Bond Yields (Current yield, YTM, Yield to Call) (Invesments N2&3)

Bond Yields

We have noted that the current yield of a bond measures only the cash income provided by the bond as a percentage of bond price and ignores any prospective capital gains or losses. We would like a measure of rate of return that accounts for both current income and the price increase or decrease over the bond’s life. The yield to maturity is the standard measure of the total rate of return. However, it is far from perfect, and we will explore several variations of this measure.

Yield to maturity

In practice, an investor considering the purchase of a bond is not quoted a promised rate of return. Instead, the investor must use the bond price, maturity date, and coupon payments to infer the return offered by the bond over its life. The YTM is defined as the interest rate that makes the present value of a bond’s payments equal to its price. This interest rate is often interpreted as a measure of the average rate of return that will be earned on a bond if it is bought now and held until maturity. To calculate the yield to maturity, we solve the bond price equation for the interest rate given the bond’s price.

Example:

Suppose an 8% coupon, 30year bond is selling at 1276.76. What average rate of return would be earned by an investor purchasing the bond at this price? We find the interest rate at which the present value of the remaining 60 semiannual payments equals the bond price. This is the rate consistent with the observed price of the bond. Therefore, we solve for r in the following equation.

$40 dollar payment 60 times + $1000 discounted at 60 time

These equations have only one unknown variable, the interest rate, r. You can use a financial calculator or spreadsheet to confirm that the solution is r= 0.03 or 3% per half-year. This is the bond’s YTM.

The financial press reports yields on an annualized basis, and annualizes the bond’s semiannual yield using simple interest techniques, resulting in an annual percentage rate (APR). Yields annualized using simple interest are also called “bond equivalent yields.” Therefore, the semiannual yield would be doubled and reported in the newspaper as a bond equivalent yield of 6%. The effective annual yield of the bond, however, accounts for compound interest. If one earns 3% interest every 6 months, then after 1 year, each dollar invested grows with interest to $1x (1.03)^2= 1.0609, and the effective annual interest rate on the bond is 6.09%.

Excel also provides a function for yield to maturity that is especially useful in-between coupon dates. It is = (YIELD settlement date, maturity date, annual coupon rate, bond price, redemption value as percent of par value, number of coupon payments per year)

The bond price used in the function should be the reported flat price, without accrued interest.

Yield to maturity differs from the current yield of a bond, which is the bond’s annual coupon payment divided by the bond price. For example, for the 8%, 30- year bond currently selling at $1276.76, the current yield would be $80/$1276.76= 0.0627 or 6.27%, per year. In contrast, recall that the effective annual yield to maturity is 6.09%. For this bond, which is selling at a premium over par value ($1276 rather than $1000), the coupon rate (8%) exceeds the current yield (6.27%) which exceeds the yield to maturity (6.09%). The coupon rate exceeds the current yield because the coupon rate divides the coupon payments by par value rather than the bond price ($1276). In turn, the current yield exceeds yield to maturity because the yield to maturity accounts of the built-in-capital loss on the bond; the bond bought today for $1276 will eventually fall in value to $1000 at maturity.

For premium bonds, coupon rate is greater than current yield, which in turn is greater than yield to maturity. For discount bonds, these relationships are reversed.

Yield to Call

 Yield to maturity is calculated on the assumption that the bond will be held until maturity. What if the bond is callable, however, and may be retired prior to the maturity date? How should we measure average rate of return for bonds subject to a call provision?

1. Coupon
"= yearly interest ($ value)/ FV"
2. Current yield
yearly interest/BP
3. YTM
"capital gains and loss+ yearly interest / BP"

	Semiannual coupons		Annual coupons
Settlement date	2000-01-01		2000-01-01
Maturity date	2030-01-01		2030-01-01
annual coupon rate	0.08		0.08
Bond price	127.676		127.676
Redemption value	100		100
Coupon payments per year	2		1
Yield to maturity	0.05999974		0.059912507
	the function used here is "YIELD"

Current yield does not take the capital gain and loss into account. that is the trick.

	1 year	2 year
Discount rate	10%	15%
Cash flow	10	110
PV	9.090909	83.1758034
Sum of PV		92.26671249	this is the price of the bond
14.70%	92.32986
this is the solution for the "equal rate" that would solve for r
This is the YTM

Why do bond prices go down when interest rates go up? Don't lenders like higher interest rates?

A bond's coupon interest payments and principal repayment are not affected by changes in market rates. Consequently, if market rates increase, bond investors in the secondary markets are not willing to pay as much for a claim on a given bond's fixed interest and principal payemnts as they would if market rates were lower.

This relationship is apparent from the inverse relationship between interest rates and prevent value. An increase in the discount rate (the market rate) decreases the present value of the future cash flows.