Introduction to Par Yield and Yield to Maturity

In the previous chapters of the Term structure, duration and immunisation section, we looked at how interest rates change depending on the length of time you invest (the term structure). We explored spot rates and forward rates, which give us a precise way to value individual cashflows.

However, in the real world, investors often want a single "summary" number to describe the return on a bond or a project. This is where Yield to Maturity (YTM) and Par Yield come in. These concepts help us bridge the gap between complex spot rates and the actual prices we see in the market. Don't worry if this seems a bit abstract at first; we will break it down into simple equations of value!

Note: This chapter assumes you are comfortable with the concept of an equation of value and the basic definitions of spot rates.

Yield to Maturity (YTM)

The Yield to Maturity (often called the Redemption Yield) is the single constant annual effective rate of interest that makes the present value of all future cashflows from a bond equal to its current market price.

Think of it as the "internal rate of return" (IRR) for a bond. If you buy a bond today, hold it until it matures, and reinvest all coupons at that same rate, the YTM is the annual return you would achieve.

The Equation of Value for YTM

If a bond has a price \(P\), annual coupons \(C\), and a redemption value \(R\) at time \(n\), the YTM (let's call it \(y\)) is the solution to:

\(P = C \cdot a_{\overline{n}|y} + R \cdot v_y^n\)

Where:

  • \(a_{\overline{n}|y} = \frac{1 - (1+y)^{-n}}{y}\)
  • \(v_y = (1+y)^{-1}\)

Important Distinction: Unlike spot rates (where we use a different interest rate for each year’s cashflow), YTM uses the same rate \(y\) for every single cashflow, regardless of when it occurs.

Why do we use YTM?

It is the standard way to quote bond returns in the news and on trading floors. It allows investors to compare a 5-year bond and a 10-year bond using a single percentage figure, even if they have different coupon rates.

Key Takeaway: YTM is the "average" interest rate implied by a bond's current market price. If the price goes UP, the YTM goes DOWN.

Par Yield

The Par Yield is the specific coupon rate for which a bond’s market price is exactly equal to its nominal (par) value. In other words, if a bond is "trading at par," its price is 100% of its face value.

In actuarial exams, you are often asked to calculate the par yield for a specific term \(n\) based on a given set of spot rates.

The Logic of Par Yield

Imagine a bond with a face value of 1. It pays a coupon \(c\) at the end of every year for \(n\) years and pays back the 1 at the end of year \(n\). For this bond to be worth exactly 1 today, the present value of these payments (calculated using spot rates) must equal 1.

\(1 = c \cdot v(0,1) + c \cdot v(0,2) + \dots + (c+1) \cdot v(0,n)\)

Where \(v(0,t)\) is the present value of 1 due at time \(t\) calculated using the spot rate for that specific term.

Calculating the Par Yield (\(c_n\))

By rearranging the equation above, we can solve for the par yield \(c_n\):

\(1 = c_n \sum_{t=1}^{n} v(0,t) + v(0,n)\)

\(c_n = \frac{1 - v(0,n)}{\sum_{t=1}^{n} v(0,t)}\)

Did you know? If the yield curve is flat (i.e., all spot rates are the same), the par yield will be exactly equal to that spot rate!

Key Takeaway: The par yield is the "break-even" coupon rate. If a bond's actual coupon is higher than the par yield, the bond will trade at a premium (price > 100). If the coupon is lower, it trades at a discount (price < 100).

Comparing YTM and Par Yield

It is easy to get these two confused. Here is a simple way to remember the difference:

  • Yield to Maturity: You know the Price and the Coupon; you are solving for the Interest Rate.
  • Par Yield: You know the Spot Rates (Interest Rates) and you set the Price to 100; you are solving for the Coupon Rate.

The Relationship with the Yield Curve

The par yield is a useful tool for constructing the "Par Yield Curve," which is a plot of par yields for different maturities. This is often easier for practitioners to observe than the "Spot Rate Curve" because many bonds are issued close to par value.

Summary of Key Formulas

1. Yield to Maturity (\(y\)): Solve the Equation of Value for the unknown interest rate \(y\).
\(Price = \sum PV(\text{Cashflows at rate } y)\)

2. Par Yield (\(c_n\)): The coupon rate that makes Price = Par using spot rates.
\(c_n = \frac{1 - v(0,n)}{\sum_{t=1}^n v(0,t)}\)

Quick Review: Common Pitfalls

Mistake 1: Confusing annual and p-thly rates.
If a bond pays coupons semi-annually, remember to adjust your yield calculation. If \(y\) is the annual effective yield, the semi-annual coupon should be discounted by \((1+y)^{-0.5}\). However, the syllabus usually focuses on annual cashflows for these specific definitions unless stated otherwise.

Mistake 2: Mixing up Spot Rates and YTM.
Remember: Spot rates are "time-specific" (one rate for year 1, a different rate for year 2). YTM is a "single average" applied to all years of a specific bond.

Don't worry if this seems tricky at first! The most important skill is setting up the Equation of Value. Once the equation is written down correctly, the rest is just algebra or using your calculator's solver/interpolation functions.

Key Takeaway: These yields are simply different ways of expressing the price of a bond in terms of a percentage return. Mastering them is essential for the later topics of Duration and Immunisation.