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5 Bond Pricing Essentials You Need To Know Now

Ray Yeo

Bond pricing questions are likely to come up in a market risk interview since bonds are universal instruments that have some degree of complexity. Here are 5 bond pricing essentials you need to know to better prepare yourself.

1) What is a bond and how does it work?

Before jumping into bond pricing, it is good to have some context i.e., what a bond is and how it works.

Bonds are financial instruments that allow companies (the “issuer”) to borrow money from bondholders/investors. In other words, a bond nothing but an IOU (“I Owe You”) or promissory note. As an investor that holds the bond, you are really just holding a slip of paper stating the company owes you some money that they will return in the future, with some interest.

See Investopedia for more background info on bonds.

A stream of cash flows receivable

If you focus on the structure of the bond, you will see that a bond is simply a stream of future cash flows receivable.

Suppose you are the investor, you will:
– From now till maturity: receive fixed / floating interest payments (“I”) i.e. “coupons”
– At maturity (“T”): receive the principal amount (“P”) i.e. face value

Graphically, it looks something like this.

bond pricing: bond cashflows

2) Bond Pricing / Valuation Formula

Assuming we have a bond that pays coupons semi-annually, the current price of the bond is the total present value of all cash flows receivable. In other words, discount each cash flow to today and add them all up. Mathematically, that is:

B(y)=PV_1 +PV_2 +...+PV_N = \sum_{i=1}^{N}PV_i =\sum_{i=1}^{N}\frac{C_i}{(1+\frac{y}{2})^i}

Where:
– B: Bond price
– y: Yield-to-Maturity
– N: Number of cashflows
– C: Cashflow (interim coupons + principal at maturity)

Here, let’s briefly talk about yield-to-maturity (sometimes just called “yield”). You may interpret yield as the implied annual return on the bond, given its market price and features i.e. maturity, face value, payment frequency, etc. Note that yield is implied (rather than specified) from bond price.

If you look at the above bond price formula, you’ll see that the only market risk factor is bond yield. The other terms are trade specifics that are fixed upfront as part of the bond: coupon amounts (for fixed-rate bonds) and payment dates, maturity date, principal amount i.e. face value, etc. In general, identifying the risk factors of a given product is a key part of market risk management and is a prerequisite for things like VaR.

3) Yield up, Price down, and Vice Versa in Bond Pricing

If you plot the bond price against yield on a graph, you’ll see a downward-sloping curve like the below.

bond pricing: bond price vs yield

This means three things:
1) As yield increases, price decreases (and vice versa)
2) As yield increases, price decreases at slower and slower i.e. decreasing rate
3) As yield decreases, price increases at a faster and faster i.e. increasing rate

From the investor’s perspective, this is good. As yield rises, the drop in bond price is increasingly small. Conversely, as yield falls, the gain in bond price is increasingly large.

4) Duration, Convexity, PV01 in Bond Pricing

When thinking about how to measure price risk of a bond, three ideas come into mind: Duration, Convexity, and PV01. Let’s examine them one by one.

Duration

D=-\frac{dB/B}{dy}=-\frac{1}{B}\frac{dB}{dy}

Modified Duration, mostly just called “Duration”, is the first-order sensitivity of bond price to its yield. Specifically, it is the approximate percentage change of the bond price for a 1% point increase in yield. For example, if the current price is $100 and the duration is 5, if yield goes up by 1% point, the bond price will drop by 5% to end at $95.

One thing to note about the term “Duration” is that there are actually two different types of duration that are related: Macaulay Duration and Modified Duration. Whenever one talks about bond price risk though, they are really referring to Modified Duration.

Convexity

C=\frac{1}{B}\frac{d^2B}{dy^2}

Convexity is the second-order sensitivity of price to yield or equivalently, the sensitivity of Duration to yield. Think of this quantity as an adjustment on top of Duration to give you a better estimate of the actual change in bond price when yield changes.

PV01

\text{PV01}=-BD\Delta{y}=-BD\cdot\text{1bps}=-\frac{BD}{10000}

PV01 is shorthand for “Present Value change of 1 bps increase in yield”. Essentially, this is similar to Duration in that it is also a first-order sensitivity of bond price except for two differences. First, the given yield change is 1bps rather than 1%. Second, this is a dollar change in price rather than a percentage change in price.

5) Embedded Options and Effects in Bond Pricing

Embedded options are additional rights that either bond issuer or investor has. Note that these are rights rather than obligations and so, the term “option” is apt. There are two types of embedded options in bonds: Callable (by the issuer) and Puttable (by the investor).

Callable Bonds

Callable bonds are bonds that the issuer can buy back/redeem at a certain strike price, at a certain time(s) in the future. For example, a bond may be callable by the issuer a year from now at $110. If the issuer exercises the option, they pay the investor $110 and get back the bond.

Effectively, the issuer had bought a call option from the investor and is now protected against falls in yields. As the option seller, the investor receives the option premium and therefore, better-than-market yields on the bond. However, the investor is now exposed to falling yields. Bond price is also now capped at the strike price since the issuer can exercise the option and buy the bond back at the strike price.

Bond issuers make a bond callable because it allows them to borrow money cheaper if interest rates fall. Suppose a bond issuer issues a 5Y callable bond when market yields are 3%. If yields fall to 1% tomorrow, the bond issuer can call the bond back and re-issue it at the prevailing 1% yield.

Puttable Bonds

Puttable bonds are bonds that can be put/sold back to the issuer at a certain strike price, at a certain time(s) in the future. For example, a bond may be puttable by the investor a year from now at $110. If the investor exercises the option, they sell the bond back to the issuer and receive $110.

Effectively, the investor had bought a put option from the issuer and paid an option premium, resulting in lower-than-market yields on the bond. In exchange, the investor is now protected against rising yields. Bond price is also now floored at the strike price since the investor can exercise the option and sell the bond back to the issuer at the strike price.

Bond issuers make a bond puttable because it allows them to borrow money cheaper upfront. In exchange for the optionality the issuer grants to the investor, the issuer receives an option premium that reduces their financing cost via lower-than-market coupon rates for the investor.

What else do you think is important to know about bond pricing in a market risk or counterparty risk context? Let me know in the comments below. If you know someone who’s preparing for a market risk interview, feel free to share this with them!