Bond Value Quick question

[NNath]

Q. The price of a 1,000 par value Treasury Bond (T-bond) with a 3% coupon that matures in 1.5 years is closest to:

A 1. 1010.02
A 2. 1011.85
A 3. 1013.68
A 4. 1015.51

The price is calculated as $15(0.992556) + $15(0.98224) +$1015(0.967713) = $ 1011.85
BUT
using the bond keys N=3, PMT=15, FV=1000, I/Y=1.5 CPT PV = 1000. What am I doing wrong.

 

[David Harper CFA FRM]

Hi @NNath

By keying in PMT = 15 and I/Y=1.5 you are assuming the coupon rate equals the yield, which ensures the bond will price to par (i.e., the coupon returns exactly the yield such that no capital appreciation is required). PMT = 15 * 2 = 30 which is 3.0% of the face value and I/Y = 1.5*2 = 3.0% yield with semi-annual compounding. Presumably the questions gives you either the spot rates or the discount factors. Notice:

• r(0.5) = [(1/0.992556)^(1/[0.5*2])-1]*2 = 1.50%
• r(1.0) = [(1/0.98224)^(1/[1.0*2])-1]*2 = 1.80%
• r(3.0) = [(1/0.967713)^(1/[1.5*2])-1]*2 = 2.20%

So the spot rate curve is 1.50% @ 0.5 years, 1.80% @ 1.0 years and 2.20% at 3.0 years. The answer is simply multiplying the discount factors, which is equivalent to:

• $15/(1+1.50%/2)^(0.5*2) = 14.888
  
• $15/(1+1.80%/2)^(1.0*2) = 14.734
  
• $1,015/(1+2.20%/2)^(1.5*2) = 982.229
• And 14.888 + 14.734 + 982.229 = $1,011.851

So the mistake is assuming yield is 3.0% with I/Y = 1.5. You cannot know the yield with only par, coupon rate and maturity: you need either the price (which the question is looking for) or you need to be able to infer the spot/forward curve somehow. I hope that helps!

 

[David Harper CFA FRM]

@NNath I was practicing some intro videos yesterday, including this < 5 min video on discount factors, in case it's helpful:

 

[NNath]

Thanks David, When in doubt, trust discount factor.