# Calculating discount factors using Nelson Siegel Svensson model

I am trying to understand how to calculate the discount factors $disc(TTM)$ mentioned on Page 9 of this pdf. When I'm calculating the discount factors, mentioned each bond has its own cash flow and cash flow time vector. Wouldn't this imply that since there are as many TTM Vectors as there are bonds, there will be as many $disc(TTM)$ values as there are bonds, and as a result the discount vector will have as many rows as there are bonds? However, shouldn't it have as many rows as the number of cash flows of the bond with the longest time to maturity (Only then would we be able to multiply the matrix of coupons and face values ($A$) with the discount vector ($P$) as listed here?

Edit: I apologize if I've asked a question similar to this one. But I'm trying to gather as much information as I can and ask this question thats been plaguing me as clearly as possible.

Thank You

## 1 Answer

You really don't even need to put them into matrices, which I feel is confusing you more than helping...

For each bond, you have a list of cash flows ($c_i$'s). For each cash flow, you can compute the corresponding discount factor ($d(t_i)$'s). Sum up the discounted cash flows gets you the theoretical price: $P = \sum_i c_i d(t_i)$.

Repeat this for every bond, and add up all the (weighted squared) errors. That's the thing you need to minimize.

• Thank you very much. And if I was calculating the theoretical Yields directly from the discount factors using the equation $Y(t) = P(t)^{-1/t}-1$, where $P(t)$ is the discount function and $Y(t)$ are the yields, what would I take $t$ to be given that I have the discount factor matrix you described above? – Jojo Jul 28 '15 at 14:18
• @Jojo, can you clarify why you're doing this? – Helin Jul 28 '15 at 15:09
• I want to find the NSS parameters for the UK Yield Curve and am using the Non-Linear Optimization for this. However, to get the initial parameters for the Optimization I am using the equation provided in Figure 5 of this paper--which suggests I need to use the theoretical and observed Yields. I then change the tau_1 and tau_2 parameters and choose the set of parameters associated with the lowest residual. – Jojo Jul 28 '15 at 15:45
• But you should use the theoretical and actual yields of a bond, not the zero coupon rate. – Helin Jul 28 '15 at 16:27
• @Jojo No, that's the formula for zero coupon yields. You should compute the yield to maturity of your coupon bonds using the standard price-yield formula instead. – Helin Jul 28 '15 at 20:25