Example 14.3: Yield to Maturity

Suppose an 8% coupon, 30year bond is selling at 1,276.76 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 equal the bond price. This is the rate consistent with the observed price of the bond. Therefore, we solve for r in the following equation: [pic]

1,276.76 = [pic] $40 + $1000 (1+ r)t (1+ r)60

Or equivalently,

1,276.76 = 40 × Annuity factor (r , 60) + 1,000 × PV factor (r , 60)

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 = .03, or 3% per half year. This is considered the bond ‘s yield to maturity.

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 ,or APR. Yields annualized using simple interest are also called “bond equivalent yields.”

Example 14.4: Yield to Call

Suppose the 8% coupon, 30- year maturity bond sells for $1,150 and is callable in 10 years at a call price of $1,100. its yield to maturity (YTM) and yield to call would be calculated using the following inputs

Yield to call Yield to Maturity | | |Coupon payment $40 $40 | |Number of Semiannual Periods 20 periods 60 periods | |Final Payment $1,100 $1,000 | |Price $1,150 $1,150 |

Yield to call is then 6.64% (To confirm this in our calculator, input n = 20; PV =1,150; FV = 1100; PMT= 40; Compute i as 3.32% or 6.64% bond equivalent yield).

Yield to maturity is 6.82% . (To confirm, input n = 60; PV= (-) 1,150; FV = 1000; PMT= 40 ; Compute i as 3.41% or 6.82% equivalent yield.

Example 14.5: Realized Compound yield

Consider, a 2 year bond selling at par paying a 10% coupon once a year. The YTM=10%.

If the $100 coupon is re-invested at an interest rate of 10%, the $1000 investment in the bond will grow after the 2 years to $1,210.

Reinvestment Rate=10%

[pic]

The compound growth rate of invested funds is therefore:

$1,000 (1 + y realized) 2 = $1,210

y realized = 0.1 = 10%

Thus, with a reinvestment rate=YTM, the realized compound yield= YTM

If the interest rate earned on the first coupon is less than 10%, the final value of the investment will be less than $1,210, and the realized compound yield will be less than 10%. To illustrate, suppose the interest rate at which the coupon can be invested equals 8%.

Future Value of first coupon payment with interest earnings $100 × 1.08 = $108 Cash payment in second year (final coupon plus par value) $1,100 Total value of investment with reinvested coupons $ 1,208

The realized compound yield is computed by calculating the compound rate of growth of invested funds, assuming that all coupon payments are reinvested. The investor purchased the bond for par at $ 1,000, and this investment grew to $ 1,208.

$1,000 (1 + y realized) 2 = $1,208...