Yield to maturity

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The Yield to maturity (YTM), book yield or redemption yield of a bond or other fixed-interest security, such as gilts, is the 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 will be held until maturity, and that all coupon and principal payments will be made on schedule.^[1] 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 price of the bond. The YTM is often given in terms of Annual Percentage Rate (A.P.R.), but more usually market convention is followed. In a number of major markets (such as gilts) the convention is to quote annualised 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.^[2]

Main assumptions

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

The bond will be held to maturity.
All coupon and principal payments will be made on schedule.
All the coupons are reinvested at an interest rate equal to the yield-to-maturity.^[3] 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.
The reason for the confusion is this: The YTM is equivalent to a price in the market place. You can bid a 5% YTM on a bond. In that case each cash flow will be discounted at that rate to give you a current number price for a bond. However if you take that price for the bond and annualize it at the YTM you will not get the same economic return as you would get from buying and holding the bond. For example, the paper cited above discounts the cash flows of a 5 year 5% coupon bond at a YTM rate of 5% and shows that the current price is par. However, if you buy a 5 year 5% coupon bond for $100 you would gross $125 at maturity. However if you invest $100 at a rate of 5% for 5 years you would gross: 100 * 1.05^5 = $127.63. The difference between $127.63 and $125 is the reinvestment of the coupon payments at 5%. Therefore if you want to compound the dollar amount used to purchase a bond by the YTM, you will have to reinvest the coupons at the YTM rate as well.
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

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.

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 = (25.842/5.731), giving 1.1625.

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

You buy ABC Company bond which matures in 1 year and has a 5% interest rate (coupon) and has a par value of $100. You pay $90 for the bond.

The current yield is 5.56% (5/90).

If you hold the bond until maturity, ABC Company will pay you $5 as interest and $100 par value for the matured bond. Now for your $90 investment, you get $105, so your yield to maturity is 16.67% [= (105/90)-1] or [=(105-90)/90].

Assume now that you had to pay $101 for the same bond with a 5% interest rate. When the coupon is paid, you get $5. But you had to pay an extra $1 when buying the bond compared to the par value, your real gain is $4 for an initial investment of $101. Your yield to maturity is 4/101 = 3.96%

If you had paid $105 for the same bond, you would get $5 when the coupon is paid. However you had to compensate for the extra $5 of your initial investment. Your gain is 0 and so is your yield to maturity.

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 employed to approximate the yield which renders the present value of future cash flows equal to the bond price.

References

↑ Definition of 'Yield To Maturity (YTM)'
↑ Formulae for Calculating Gilt Prices from Yields
↑ Fabozzi, Frank. The Handbook of Fixed Income Securities. McGraw-Hill, 2005, p. 87.

Mayle, Jan (1993), Standard Securities Calculation Methods: Fixed Income Securities Formulas for Price, Yield and Accrued Interest 1 (3rd ed.), Securities Industry and Financial Markets Association, ISBN 1-882936-01-9 . The standard reference for conventions applicable to US securities.

External links

Yield-to-Maturity and the Reinvestment of Coupon Payments

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