Price of a callable bond for a minimum yield

Welcome to the World of Callable Bonds!

In your Exam FM journey, you’ve already mastered standard bonds. But what happens when the bond issuer has the power to change the rules? That’s exactly what a callable bond is. In this chapter, we are going to learn how to calculate the price of these bonds so that you, the investor, are guaranteed a minimum yield no matter when the bond is called. Don't worry if this seems a bit "shifty" at first—once you learn the simple logic behind it, you'll see it’s just about planning for the "worst-case scenario."

What is a Callable Bond?

A standard bond has a fixed maturity date. A callable bond is different: the issuer (the person who borrowed the money) has the right to pay back the loan earlier than the maturity date. Usually, they do this on specific "call dates" listed in the bond contract.

Think of it this way: Imagine you lend a friend $100 and they promise to pay you 10% interest for 5 years. If interest rates in the real world drop to 2%, your friend will want to pay you back immediately so they can go borrow money elsewhere at the cheaper 2% rate. That’s a "call."

Key Term: Redemption Price (C)
\nSometimes, if the issuer calls the bond early, they have to pay a little extra "penalty" to you. This means the redemption price (\(C\)) might be different depending on when the bond is called. If it isn't specified, we usually assume \(C\) is the par value.

The Golden Rule: The "Worst-Case" Price

When an exam question asks for the "Price for a minimum yield," it is asking: "What is the most I can pay for this bond to ensure that, no matter when the issuer calls it, my actual yield is at least \(i\)?"

To find this, we look for the minimum price among all possible redemption dates. Why the minimum? Because if you pay the lowest possible price calculated across all scenarios, any other scenario that actually happens will only result in a higher yield for you!

Step-by-Step Logic: Premium vs. Discount

To find the minimum yield, we first need to look at the relationship between the coupon rate (\(r\)) and the desired yield rate (\(i\)).

1. Bonds Sold at a Premium (\(g > i\))

If the modified coupon rate (\(g = Fr/C\)) is greater than the yield rate (\(i\)), the bond is selling at a premium (Price > Redemption Value).
\n• In this case, the "worst-case scenario" for the investor is that the bond is called as early as possible.
\n• Why? Because you paid a "bonus" (premium) to get those high coupons. If the bond is called early, you don't get those high coupons for as long as you hoped.
\n• Action: Calculate the price at the earliest possible call date.

2. Bonds Sold at a Discount (\(g < i\))

If the modified coupon rate (\(g\)) is less than the yield rate (\(i\)), the bond is selling at a discount (Price < Redemption Value).
\n• In this case, the "worst-case scenario" for the investor is that the bond is called as late as possible (the maturity date).
\n• Why? You bought the bond for cheap, expecting a big gain when it's eventually paid back at par. The longer it takes to get that final payment, the lower your effective yield.
\n• Action: Calculate the price at the latest possible date (maturity).

Quick Review Box:
\n• If Coupon > Yield (Premium) \(\rightarrow\) Use Earliest Date
\n• If Coupon < Yield (Discount) \(\rightarrow\) Use Latest Date

Did You Know?

Bond issuers usually call bonds when interest rates in the market drop. It’s exactly like a homeowner refinancing their mortgage to get a lower monthly payment. As an investor, this is "reinvestment risk"—you get your money back right when it's hardest to find a new good investment!

How to Handle Multiple Call Dates and Different Redemption Values

Sometimes the Exam FM questions get a little trickier. What if the redemption value (\(C\)) changes? For example, the issuer might pay \(105\) if called in year 5, but only \(100\) if called in year 10.

The "Safe" Strategy:
\nIf you aren't sure which date is the "worst," or if the redemption values are changing, follow these steps:
\n1. Identify all possible call dates (and the maturity date).
\n2. Calculate the Price (\(P\)) for each date using the formula:
\n\(P = Fr \cdot a_{\overline{n}|i} + C \cdot v_i^n\)
\n3. The lowest (minimum) price you calculated is the price for the minimum yield.

Example Walkthrough

Scenario: A $1,000 par value bond with 6% semiannual coupons is callable at the end of years 5 through 10 at par. Find the price to yield a minimum of 8% convertible semiannually.

Step 1: Identify the variables.
\(F = 1000\)
\(r = 0.03\) (per half-year)
\(i = 0.04\) (per half-year)
\(C = 1000\)

Step 2: Compare \(g\) and \(i\).
Here, the coupon rate (3%) is less than the yield rate (4%). This is a discount bond.

Step 3: Choose the date.
For a discount bond, the minimum yield occurs at the latest possible date. That is \(n = 20\) half-years (10 years).

Step 4: Calculate.
\(P = 30 \cdot a_{\overline{20}|0.04} + 1000 \cdot (1.04)^{-20}\)
\(P = 30 \cdot (13.5903) + 1000 \cdot (0.4564)\)
\(P = 407.71 + 456.39 = 864.10\)

Key Takeaway: If you pay $864.10, your yield will be exactly 8% if the bond lasts 10 years. If the issuer calls it earlier (at 5 years), your yield will actually be higher than 8% because you earned that discount faster!

Common Mistakes to Avoid

1. Mixing up the dates: Students often reflexively pick the earliest date. Remember: Earliest for Premium, Latest for Discount!

2. Ignoring changing Redemption Values: If the redemption value \(C\) is higher in the early years, it might offset the "premium" effect. If the question gives you different \(C\) values for different years, calculate the price at every single change point to be safe.

3. Forgetting to divide by 2: Most Exam FM bonds are semiannual. Always make sure your \(n\), \(i\), and \(r\) are all adjusted for the same time period.

Summary Checklist

• Callable Bond: Issuer can pay back early.
• Minimum Yield: The "worst-case" return for the investor.
• Premium (\(r > i\)): Price is minimized at the earliest call date.
• Discount (\(r < i\)): Price is minimized at the latest call date (maturity).
• The Goal: Calculate the price for all "suspect" dates and choose the lowest price.