend on the price of the bond in a year. If interest rates rose during this time and the bond price were to drop to $950, then the profit is \( $100 + $950 - $1000 = $50 \), and so the actual return is \( \frac{ $50 } { $1000 } = 5. 0 \% \).

An investor bought a bond, with a constant coupon rate, for $1000. The current yield of the bond is 10%. If the yield rises to 11%, what happens to the price?

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## Stated yield

The **stated yield**, or nominal yield, is calculating by dividing the amount of interest paid by the face value. It represents the return paid on the face value of the bond. This yield is independent of the price of the bond. For a fixed coupon bond, the stated yield is equal to the coupon rate.

\[ \text{ Stated yield} = \frac{ \text{ Annual Interest} } { \text{ Face Value } } \]

## Yield to maturity

The **yield to maturity (YTM)** is the internal rate of return earned by an investor who purchases the bond at market price and holds it to maturity, assuming that all coupon and principal payments are made on schedule. It is the single rate \( y \), such that the present value of all repayments discounted at the rate \( y \), would be equal to the price of the bond.

The Yield to Maturity of a bond is the positive value \( y \) which satisfies the equation

\[ P = \frac{ F } { ( 1 + y ) ^ T} + \sum_{ i = 1 } ^ T \frac{ C _ i } { ( 1 + y) ^ i}. \]

### Zero coupon bonds

If a bond does not pay a coupon, then the equation that we have to solve simplifies to \( P = \frac{ F} { ( 1 + y) ^ T } \). As such,

\[ y = \sqrt[ T ] { \frac{ F}{P} } -1 . \]

## Consider a 10-year zero-coupon bond with a face value of $1000. If the current price of the bond is $500, what is the yield to maturity?

Using the above formula,

\[ YTM = \sqrt[ 10 ] { \frac{ $1000 } { $500 } } - 1 = 7.18 \% . _\square \]

Note: This is in line with the Rule of 72, which tells us that it would take an approximate interest rate of \( \frac{ 72}{10} = 7.2 \% \) to double a sum of money over a period of 10 years.

### Coupon-bearing bonds

It is not generally possible to algebraically solve for the YTM in terms of the price. Instead, numerical methods like Newton Raphson is used. Often, we rely on the solver function on a spreadsheet to find the solution.

If coupon payments are non-negative (which is the usual case for bonds), then we are guaranteed a unique positive solution to this equation. See this note for a proof.

In the rare case that the yield curve is constant (equal to \(Y\)) over the life of the bond, then the yield of the bond would be equal to \(Y\). This is because the price of the bond is

\[ P = \frac{ F } { ( 1 + Y ) ^ T} + \sum_{ i = 1 } ^ T \frac{ C _ i } { ( 1 + Y) ^ i}, \]

so when we solve for \( y \) in

\[ P = \frac{ F } { ( 1 + y ) ^ T} + \sum_{ i = 1 } ^ T \frac{ C _ i } { ( 1 + y) ^ i},\]

then \( y = Y \) is clearly a solution.

If you win a Mega Millions Jackpot of $500 million, you can either take a lump-sum payment of $360 million, or take an annual payout of $19 million a year over 26 years. In both cases, you will receive a payment today.

What discount rate (in %) would make you indifferent between these two payment options?

Ignore taxation rates.

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## Yield to call

For a callable bond, the **yield to call** is the internal rate of return on the bond's cash flows, assuming that it is called at the first opportunity, instead of being held to maturity.

For a callable bond that can be called at time \( T^ * \), with a call premium of \( C^ * \), then the yield to call is the positive value \( y \) which satisfies the equation

\[ P = \frac{ F + C^* } { ( 1 + y ) ^ {T^*}} + \sum_{ i = 1 } ^ {T^*} \frac{ C _ i } { ( 1 + y) ^ i}. \]