Yield Curves: Common Patterns in Prices of Fixed-Income Securities

Summary

• Interest rates and yield curves are not observable, but need to be estimated from prices of fixed-income securities.
• Common patterns in prices of fixed-income securities can be expressed in three ways: yield curves, forward rate curves or discount functions.
• When working with interest rates, we need to know the exact unit of measurement in order to know the correct formula to compute present values.
• This post is the first one in a series on fixed-income securities and interest rates.

Yield Curve Estimation

Whenever we deposit money at a bank account, this means that we effectively lend money to the bank. The bank will take most of the deposit and try to earn some money with it by investing it in the financial markets or lending it out as loans. As the bank expects to earn money with it, it usually is also willing to pay some compensation to us, which is generally quoted in terms of interest rates. Aside from lending money to a bank through deposits, the compensation that we get for lending in the financial markets is not usually quoted in terms of interest rates. Instead, fixed-income securities just define a series of future cash-flows at future points in time. Lending money to a company or government simply means buying the entitlement to a series of these cash-flows, and depending on the price that we pay for this entitlement, we will either get higher or lower compensation for lending the money. Given that many fixed-income instruments are actively traded in secondary markets, we usually have a representative price for a lot of fixed-income instruments. When prices, size, and dates of all future cash-flows are known, one can infer an estimate for the implicitly given level of compensation, e.g. by computing the internal rate of return (i.e. the yield of the security). By doing this one usually finds that bonds of roughly similar maturity tend to have similar rates of return. In other words, there is some hidden structure in the prices of fixed-income securities.

The most popular way of representing this common underlying structure is by estimating a relationship between maturity and compensation, a so-called yield curve. For each maturity it determines the annualised interest rate that is used by the market to discount a future cash-flow. However, annualised interest rates given by a yield curve cannot be directly observed in financial markets. They are estimated values that are inferred from fixed-income prices through the help of assumptions that determine how smoothly interest rates should change for varying maturities. Two of the most popular methodologies for this are the Svensson yield curve model (Svensson 1994) or a model based on a quasi-cubic hermite spline function which is for example used by the US Treasury¹. Given the structural assumptions that are implicitly defined by the yield curve model that is used, any given estimated yield curve will never fully match all fixed-income prices perfectly. Variable parameters of a yield curve model are determined such that derived fixed-income prices match observed market prices as closely as possible. This, however, is only meaningful if fixed-income securities have roughly similar characteristics. Most importantly, they should be aligned with regards to credit quality. As future cash-flows will only occur when the debtor is able to repay, all cash-flows do come with a (sometimes negligible) probability of default. For the US government (and other trusted developed market countries) this probability is usually assumed to be zero. But as soon as default probabilities are not perceived to be negligible, the market requires some additional compensation for the credit risk that is inherent. This compensation will lead to market prices trading below those of comparable risk-free securities, like with corporate bonds or emerging market government bonds. This way two bonds with equal future cash-flows, but differing credit quality, would be traded at different market prices. Hence, it does not make sense to estimate yield curves for a group of bonds with different credit quality, because it would be impossible to match all prices reasonably closely. Yield curves will generally be estimated for buckets of similar credit quality, such that each credit quality gets its own yield curve.

Given that we have an estimate for a yield curve, we can now compute the present value for any future cash-flow. However, one additional piece of information is necessary: the unit of measurement of interest rates so to speak. Interest rates quantify the magnitude of compounding, but the actual frequency of compounding also needs to be defined. So any quoted interest rate could lead to different aggregate compounding depending on whether interest payments occur once per year (annually), twice per year (bi-annually) or even infinitesimally frequently (continuous compounding). While annual and bi-annual compounding actually might occur in reality for bank accounts, you will usually never encounter continuous compounding in real life. However, stating yields in terms of this unit of measurement happens quite frequently, because it allows for straightforward transformations and simple computations of present values for given interest rates. With given interest rates of 6% we will get the following different market prices $V_{t}$ for a cash-flow of 100 EUR with maturity of one year:

• annual compounding: $V_{t}=$
  frac{100}{1.06}
  approx 94.3396
• bi-annual compounding: $V_{t}=$
  frac{100}{1.03^{2}}
  approx 94.2596
• continuous compounding: $V_{t}= 100$
  exp{(-0.06)}
  approx 94.1765

Yield Curve Representation

The information of yield curves can be expressed in three ways:

• yields
• forward rates
• discount functions

All three fundamentally represent the same information, but each of them through a different perspective.

Discount functions are a kind of inverse perspective on interest rates and compounding. Instead of telling you how much interest rates you get for a given amount of money that you invest today, discount functions give you an answer to the opposite direction: 1 EUR in the future is worth how much today? The nice thing about them is: there is only one unit of measurement, so they are uniquely defined. Yields and forward rates are somewhat auxiliary constructs to identify common patterns in fixed-income security prices and as such they depend on the unit of measurement (e.g. annual or bi-annual compounding). Discount functions, in contrast, can be directly observed in the form of market prices for some durations, and there is only one unit of measurement for market prices. However, the downside of discount functions is that they are not easily interpretable. Saying that the discount function for maturity 3 years has a value of 0.86 is much harder to interpret than saying that annualised interest rates for such a duration are 5%.

Forward rates express compounding of money during time intervals in the future. In forward rate curves, these time intervals become infinitesimally small, such that each point represents instantaneous compounding at some future point in time. This is not to say that this value will really materialise in the future, but it is the level of compounding that one could lock in today.

Yields express compounding of money for different durations when money is invested today. As such, the yield at time $t=0$ with maturity given by $T$ is the average of forward rates for the interval $[0,T]$. Yields are normalised to some unit of measurement, however. In most cases they are annualised in order to make yields of different maturities better comparable.

An example for how yield curves and matching forward rate curves and discount functions look like can be seen in Figure 1. In this chart, yields and forward rates are quoted in terms of continuous compounding. Using continuous compounding, there is a straightforward relationship between yields and forward rates. For any given duration, the yield is equal to the average of all forward rates from zero to that duration:

$$y_{t}(n)=$$
frac{1}{n}
int_{0}^{n}{f_{t}(x)}{
text{dx}}