# Deriving spot rates from treasury yield curve

I've been experimenting with bond pricing using easily available data (treasury auction prices and treasury yield curves on treasury direct).

At first I assumed that I could use the components yield curve to price any (risk free) series of cash flows, but that doesn't work because the curve is of the yield-to-maturity of instruments with multiple payments over time, not spot rates for cash on a date.

I feel like I should be able to approximate the spot rate curve from the treasury curve, but I'm not sure how. I feel like I need more than just the yield-to-maturity - wouldn't I also need the coupon rate for an instrument? Otherwise the equation

$$PV_{\text{payments discounted with IRR}} = PV_\text{payments discounted at spot rates}$$ $$par*df_{IRR}+\sum_{c=coupons}c*df_{IRR} = par*df_{spot}(t_{par})+\sum_{c = coupons}c*df_{spot}(t_c)$$

has too many unknowns.

Is there published coupon rates that go along with the treasury yield curve? Am I missing something fundamental here? Am I just working with the wrong data-set?

I don't have any formal background in finance, so if the answer is "read book \$x" then that is okay with me.

## 1 Answer

First step is to decide what instruments you want to include in your process for estimating the spot curve.

You want to look at the following instruments for inclusion - treasury coupon strips, on-the-run treasury issues, and some off-the-run treasury issues (those not trading at liquidity discounts), or all treasury coupon securities and bills.

You want to ensure your selected instruments are not biased by credit risk, embedded options, pricing errors, or liquidity as noted above. You might want to filter securities that are on "special" (i.e. trading at a lower yield than their true yield in the repo market.)

1. If you select only Treasury coupon strips, then the process is simple since the yields on the coupon strips are by definition spot rates.

2. If you select on-the-run Treasury issues with or without off-the-run Treasury issues then you use the bootstrap approach. Given the par yield curve, linear interpolation is used to fill in gaps for missing maturities. Bootstrapping is then used to construct the theoretical spot yield curve. Bootstrapping is a technique that repetitively applies a no-arbitrage implied forward rate equation to yields on the estimated Treasury par yield curve. This post on bootstrapping does a nice job illustrating several of these concepts.

3. If you select all Treasury coupon securities and bills, then you must use techniques more complex than bootstrapping. See Vasicek and Fong, "Term Structure Modelling Using Exponential Splines", Journal of Finance, (May 1982).

Frank Fabozzi's Handbook of Fixed Income Securities can help you understand the other methods as well.

• It looks like, then, I do need coupon rates to go along with the published yield curve IRR rates. Is that a correct reading? Apr 20 '12 at 16:30
• If by IRR rates you mean the YTM on securities, then it looks like you mean to use all Treasury Copuon Securities and Bills which is the most difficult method of the three. I updated my answer so you can see the alternatives. Hopefully this helps Apr 20 '12 at 16:54
• In summary - I'm probably looking at the wrong data - It sounds like it would be easier to start with trading prices of treasuries, that way I know both the PV, face value and coupon rate of each instrument. I'll see if Yahoo or someone else lets me dig those out for free. Apr 20 '12 at 17:02
• Okay, from the link provided in your #2, when I read the yield curve I have to assume that the YTM equals the coupon rate - that is, the curve is for hypothetical instruments trading at par. If that is true, I think it gives me the piece of information I was missing. Apr 20 '12 at 19:44
• Hello Antoine Latter, would separating software (electronic, credit cards, cheques) and non-software cash flows, and using just the completly netted non-software cash flows together with the treasury interest rates, make any difference on your results? Aug 28 '12 at 13:27