Question

A 20-year, 4% quarterly coupon, $1,000 par value bond is selling for $1,057.31 with. Find its YTM in (a) APR and (b) EAR.

Answer 1

a) Par value or Face value = $1000, Current selling price = $1057.31, Coupon rate = 4%, No of years to maturity = 20

Since the bond pays coupon quarterly, therefore

Quarterly coupon payment = (Coupon rate x par value) / no of quarters in a years = (4% x 1000) / 4 = 40 / 4 = $10

No of quarters to maturity = No of quarters in a year x No of years to maturity = 4 x 20 = 80 quarters

We know that YTM in APR = No of quarters in a years x quarterly yield to maturity = 4 x quarterly yield to maturity

We can find the quarterly rate to maturity using RATE function in excel

Formula to be used in excel: =RATE(nper,-pmt,pv,-fv)

Using RATE function in excel, we get quarterly yield to maturity = 0.8992%

YTM in APR = 4 x quarterly yield to maturity = 4 x 0.8992% = 3.5968% = 3.60% (rounded to two decimal places)

Hence YTM in APR = 3.60%

b) YTM in EAR = (1 + Quarterly rate)^{no of quarters in a year} - 1 = (1 + 0.8992%)⁴ - 1 = (1.008992)⁴ - 1 = 1.0364560 - 1 = 0.0364560 = 3.64560% = 3.65% (rounded to two places of decimal)

Hence YTM in EAR = 3.65%