# Are bond coupons reinvested at YTM?

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.

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.

• YTM is the hypothetical yield you get by re-investing the coupon at the prevailing interest rate at issuance. The bond will always yield x% only if you're able to re-invest at x%. Oct 1, 2013 at 23:36
• @jeffm can you explain where in the calculation of YTM are we accounting for reinvestment of coupons Oct 2, 2013 at 9:26
• And if that bond was pricing at \$975?
– user18982
Jul 10, 2014 at 20:56

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

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

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.