14.1. Return

Here the most general expression for nominal and real yield are derived as a function of prices, face value of coupon, taxation and year to maturity, both in case of coupon reinvestment or not (reinvestment not always possible); a closed form solution is then derived under some assumptions, like constant (or average) rates; the effect on price and yield of credit rating and rating change, coupon, year to maturity are discussed on both examples and real-world cases.

Extra:

• definition of duration

• risks: inflation; reinvesment (at lower rates) for bonds with same maturity and different coupons

• inflation linked

14.1.1. Constant coupon bonds

14.1.1.1. W/o reinvestment

At time \(t_0\) the unit price of a bond is \(p_0\); investing \(Y_0\) allows to buy \(N_0 = \frac{Y_0}{p_0}\) titles; each title has the right of receiving net coupon \(C (1 - t)\), with \(t\) taxation rate, per period (here assumed 1-year coupon range).

\[N_0 = \dfrac{Y_0}{p_0} = \dfrac{Y_0}{p_{in}} \dfrac{p_{in}}{p_0}\]

W/o reinvestment, the number of titles hold is constant and equal to \(N_0\). As capital \(Y_i\) can be written as the product of unit price and number of bond in portfolio, the DCF of a bond w/o coupon reinvestment reads

\[\begin{split}\begin{aligned} \widetilde{DCF} = & = - Y_0 + Y_N \prod_{k=1}^N ( 1 + r_k )^{-1} + \sum_{k=1}^{N} N_0 C (1-t) \prod_{j=1}^{k} (1 + r_j )^{-1} \\ & = N_0 \left[ - p_0 + p_N \prod_{k=1}^N ( 1 + r_k )^{-1} + C (1-t) \sum_{k=1}^{N} \prod_{j=1}^{k} (1 + r_j )^{-1} \right] \ , \end{aligned}\end{split}\]

This DCF must be corrected a CF at time \(t_N\) corresponding to tax on capital gain if \(p_n > p_0\), discounted as

\[- N_0( p_N - p_0) \, t \, \prod_{k=1}^{N} (1+r_k)^{-1} \qquad (\text{only if $p_N > p_0$})\]

The cumulative real return (if the discount ratio is inflation) is the ratio between the \(DCF\) and the actual value of the investment \(Y_0\),

\[\widetilde{\dfrac{DCF}{Y_0}} = - 1 + \dfrac{p_N}{p_0} \prod_{k=1}^N ( 1 + r_k )^{-1} + \dfrac{C}{p_0} (1-t) \sum_{k=1}^{N} \prod_{j=1}^{k} (1 + r_j )^{-1} \]

If the discount rate is constant, or the average (which average) discount rate is used, the expression of the cumulative return reads

\[\begin{aligned} \dfrac{\widetilde{DCF}}{Y_0} = - 1 + \dfrac{p_N}{p_0} ( 1 + r )^{-N} + \dfrac{C}{p_0} (1-t) \sum_{k=1}^{N} (1 + r )^{-k} \end{aligned}\]

14.1.1.2. W/ reinvestment

Time	Cashflows	\(\Delta\)Quantity	Quantity	DF
\(0\)	\(-Y_0\)	\(N_0 = \frac{Y_0}{p_0}\)	\(N_0 = \frac{Y_0}{p_0}\)	1
\(1\)	\(+N_0 C ( 1-t )\)			\((1+r_1)^{-1}\)
\(1\)	\(-N_0 C ( 1-t )\)	\(N_1 = \frac{N_0 C (1-t)}{p_1}\)	\(N_{0:1} = N_0+N_1\)	\((1+r_1)^{-1}\)
\(2\)	\(+N_{0:1} C ( 1-t )\)			\((1+r_1)^{-1} (1+r_2)^{-1}\)
\(2\)	\(-N_{0:1} C ( 1-t )\)	\(N_2 = \frac{N_{0:1} C (1-t)}{p_2}\)	\(N_{0:2} = N_0+N_1 + N_2\)	\((1+r_1)^{-1} (1+r_2)^{-1}\)
…
\(T-1\)	\(+N_{0:T-2} C ( 1-t )\)			\(\prod_{k=1}^{T-1} (1+r_k)^{-1}\)
\(T-1\)	\(-N_{0:T-2} C ( 1-t )\)	\(N_{T-1} = \frac{N_{0:T-2} C (1-t)}{p_{T-1}}\)	\(N_{0:T-1} = \sum_{k=0}^{T-1} N_k\)	\(\prod_{k=1}^{T-1} (1+r_k)^{-1}\)
\(T\)	\(+N_{0:T-1} C ( 1-t )\)			\(\prod_{k=1}^{T } (1+r_k)^{-1}\)
\(T\)	\(+N_{0:T-1} p_T\)			\(\prod_{k=1}^{T } (1+r_k)^{-1}\)

