Bond and Stock Valuation

Bond valuation and duration concepts are among the most calculation-intensive topics on the CFP exam. Understanding how to price bonds, measure interest rate sensitivity, and implement portfolio strategies is essential for both exam success and client portfolio management.

Bond Pricing Formula

The value of a bond equals the present value of all future cash flows--coupon payments plus the return of principal at maturity.

Bond Price Formula:

Bond Value = Sum of [Coupon / (1 + r)^t] + [Par Value / (1 + r)^n]

Where:

• Coupon = Annual coupon payment (Coupon Rate x Par Value)
• r = Required yield (YTM) per period
• t = Time period for each cash flow
• n = Number of periods to maturity
• Par Value = Face value (typically $1,000)

Calculator Approach (Semi-Annual Bonds):

Most bonds pay semi-annual coupons. To calculate bond price on a financial calculator:

Worked Example:

Calculate the price of a 10-year bond with a 6% coupon rate and 8% yield to maturity (semi-annual payments):

• N = 10 x 2 = 20 periods
• I/Y = 8% / 2 = 4% per period
• PMT = $60 / 2 = $30 per period
• FV = $1,000

PV = $864.10

Since the YTM (8%) exceeds the coupon rate (6%), the bond trades at a discount to par.

Price-Yield Relationship

The inverse relationship between bond prices and yields is fundamental:

Key Insight: The price-yield relationship is not linear--it's convex. This means price increases from falling yields are larger than price decreases from rising yields of the same magnitude.

---

Duration: Measuring Interest Rate Sensitivity

Duration measures a bond's price sensitivity to interest rate changes. There are two main types: Macaulay Duration and Modified Duration.

Macaulay Duration

Macaulay Duration is the weighted average time until an investor receives all cash flows from a bond, with weights based on the present value of each cash flow.

Macaulay Duration Formula (on CFP Exam Formula Sheet):

Macaulay Duration = [1 + y] / y - [1 + y + t(c - y)] / [c((1 + y)^t - 1) + y]

Where:

• y = Yield to maturity per period
• c = Coupon rate per period
• t = Number of periods to maturity

For a zero-coupon bond: Macaulay Duration = Time to Maturity (no interim cash flows to weight)

Modified Duration

Modified Duration adjusts Macaulay Duration to directly estimate percentage price changes for a given yield change.

Modified Duration Formula:

Modified Duration = Macaulay Duration / (1 + YTM/n)

Where n = number of compounding periods per year (typically 2 for semi-annual bonds)

Estimating Price Changes with Duration

Percentage Price Change Formula (on CFP Exam Formula Sheet):

% Price Change = -Modified Duration x Change in Yield

Worked Example:

A bond has a Modified Duration of 7.5 years. If interest rates rise by 0.50%, what is the approximate price change?

% Price Change = -7.5 x 0.005 = -0.0375 = -3.75%

If the bond was priced at $1,000, the new price would be approximately $962.50.

Factors Affecting Duration

CFP Exam Tip: Remember the inverse relationships--coupon rates and YTM have an INVERSE relationship with duration. "IN-terest rates are IN-versely related to duration."

---

Convexity: Improving Duration Estimates

Duration provides a linear approximation of the price-yield relationship, but the actual relationship is curved (convex). Convexity measures this curvature and improves price estimates for larger yield changes.

Price Change with Convexity Adjustment:

% Price Change = (-Modified Duration x Delta Y) + (0.5 x Convexity x (Delta Y)^2)

The first term is the duration effect (negative for rising rates). The second term is the convexity adjustment (always positive for option-free bonds).

Why Convexity Matters:

Convexity Benefit:

Bonds with higher convexity outperform in both rising and falling rate environments:

• When rates rise, they lose less than duration predicts
• When rates fall, they gain more than duration predicts

Factors Affecting Convexity:

CFP Exam Tip: Convexity is always a benefit for option-free bonds. Investors should prefer bonds with higher convexity, all else equal.

---

Bond Immunization Strategies

Immunization is a strategy that protects a portfolio against interest rate changes by matching the portfolio's duration to the investor's time horizon.

How Immunization Works

Interest rate risk and reinvestment risk work in opposite directions:

When portfolio duration equals the investment time horizon, these two effects offset each other, "immunizing" the portfolio from rate changes.

Requirements for Effective Immunization:

1. Portfolio duration must equal the investment time horizon
2. The portfolio must be rebalanced periodically as duration changes
3. Yield curve shifts should be parallel (limitation of basic immunization)

Worked Example:

A client needs $50,000 in exactly 5 years for a child's education. To immunize:

• Select a bond or portfolio with a duration of 5 years
• If rates rise, the bond price falls but reinvested coupons earn more
• If rates fall, the bond price rises but reinvested coupons earn less
• At year 5, the total value should meet the goal regardless of rate changes

---

Bond Portfolio Strategies

Bond Laddering

A laddered portfolio holds bonds with staggered maturities at regular intervals (e.g., 1, 2, 3, 4, 5 years).

Advantages:

• Provides regular liquidity as bonds mature
• Averages exposure across different rate environments
• Simple to implement and rebalance
• Reduces timing risk

Example Ladder Structure:

When the 2026 bond matures, proceeds are reinvested in a new 5-year bond maturing in 2031, maintaining the ladder.

Barbell Strategy

A barbell strategy concentrates holdings in short-term and long-term bonds with little in intermediate maturities.

Structure Example:

• 50% in 1-3 year bonds (liquidity, low duration)
• 0% in 4-7 year bonds
• 50% in 8-10+ year bonds (higher yield, higher duration)

Advantages:

• Short-term bonds provide liquidity and flexibility
• Long-term bonds provide higher yields
• Can benefit from yield curve steepening

Disadvantages:

• More volatile than a laddered portfolio
• Requires active rebalancing
• Higher transaction costs

Bullet Strategy

A bullet strategy concentrates all bond maturities around a single target date.

Structure Example:

• All bonds mature in years 4-6
• Portfolio duration closely matches target horizon

Best Used For:

• Matching a specific future liability
• Pension fund obligations
• Education funding with a known date

Comparison of Strategies:

---

Key Duration and Convexity Relationships Summary