Choosing Yield Curves and Zero Rates to Price Bonds

# Yield curve and bond price


# Yield curve and bond price












I am confused about how the yield curve is built and how it relates to the pricing of bonds. First what I don't understand is that when people talk about the yield curve, which yield curve are they talking about?

Because from what I understand we can take a basket of bonds with maturities ranging from $1$ to $30$ years and build the yield curve using boostrapping:

- we observe the price of the bond $1$y in the market. Using that price and the coupon we find the yield to maturity of that bond, which would be the point on the yield curve corresponding to $1$

- we use the price of the bond with maturity $n$ years in the market and all the $n-1$ previous yield calculated and find the unique yield such that the discounted cash flows equal the current price in the market

The problem with all of this is that it depends on the basket of bonds we choose at the beginning since bonds of the same maturity cab have different prices, different coupon, different liquidity. So when solving the equations for the yield we will find different values depending on the bonds choosen.

So when talking about the yield curve, which basket of bond are we choosing to calculate it?

and also when we have the yield curve we can calculate the faire value of a bond by discounting the cash flows and can then compare this fair value to the price of the bond given by the market to see if the bond is overpriced or mispriced. Yet to apply this formula for the price of the bond which yield curve are we using? since I believe for a 2% coupon bond and a 6% coupon bond different yield curve should be used?

## Answer by D Stanley (score 1)

https://quant.stackexchange.com/a/78511

> So when talking about the yield curve, which basket of bond are we choosing to calculate it?

There is not one "yield curve". Generally a curve is generated from multiple bonds with a shared set of characteristics, but different maturities. The US Treasury yield curve, for example, is constructed from multiple US Treasury bonds (and notes and bills) of different maturities.

While bootstrapping is one method that can be used to create a curve, as you mentioned different bonds of the same maturity can have slightly different yields (coupon is not a concern as noted below), so in practice all applicable bonds are used, and interpolation methods are used to find a curve that most closely matches the full range of bonds.

You might also construct a yield curve for a specific company if they have enough bonds of different maturities traded to create a meaning curve.

There are also curves for different market segments (e.g. Energy, Tech), credit ratings (AAA, B), etc. Whomever is constructing the yield curve decides what bonds meet the criteria to use for construction.

> to apply this formula for the price of the bond which yield curve are we using?

You use a yield curve that matches the characteristics of the bond that you are valuing. You would not use a Treasury curve, for example, to value a corporate bond since a treasury curve would not take credit risk into account. For a corporate bond, you could use a curve built from bonds from that company, from its credit rating, industry, or whatever curve you can find that is most specific to that bond.

> since I believe for a 2% coupon bond and a 6% coupon bond different yield curve should be used?

No. Different coupons are not a concern, since it's the yield of each bond that is used to construct the curve, which is a function of the price and coupon (among other things). The same yield curve is used to value bonds with differing coupons.

## Answer by AlRacoon (score 1)

https://quant.stackexchange.com/a/78512

The ultimate goal of any yield curve construction is a zero curve by which you can derive discount factors to price any cash flow at any given time. Market traded bonds will be the source of prices and therefore interest rates. It is preferable to use the most liquid and traded bonds in the market as these will provide the most up-to-date prices. Ideally, the bonds you choose should be of similar risk of the cash flows you will be attempting to price. However this can be a challenging task to find bonds over all the maturities with similar risk to your cash flows. As such, many start with building risk-free yield curve based on US Treasuries, perhaps the most liquid and deepest bond markets, and then add a risk premium to this curve, or rates derived from this curve to discount their risky cash flows.

In constructing a UST zero curve, in the front part of the curve, you will find a series of liquid zero coupon instruments called Treasury bills. In general, 1M, 3M, 6M, 1Yr T-bills are the source of interest rates for this part of the curve. Bootstrapping coupon Treasury notes and bonds, you will calculate zero rates for other knot points of different maturities. You can bootstrap using the on-the-run Treasury Notes and Bonds (2Yr, 5Yr, 7Yr, 10Yr, 20Yr, 30Yr, etc) as these are the most liquid Notes and Bonds.

Once you have bootstrapped these zero rates for your knot points, you will then use some curve fitting mechanism to connect the knot points such that you will be able to arrive at a zero rate to derive a discount factor for dates between these knot points. The most rudimentary is linear interpolation but this method will not give you very accurate rates in between knot points and will result in forward rates that are unusable for dates that go over a knot point as the slopes of the lines will be drastically different between different knot points. There are more advanced curve fitting procedures, such as cubic spline etc. These interpolation techniques will generate polynomial equations that describe the rates between the knot points, which you can then use to derive interest rates and discount factors.

As most cashflows are not risk free, you can add a risk premium to the rates you derive from these risk free rates to discount your cash flows (or create a curve for your risky cash flows by adding a premium to the knot points, followed by interpolating between these points.

As most of the cashflows in the markets are traded between intermediary banks and bank customers, curves are generated with "bank risk". The banks are typically AA-rated entities. You can take a similar approach to constructing a curve for these counterparties with instruments that are frequently traded between banks. Historically, this has been LIBOR (London Interbank Offer Rate). A LIBOR curve was generated with various LIBOR instruments such as 3M, 6M, 1Y LIBOR, Eurodollar futures for up to 2YR rates, and Swap rates for longer periods. With LIBOR being supplanted with SOFR, there are similar liquid instruments across the maturities that can be used to construct a SOFR curve. The same methodology can be applied to build other sovereign rate curves etc.

From these zero curves, you will be able to price other bonds by discounting their coupons and principal payments and then compare this price to those you see traded, to make an assessment of whether or not these bonds are trading rich or cheap, after adjusting for credit risk, liquidity risk etc.