YTM is yield achieved irrespective of reinvestment of coupons.

Consider a $1000 bond with 10% YTM that pays 10% coupons for 3 years.

Case 1: With reinvestment of coupons

Total return = $121 + $110 + $1100 = $1331
(where $1100 is the final coupon + par value).

FV = PV (1 + r)ⁿ

Since FV = $1331, PV = $1000, n = 3:

Annualized return, r = (1331/1000)^1/3 - 1 = 0.1 = 10% = YTM

Case 2: No reinvestment of coupons

Total return = $100 + $100 + $1100 = $1300
(where $1100 is the final coupon + par value).

But now we cannot use the compound interest formula as the coupons were not reinvested, so we use simple interest:

FV = PV (1 + r * n)

Since FV = $1300, PV = $1000, n = 3:

r = ((1300/1000) - 1) / 3 = 0.1 = 10% = YTM

Thus, the bond will always return the YTM, whether or not the coupons are reinvested.

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

YTM is yield achieved irrespective of reinvestment of coupons.

Consider a $1000 bond with 10% YTM that pays 10% coupons for 3 years.

Case 1: With reinvestment of coupons

Total return = $121 + $110 + $1100 = $1331
(where $1100 is the final coupon + par value).

FV = PV (1 + r)ⁿ

Since FV = $1331, PV = $1000, n = 3:

Annualized return, r = (1331/1000)^1/3 - 1 = 0.1 = 10% = YTM

Case 2: No reinvestment of coupons

Total return = $100 + $100 + $1100 = $1300
(where $1100 is the final coupon + par value).

But now we cannot use the compound interest formula as the coupons were not reinvested, so we use simple interest:

FV = PV (1 + r * n)

Since FV = $1300, PV = $1000, n = 3:

r = ((1300/1000) - 1) / 3 = 0.1 = 10% = YTM

Thus, the bond will always return the YTM, irrespective of reinvestment of coupons.

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

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.

Consider a $1000 bond with 10% YTM that pays 10% coupons for 3 years.

Case 1: With reinvestment of coupons

Total return = $121 + $110 + $1100 = $1331
(where $1100 is the final coupon + par value).

FV = PV (1 + r)ⁿ

Since FV = $1331, PV = $1000, n = 3:

Annualized return, r = (1331/1000)^1/3 - 1 = 0.1 = 10% = YTM

Case 2: No reinvestment of coupons

Total return = $100 + $100 + $1100 = $1300
(where $1100 is the final coupon + par value).

But now we cannot use the compound interest formula as the coupons were not reinvested, so we use simple interest:

FV = PV (1 + r * n)

Since FV = $1300, PV = $1000, n = 3:

r = ((1300/1000) - 1) / 3 = 0.1 = 10% = YTM

Thus, the bond will always return the YTM, whether or not the coupons are reinvested.