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.

Answer 1

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

1st year coupon	2nd year coupon	Coupon reinvested at 10%	3rd year value of old coupons
$100	-	100 * 1.12	= $121
-	$100	100 * 1.11	= $110

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

1st year coupon	2nd year coupon	3rd year value of old coupons
$100	-	= $100
-	$100	= $100

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

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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

Answer 2

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.