# Normalizing with Sum of Zero-Coupon Bond Prices

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:

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

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.
• 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 $i$? Mar 18 '20 at 2:40
• @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 $K$. Mar 21 '20 at 17:03
• 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? Mar 25 '20 at 22:58