difference between YTM and return on bonds

[archived_user]

Hello,
In a mock question, we are asked to say if the global full life return on a bond will be equal, higher or lower than the YTM, if the spot curve stay the same during the full life of the bond.
The answer is :
[question removed by moderator]
What i do not understand is that YTM is computed from the spot curve (actually it is a ”complex” mean of the spot curve that accounts for lower and higher rate on the spot curve), so if the spot curve did not change, shoudn’t it be the same to reinvest at the different forwards or at the unique YTM ?
Shoudn’t YTM of the bond be equal to the global return if the spot curve DO NOT MOVE ?
Thanks a lot !

 

[archived_user]

Apparently the spot curve slopes upward; the answer says that, but your question didn’t.
Their answer is wrong.
Consider this par curve;

• ½-year par rate: 2.0000%
• 1-year par rate: 2.4000%
• 1½-year par rate: 2.7600%

The corresponding spot curve is:

• ½-year spot rate: 2.0000%
• 1-year spot rate: 2.4024%
• 1½-year spot rate: 2.7669%

You buy a 1½-year bond paying a 2.76% coupon, so you pay $1,000. Let’s see what happens to your cash flows over the next 1½ years, and what your investment is worth in 1½ years. We’re assuming that the par rates (and, therefore, the spot) rates don’t change during that time.

• Six months from today you get a coupon payment of $13.80 (= $1,000 × 2.76% × ½). As there is one year left until the bond matures, you can reinvest that coupon at the 1-year spot rate of 2.4024%. In one year it will be worth $13.80(1 + 2.4024%/2)2 = $14.13.
• Twelve months from today you get a coupon payment of $13.80 (= $1,000 × 2.76% × ½). As there are six months left until the bond matures, you can reinvest that coupon at the ½-year spot rate of 2.0000%. In six months it will be worth $13.80(1 + 2.0000%/2)2 = $13.94.
• Eighteen months from today you get a coupon payment of $13.80 plus a principal payment of $1,000.

At the end of 18 months, your investment is worth $14.13 + $13.94 + $13.80 + $1,000 = $1,041.87. Your semiannual effective return is ($1,041.87 / $1,000)1/3 − 1 = 0.0138 = 1.38%. Your annual return is 2 × 1.38% = 2.76%, which is exactly the YTM.

 

[archived_user]

Hi S2000magician,
I’m a little bit confused here.
From your response, “Six months from today you get a coupon payment of $13.80 (= $1,000 × 2.76% × ½). As there is one year left until the bond matures, you can reinvest that coupon at the 1-year spot rate of 2.4024%.”, six months later, you get the coupon payment, but why are we using reinvestment rate of 2.4024%, which is 1-year spot rate? Shouldn’t it be 1 year forward rate after 6 months?
Regards,
Brockman

 

[archived_user]

Brockman wrote: Hi S2000magician,
I’m a little bit confused here.
From your response, “Six months from today you get a coupon payment of $13.80 (= $1,000 × 2.76% × ½). As there is one year left until the bond matures, you can reinvest that coupon at the 1-year spot rate of 2.4024%.”, six months later, you get the coupon payment, but why are we using reinvestment rate of 2.4024%, which is 1-year spot rate? Shouldn’t it be 1 year forward rate after 6 months?
Regards,
Brockman

I cannot be held responsible for anything I wrote over 2½ years ago.
However, this one time I think that I’m correct: we’re told that the spot curve doesn’t change during the life of the bond, so 6 months from now the 1-year spot rate will be the same as it is today: 2.4024%.