Question

Suppose a bond with a 10% coupon rate and semiannual coupons, has a face value of $1000, 20 years to maturity and is selling for $1197.93.

◦Is the YTM more or less than 10%?

1197.93 = 50[1 – 1/(1+r)40] / r + 1000 / (1+r)40

i know the answer is 4% but how can i calculated the r by hand?

Answer 1

Face value=   1000  
time (n) semiannual periods =20*2=   40  
price =   1,197.9300  
Coupon = 1000*10%*1/2=   50  
      
Bond price formula = Coupon amount * (1 - (1/(1+r)^n)/r + face value/(1+r)^n      

Method of calculating r by hand is trial and error method. In which we Assume r and calculate price of bond assuming r is the relevant period yield.

Assume r=3%      
50*(1-(1/(1+0.03)^40))/0.03+ 1000/(1+0.03)^40      
=1462.295439      

Assume r=4%      
50*(1-(1/(1+0.04)^40))/0.04 + 1000/(1+0.04)^40      
=1197.927739      

Price of bond is $1197.93 at r of 4%. SO Semiannual yield is 4% and annual yield is 8%.

Note : important

Here price result is exacatly $1197.93 at r of 4%. If price calculated were lower by this amount. Then r can be calculated by below mentioned interpolation formula

interpolation formula = lower rate + (uper rate - lower rate)*(Uper price - bond actual price)/(uper price - lower price)      
3% + ((4%-3%)*(1462.295439-1197.93)/(1462.295439-1197.93))
=4.00%