Suppose you are receiving a payment $K$ at time $t_m$.

Let $p(0,t_i)$ be the maturity-$t_i$ zero-coupon bond price at $t=0$.

If we consider a discrete time $\{0,...,t_m\}$, what would it mean to normalize the payment $K$ by the sum of the zero-coupon bond prices? In other words, is there an economic meaning behind:

$$\frac{K}{\sum\limits_{i=0}^m p(0,t_i)}$$


1 Answer 1


Let: $$c=\frac{K}{\sum\limits_{i=0}^mp(0,t_i)}$$ Then: $$K=\sum_{i=1}^mp(0,t_i)c$$ $K$ is the present value of an annuity paying a constant coupon $c$ over the period $\{t_0,\dots,t_m\}$. So it would consist on a transaction in which you pay a flow of coupons $c$ in exchange for a set payment at $t_m$ equal to $K$. This is analogous to some sort of fixed-fixed zero-coupon swap.

  • $\begingroup$ Hi Daneel, thanks for the response. How do you justify that $K$ is the PV of an annuity paying a constant coupon over the period? Do you mean it pays in each $t_i$ for all $i$? $\endgroup$ Mar 18, 2020 at 2:40
  • $\begingroup$ @FrankSwanton The sum in my 2nd equation is simply the present value of a series of coupons, all equal to $c$, paid over $t_0$, $t_1$, ..., $t_m$. So $K$ could be interpreted as the present value, ie. at $t_0$, of that stream of cash flows. Hence the economic meaning of your quantity of interest is (somehow) the coupon paid in an annuity whose present value is $K$. $\endgroup$ Mar 21, 2020 at 17:03
  • $\begingroup$ Hi Daneel, when you equate the summation part as the PV of a series of coupons, how do you justify this? The way I defined the sum is the sum of the PRICES of zero-coupon bonds that pay a sure dollar at $t_i$ for the index sum $i=1,...,m$. Can you explain? $\endgroup$ Mar 25, 2020 at 22:58
  • $\begingroup$ @FrankSwanton sorry for the late response. Not sure I understand, the pv and the price of a zero-coupon bond is the same thing. $\endgroup$ Nov 3, 2020 at 16:41

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