# Yield to maturity

The yield to maturity (YTM), book yield or redemption yield of a bond or other fixed-interest security, such as gilts, is the (theoretical) internal rate of return (IRR, overall interest rate) earned by an investor who buys the bond today at the market price, assuming that the bond is held until maturity, and that all coupon and principal payments are made on schedule. Yield to maturity is the discount rate at which the sum of all future cash flows from the bond (coupons and principal) is equal to the current price of the bond. The YTM is often given in terms of Annual Percentage Rate (A.P.R.), but more often market convention is followed. In a number of major markets (such as gilts) the convention is to quote annualized yields with semi-annual compounding (see compound interest); thus, for example, an annual effective yield of 10.25% would be quoted as 10.00%, because 1.05 × 1.05 = 1.1025 and 2 × 5 = 10.

## Main assumptions

The main underlying assumptions used concerning the traditional yield measures are:

• The bond is held to maturity.
• All coupon and principal payments are made on schedule.
• The yield to maturity is the single interest rate that equates the present value of a bond's cash flows to its price. A common misconception is that the coupons must be reinvested at the yield to maturity. However, the paper Yield-to-Maturity and the Reinvestment of Coupon Payments says making this assumption is a common mistake in financial literature and coupon reinvestment is not required for YTM formula to hold. And, as Fabozzi states: "The implication is that the yield to maturity measure tells little about the potential yield that an investment may realize if held to maturity."
• The yield is usually quoted without making any allowance for tax paid by the investor on the return, and is then known as "gross redemption yield". It also does not make any allowance for the dealing costs incurred by the purchaser (or seller).

## Coupon rate vs. YTM and parity

• If a bond's coupon rate is less than its YTM, then the bond is selling at a discount.
• If a bond's coupon rate is more than its YTM, then the bond is selling at a premium.
• If a bond's coupon rate is equal to its YTM, then the bond is selling at par.

## Variants of yield to maturity

As some bonds have different characteristics, there are some variants of YTM:

• Yield to call (YTC): when a bond is callable (can be repurchased by the issuer before the maturity), the market looks also to the Yield to call, which is the same calculation of the YTM, but assumes that the bond will be called, so the cashflow is shortened.
• Yield to put (YTP): same as yield to call, but when the bond holder has the option to sell the bond back to the issuer at a fixed price on specified date.
• Yield to worst (YTW): when a bond is callable, puttable, exchangeable, or has other features, the yield to worst is the lowest yield of yield to maturity, yield to call, yield to put, and others.

## Consequences

When the YTM is less than the (expected) yield of another investment, one might be tempted to swap the investments. Care should be taken to subtract any transaction costs, or taxes.

## Calculations

### Formula for yield to maturity for zero-coupon bonds

${\text{Yield to maturity(YTM)}}={\sqrt[{\text{Time period}}]{\dfrac {\text{Face value}}{\text{Present value}}}}-1$ #### Example 1

Consider a 30-year zero-coupon bond with a face value of $100. If the bond is priced at an annual YTM of 10%, it will cost$5.73 today (the present value of this cash flow, 100/(1.1)30 = 5.73). Over the coming 30 years, the price will advance to $100, and the annualized return will be 10%. What happens in the meantime? Suppose that over the first 10 years of the holding period, interest rates decline, and the yield-to-maturity on the bond falls to 7%. With 20 years remaining to maturity, the price of the bond will be 100/1.0720, or$25.84. Even though the yield-to-maturity for the remaining life of the bond is just 7%, and the yield-to-maturity bargained for when the bond was purchased was only 10%, the return earned over the first 10 years is 16.25%. This can be found by evaluating (1+i) from the equation (1+i)10 = (25.882/5.7389), giving 0.1625.

Over the remaining 20 years of the bond, the annual rate earned is not 16.25%, but rather 7%. This can be found by evaluating (1+i) from the equation (1+i)20 = 100/25.84, giving 1.07. Over the entire 30 year holding period, the original $5.73 invested increased to$100, so 10% per annum was earned, irrespective of any interest rate changes in between.

#### Example 2

An ABCXYZ Company bond that matures in one year, has a 5% yearly interest rate (coupon), and has a par value of $100. To sell to a new investor the bond must be priced for a current yield of 5.56%. The annual bond coupon should increase from$5 to $5.56 but the coupon can't change as only the bond price can change. So the bond is priced approximately at$100 - $0.56 or$99.44 .

If the bond is held until maturity, the bond will pay $5 as interest and$100 par value for the matured bond. For the $99.44 investment, the bond investor will receive$105 and therefor the yield to maturity is 5.56 / 99.44 for 5.59% in the one year time period. Then continuing by trial and error, a bond gain of 5.53 divided by a bond price of 99.47 produces a yield to maturity of 5.56%. Also, the bond gain and the bond price add up to 105.

Finally, a one-year zero-coupon bond of \$105 and with a yield to maturity of 5.56%, calculates at a price of 105 / 1.0556^1 or 99.47.

### Coupon-bearing Bonds

For bonds with multiple coupons, it is not generally possible to solve for yield in terms of price algebraically. A numerical root-finding technique such as Newton's method must be used to approximate the yield, which renders the present value of future cash flows equal to the bond price.

### Varying coupon

With varying coupons the general discounting rule should be applied.

## Subscriber Yield

A term used in Japan, this is simply the Yield to Maturity at time of issue: in other words the Yield to Maturity enjoyed by the buyer (subscriber) in the primary market.