Price, book value, and market value

Welcome to the World of Bond Valuation!

Hello future actuaries! Today, we are diving into one of the most practical and frequently tested parts of the Exam FM syllabus: Price, Book Value, and Market Value of bonds. Don't worry if these terms sound like accounting jargon; by the end of this guide, you'll see they are just different ways of looking at how much a bond is worth as time passes. Think of this as learning how to track the "balance" of a loan from the perspective of the person who lent the money.

1. Determining the Price of a Bond

Before we can talk about book value, we need to know how a bond is priced. In simple terms, the Price of a bond is the Present Value (PV) of all the cash it will give you in the future.

When you buy a bond, you get two things:
1. A series of regular payments called Coupons.
2. A final lump sum at the end called the Redemption Value (usually denoted as \(C\)).

The Basic Pricing Formula

The standard formula used in Exam FM is:
\(P = Fra_{\overline{n}|i} + Cv^n\)

Where:
\(P\) = Price
\(F\) = Face Value (used to calculate the coupon amount)
\(r\) = Coupon rate per period
\(i\) = Yield rate (the interest rate the investor actually earns)
\(n\) = Number of periods until maturity
\(C\) = Redemption value (often the same as Face Value, but not always!)

Quick Tip: If the problem says "a $1,000 bond," that usually means both \(F\) and \(C\) are $1,000 unless stated otherwise. If the redemption value is different, the problem will call it a "bond redeemable at \(C\)."

Analogy: The Fruit Tree

Imagine buying a fruit tree. The "Price" is what you pay today. The "Coupons" are the baskets of fruit you get every year. The "Redemption Value" is what you can sell the wood for once the tree stops producing fruit. To find the fair price, you just find the value of all that future fruit and wood in today's dollars!

Key Takeaway: Price is simply the sum of the discounted coupons and the discounted redemption value.

2. Premium vs. Discount

You will often hear that a bond is selling at a Premium or a Discount. This depends entirely on the relationship between the coupon rate (\(r\)) and the yield rate (\(i\)).

Premium (\(P > C\)): This happens when the coupon rate is higher than the market interest rate (\(r > i\)). Investors love high coupons, so they are willing to pay more than the redemption value to get them.
Discount (\(P < C\)): This happens when the coupon rate is lower than the market interest rate (\(r < i\)). Because the coupons are "low," the bond must be cheaper to attract buyers.

Memory Aid: "If the coupon is Great, you pay a higher Rate (Premium). If the coupon is Low, the price must Go (Discount)."

3. Book Value (\(B_t\))

The Book Value of a bond at time \(t\) is the value of the bond as recorded on the investor's balance sheet. It is essentially the "outstanding balance" of the bond at any point in time.

How to Calculate Book Value

There are two ways to find the Book Value at time \(t\), but the Prospective Method is usually the easiest for Exam FM:

Prospective Method:
\(B_t = Fra_{\overline{n-t}|i} + Cv^{n-t}\)

In plain English: To find the Book Value today, just calculate the Present Value of all the payments that are left to be paid. If a 10-year bond has already existed for 4 years, there are 6 years left. You would find the PV of those remaining 6 years of coupons and the redemption value.

Did you know? At the very start (time 0), the Book Value is just the Price (\(B_0 = P\)). At the very end (time \(n\)), the Book Value is exactly the Redemption Value (\(B_n = C\)).

4. Amortization of Premium and Accumulation of Discount

As time goes by, the Book Value moves from the original Price toward the final Redemption Value. This movement is called amortization (if the value goes down) or accumulation (if the value goes up).

The Basic Relationship

For any period, the following is always true:
Interest Earned = (Book Value at start of period) \(\times\) (Yield rate \(i\))
Principal Adjustment = Coupon - Interest Earned

1. Amortization of Premium: If you paid a Premium, the Book Value will decrease every period until it hits \(C\). The amount it decreases by each period is the "Amortization of Premium."
2. Accumulation of Discount: If you bought at a Discount, the Book Value will increase every period until it hits \(C\). This increase is the "Accumulation of Discount."

Step-by-Step: The Amortization Schedule Logic

1. Start with the Book Value at the beginning of the period (\(B_{t-1}\)).
2. Calculate the Interest Earned: \(I_t = i \cdot B_{t-1}\).
3. Look at your Coupon: \(Fr\).
4. The difference \((Fr - I_t)\) is the amount the Book Value changes.
5. New Book Value: \(B_t = B_{t-1} - (Fr - I_t)\).

Common Mistake: Don't confuse the Coupon with the Interest Earned. The Coupon is the cash you receive; the Interest Earned is the theoretical growth of your investment based on the yield rate.

Key Takeaway: The Book Value always "gravitates" toward the Redemption Value over time.

5. Market Value

While Book Value is based on the yield rate at the time the bond was purchased, Market Value is what the bond would sell for today based on current market interest rates.

The Calculation: Market Value is calculated exactly like Book Value, but you use the current market interest rate (\(i'\)) instead of the original yield rate (\(i\)).

\(MV_t = Fra_{\overline{n-t}|i'} + Cv_{i'}^{n-t}\)

Analogy: Imagine you bought a house with a fixed-rate mortgage. Your "Book Value" is like your remaining loan balance based on your contract. The "Market Value" is what someone would pay for your house today if you tried to sell it in the current real estate market. They might be very different!

6. Summary and Quick Review

Quick Review Box

- Price (\(P\)): PV of all future cash flows at time 0.
- Book Value (\(B_t\)): PV of remaining cash flows at time \(t\), using the original yield.
- Market Value (\(MV_t\)): PV of remaining cash flows at time \(t\), using the current market yield.
- Premium: When \(r > i\), Price \(>\) Redemption Value.
- Discount: When \(r < i\), Price \(<\) Redemption Value.

Final Encouragement

Don't worry if the formulas for amortization schedules feel overwhelming at first. Just remember the core concept: A bond is just a series of payments. Whether you are finding the Price at the start or the Book Value in the middle, you are just doing a Present Value calculation. Master the "Prospective Method," and you will be well on your way to acing this section of Exam FM!