All the cashflows from coupons are immediately reinvested so the DCF is

\[\begin{split}\begin{aligned} DCF & = - Y_0 + \underbrace{N_{0:T-1} \left( p_T + C (1-t)\right)}_{Y_T} \, \underbrace{ \prod_{k=1}^T (1+r_k)^{-1} }_{DF_T} = \\ & = - Y_0 + Y_T \, DF_T \ , \end{aligned}\end{split}\]

with

\[\begin{split}\begin{aligned} N_{0:T-1} & = N_{0:T-2} + N_{T-1} = N_{0:T-2} + N_{0:T-2} \frac{ C (1-t)}{p_{T-1}} = N_{0:T-2} \left[ 1 + \frac{ C (1-t)}{p_{T-1}} \right] = \\ & = N_{0:T-3} \left[ 1 + \frac{ C (1-t)}{p_{T-2}} \right] \left[ 1 + \frac{ C (1-t)}{p_{T-1}} \right] = \\ & = \dots = \\ & = N_{0:1} \prod_{k=2}^{T-1} \left[ 1 + \frac{ C (1-t)}{p_{k}} \right] = \\ & = N_{0 } \prod_{k=1}^{T-1} \left[ 1 + \frac{ C (1-t)}{p_{k}} \right] \ . \end{aligned}\end{split}\]

Cumulative discounted return reads

\[\begin{split}\begin{aligned} \dfrac{DCF}{Y_0} & = - 1 + \dfrac{Y_T}{Y_0} DF_{T} = \\ & = - 1 + \dfrac{N_0}{N_0 \, p_0} \prod_{k=1}^{T-1} \left( 1+ \dfrac{C(1-t)}{p_k} \right) \, ( p_T + C(t-1) ) \, DF_T \\ & = - 1 + \dfrac{p_T}{p_0} \prod_{k=1}^{T} \left( 1+ \dfrac{C(1-t)}{p_k} \right) \, DF_T \\ & = - 1 + \dfrac{p_T}{p_0} \prod_{k=1}^{T} \left( \dfrac{ 1+ \frac{C(1-t)}{p_k} }{1+r_k} \right) \ . \end{aligned}\end{split}\]

Composite discounted return is obtained, after writing the diiscounted cashflow as the difference between discounted cashflow at time \(t_T\) and \(t_0\), \(DCF = Y_T \ DF_T - T_0\),

\[(1 + DCAGR)^T = \dfrac{Y_T \, DF_T}{Y_0} = \dfrac{DCF}{Y_0} + 1 = \dfrac{p_T}{p_0} \, \prod_{k=1}^{T} \left( \dfrac{ 1+ \frac{C(1-t)}{p_k} }{1+r_k} \right)\]

\[DCAGR = \left( \dfrac{p_T}{p_0} \, \prod_{k=1}^{T} \dfrac{ 1+ \frac{C(1-t)}{p_k} }{1+r_k} \right)^{\frac{1}{T}} - 1\]

If1 price of the bond is constant throughout its whole life, \(p_k = 1\), \(\forall k=0:T\), and discount rate \(r\) is constant, the number of held bonds at time \(T-1\) is

\[N_{0:T-1} = N_0 \left( 1 + C(1-t) \right)^{T-1} \ ,\]

the discounted cashflow is

\[\begin{split}\begin{aligned} DCF & = - N_0 + N_0 \left( 1 + C(1-t) \right)^{T-1} ( 1 + C(1-t) ) \left( 1 + r \right)^{-T} = \\ & = N_0 \left[ - 1 + \left( \dfrac{ 1 + C(1-t) }{ 1 + r } \right)^{T} \right] \ , \end{aligned}\end{split}\]

cumulative discounted return

\[\dfrac{DCF}{Y_0} = - 1 + \left( \dfrac{ 1 + C(1-t) }{ 1 + r } \right)^{T}\]

and the composite discounted return reads

\[DCAGR = \dfrac{1 + C(1-t)}{1+r} - 1 \ .\]

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1

It’s a big if. Even if credit rating and inflation are constant throughout the life of the bond, years to maturity decreases and thus - usually - the required rate decreases as well.