Welcome to the World of Bond Math!
Hello there! Welcome to one of the most fundamental chapters in your FRM Part I journey: Pricing Conventions, Discounting, and Arbitrage. Think of this chapter as the "language" of fixed income. Before you can build complex risk models, you need to understand how professional traders quote prices, how time affects the value of money, and how to spot a "free lunch" (arbitrage) when it appears. Don't worry if the formulas look a bit intimidating at first—we're going to break them down piece by piece until they feel like second nature!
1. Day Count Conventions: Measuring Time
In the real world, "one month" or "one year" can be tricky. Does a year have 365 days or 366? Does a month have 30 days or 31? To keep everyone on the same page, the financial world uses Day Count Conventions. These are rules for calculating the amount of interest that accrues over a period of time.
The standard format is X/Y, where X defines how days between dates are counted, and Y defines how many days are in a year.
The Three Main Players:
• Actual/Actual (in period): Used for U.S. Treasury Bonds. You count the actual number of days between dates and divide by the actual number of days in the coupon period. This is the most "fair" but requires a calendar handy!
• 30/360: Used for U.S. Corporate and Municipal Bonds. We pretend every month has 30 days and every year has 360 days. It’s an old-school method that makes manual calculations much easier.
• Actual/360: Used for Money Market instruments (like T-bills). You count the actual days that passed but assume a 360-day year. Did you know? This convention actually results in a slightly higher interest payment than Actual/365 because the daily rate is slightly larger!
Quick Review Box:
Treasury Bonds = Actual/Actual
Corporate Bonds = 30/360
Money Market = Actual/360
2. Bond Pricing: Clean vs. Dirty
If you buy a bond between coupon payment dates, who gets the interest earned during that gap? The seller! This leads us to two different types of prices.
The Clean Price (Quoted Price): This is the price you see on a TV screen or in a newspaper. It does not include the interest that has built up since the last payment.
The Dirty Price (Cash Price): This is the price you actually pay. It is the Clean Price plus Accrued Interest.
The Logic: If you buy a bond halfway through a coupon period, the seller has "earned" half of that next coupon. Since you (the buyer) will receive the full coupon from the issuer later, you must compensate the seller for their share today.
Formula:
\( \text{Dirty Price} = \text{Clean Price} + \text{Accrued Interest} \)
\( \text{Accrued Interest} = \frac{\text{Days since last coupon}}{\text{Days in coupon period}} \times \text{Coupon Amount} \)
Common Mistake: Students often forget that the "quoted" price is the Clean Price. Always remember: "The Dirty Price is the reality of your wallet!"
3. Discounting and Compounding
Discounting is simply the process of finding out what a future payment is worth today. To do this, we need to know how often interest is compounded.
Compounding Frequencies:
Interest can be compounded annually, semi-annually, quarterly, or even continuously. In the FRM exam, continuous compounding is very common because it makes the calculus of risk models much smoother.
Key Formulas:
For discrete compounding: \( PV = FV \times (1 + \frac{r}{m})^{-mn} \)
For continuous compounding: \( PV = FV \times e^{-rt} \)
(Where r = rate, t = time in years, m = compounding periods per year, e is the exponential constant approx 2.718)
Memory Aid: Think of continuous compounding like a snowball rolling down a hill that gets bigger every single millisecond, rather than just getting bigger once a month.
Key Takeaway: The more frequent the compounding, the higher the Effective Annual Rate (EAR). If you are offered 10% compounded annually vs. 10% compounded daily, take the daily option—you'll end up with more money!
4. Treasury Bill Pricing (T-Bills)
T-bills are a bit different because they don't pay coupons. They are zero-coupon bonds sold at a discount. However, their price is quoted using a Discount Rate, not a yield.
The Formula for T-Bill Price:
\( P = 100 \times [1 - \frac{n}{360} \times Y_d] \)
(Where \( Y_d \) is the quoted discount rate and n is days to maturity)
Wait! This discount rate is a bit "sneaky" because it calculates the return based on the Face Value, not the price you actually paid. To compare a T-bill to a bond, you need to calculate the Bond Equivalent Yield (BEY), which uses the price you paid and a 365-day year.
5. Arbitrage and the Law of One Price
Arbitrage is the holy grail of finance. It is an investment strategy that guarantees a positive profit with no risk and no net investment.
The Law of One Price: In a well-functioning market, two identical assets (or sets of cash flows) must have the same price. If they don't, an arbitrageur will step in.
Example:
If a "Gold Bar" costs \$1,000 in New York and \$1,010 in London, and it costs nothing to transport it, you would:
1. Borrow money.
2. Buy gold in New York for \$1,000.
\n3. Simultaneously sell it in London for \$1,010.
4. Pocket the \$10 profit risk-free!
As everyone does this, the price in NY will go up (more buyers) and the price in London will go down (more sellers) until the prices are equal.
Quick Summary: Most of our bond pricing models are "No-Arbitrage" models. This means we assume the price of a coupon bond must be equal to the sum of its parts (the individual cash flows discounted at zero-coupon rates).
6. Bootstrapping: Building the Zero Curve
How do we find the "correct" discount rates for different time periods? We use a process called Bootstrapping. We start with the shortest-term instruments (like T-bills) to find the 3-month or 6-month zero rates. Then, we use those rates to "solve for" the next rate using a longer-term coupon bond.
Step-by-Step Logic:
1. Take a 6-month T-bill. Its yield is the 6-month zero rate.
2. Take a 1-year bond that pays a coupon at 6 months.
3. You already know the "discount factor" for the 6-month coupon.
4. Use the bond's current market price to solve for the missing piece: the 1-year zero rate.
5. Repeat for 1.5 years, 2 years, and so on.
Don't worry if this seems tricky at first! Bootstrapping is just like solving a puzzle where you use the piece you just found to help you find the next one.
Final Summary of Key Points
• Day Counts: Treasuries (ACT/ACT), Corporates (30/360), Money Market (ACT/360).
• Cash Price: What you pay (Clean Price + Accrued Interest).
• Compounding: Continuous compounding \( e^{rt} \) is the FRM standard.
• Arbitrage: "No free lunch." Prices must align so no riskless profit exists.
• Zero Rates: The foundation of all valuation. Use bootstrapping to find them from coupon bonds.
Keep practicing these calculations! Once you master the conventions, the rest of fixed income valuation becomes much clearer. You've got this!