Are bond coupons reinvested at YTM?

Question

I fail to understand reinvestment of coupons to calculate yield to maturity (YTM). I understand that YTM is the rate at which coupon payments and par value of the bond are discounted to today. i.e.
C = Coupon
T = Time

If a bond pays

C1 @ T1,
C2 @ T2,
C3 + par value @ T3

then

YTM is rate at which the price of the bond (determined by market) equals 
the present value of (C1, C2, C3 + par value) at respective times

There is no reinvestment of C1, C2, C3 but reinvestment of the interest earned on these coupons at compounding intervals T1, T2 & T3.
Link: Yield-to-Maturity and the Reinvestment of Coupon Payments
So my assumption is that if a bond is bought @ x% YTM, it will always yield x% if held till maturity irrespective of YTM in the future when the coupons are paid.

Pranay Warke · Answer

YTM is yield achieved irrespective of reinvestment of coupons.

Suppose $1000 bond with 10% YTM paying 10% coupons for 3 yrs(n)

Case 1 : Reinvestment of coupons

1 yr      2nd yr     reinvested @ 10%    3rd yr Final Value

$100        -       100 * 1.1^2              121
   -         $100    100 * 1.1^1              110

So total return = 121 + 110 + 1100( final coupon + par value )  = 1331
 annualized return = (1331/1000)^(1/3) - 1      FV = PV ( 1 + r ) ^ n  --> rearranged
 = 0.1 = 10% = YTM

No reinvestment of coupons

Case 2 : No reinvestment

1 yr      2nd yr      ! reinvested @ 10%    3rd yr Final Value

$100        -                -               100
   -         $100             -               100

So total return = 100 + 100 + 1100( final coupon + par value )  = 1300
   But now we cannot use the compound interest formula as interest was not reinvested,
   hence its simple interest
   FV = PV ( 1 + r * n )
   1300 = 1000 ( 1 + r*3)
   r =10%

Thus irrespective of coupon re-invested or not, bond will always return at YTM.

I think there are just different perspectives to look at it.

The investor is definitely at loss if coupon payments are not reinvested,
but the YTM is always delivered by the bond as promised during initial investment.

References:

http://www.economics-finance.org/jefe/econ/ForbesHatemPaulpaper.pdf

http://www.economics-finance.org/jefe/econ/CebulaYangpaper.pdf

Michael Harris · Answer

But the two metrics are different. Two different things cannot be the same thing.

I believe that the confusion arises when equating YTM to CAGR. In that case, it assumes reinvestment at the YTM rate. Note that your second calculation is not CAGR. In that case CAGR is 9.139% So apparently, it's a matter of definition/use of a metric and not an error as the authors in the quoted paper claim. Regardless, maximization of bond investment wealth requires reinvestment of coupons.

Are bond coupons reinvested at YTM?

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