MBUS 813 — Session 2 Prep: Bonds & Dividend Discount Model

1

Bond Fundamentals

The Global Bond Market

Global bond market ≈ $120 trillion — larger than global equity markets (~$100T)
Bonds are the primary financing tool for governments and investment-grade corporations
Bonds are debt: the issuer borrows, pays periodic interest (coupon), and repays principal at maturity

Bond Terminology

Term	Definition
Face Value / Par	Principal amount repaid at maturity (typically $1,000)
Coupon Rate	Annual interest as a % of face value; fixed at issuance
Coupon Payment	= Coupon Rate × Face Value (paid semi-annually in North America)
Maturity	Date principal is repaid; term = years to maturity
YTM (Yield to Maturity)	IRR of the bond — discount rate that makes PV of all cash flows = market price
Current Yield	Annual coupon / current market price (ignores capital gains)
Premium Bond	Coupon rate > YTM → price > par
Discount Bond	Coupon rate < YTM → price < par
Par Bond	Coupon rate = YTM → price = par

Types of Bonds

Zero Coupon Bond

No periodic interest payments. Sold at deep discount to face value. All return comes from price appreciation to par at maturity. Examples: T-bills, stripped bonds.

Coupon Bond

Periodic interest payments (semi-annual in North America) plus face value at maturity. Most corporate and government bonds.

Fixed Rate Bond

Coupon rate fixed for life of bond. Most common type for corporate issuers.

Floating Rate Note (FRN)

Coupon resets periodically based on a benchmark (e.g., SOFR/CORRA + spread). Lower duration than fixed-rate bonds.

2

Bond Pricing & YTM

Zero Coupon Bond Pricing

Price (PV) PV = FV / (1 + r)^n Yield (solving for r) r = (FV / PV)^(1/T) − 1

      Example: Zero coupon bond, $1,000 face, 5 years, 6% YTM:

      PV = 1,000 / (1.06)^5 = $747.26

Coupon Bond Pricing

Coupon Bond Price (Annual) B = C × [1 − (1+YTM)^-n / YTM] + F/(1+YTM)^n = Annuity PV of coupons + PV of face value Where: C = annual coupon, F = face value, n = periods

Example 1: Premium Bond

2-yr, $1,000 FV, 10% coupon, 6% YTM C = $100/yr; n = 2; YTM = 6% B = 100/(1.06) + 100/(1.06)^2 + 1,000/(1.06)^2 = $1,073.34 (premium: coupon > YTM)

Example 2: Discount Bond

10-yr, $1,000 FV, 5% coupon, 6% YTM C = $50/yr; n = 10; YTM = 6% B = 50 × [1−(1.06)^-10/0.06] + 1,000/(1.06)^10 = $926.40 (discount: coupon < YTM)

Semi-Annual Coupon Bonds (Standard in North America)

Adjustments for Semi-Annual Coupon per period = Annual Coupon / 2 Number of periods = Years × 2 YTM per period = Annual YTM / 2

      Example: Same 10-yr, $1,000, 5% coupon, 6% YTM — semi-annual:

      C = $25 per period; n = 20; r = 3%

      B = 25 × [1−(1.03)^-20/0.03] + 1,000/(1.03)^20 = $925.61

      (Slightly different from annual due to more frequent compounding)

Exam trap: Always check whether a bond pays annual or semi-annual coupons. The formula structure is identical — just halve the coupon and rate, double the periods.

Solving for YTM

YTM is the IRR of a bond — the single discount rate that equates the PV of all future cash flows to the current market price.

Excel / Calculator Excel: =RATE(nper, pmt, pv, fv, type) Calculator: Enter N, PV (negative), PMT, FV → compute I/Y Example: 10yr, $1,000 FV, $50 semi-annual coupon, price = $925.61 =RATE(20, 50, -925.61, 1000, 0) = 3% per period → 6% annual YTM

There is no closed-form algebraic solution for YTM in coupon bonds — must be solved iteratively (trial and error, financial calculator, or Excel).

3

Bond Price Dynamics & Volatility Rules

Six Bond Price Volatility Rules

Rule 1 — Inverse Relationship

Bond prices move inversely to interest rates (YTM). When yields rise, prices fall; when yields fall, prices rise. Coupon > YTM → premium; Coupon < YTM → discount; Coupon = YTM → par.

Rule 2 — Maturity Effect

For a given change in yield, longer maturity bonds have greater price sensitivity than shorter maturity bonds. The further into the future cash flows occur, the more heavily they are discounted by a rate change.

Rule 3 — Diminishing Sensitivity with Maturity

Price sensitivity to yield changes increases with maturity, but at a decreasing rate. Going from 1yr to 5yr adds much more sensitivity than going from 20yr to 25yr.

Rule 4 — Coupon Effect

For the same maturity, lower coupon bonds have greater price sensitivity than higher coupon bonds. A zero coupon bond is maximally sensitive — all cash flow is at maturity.

Rule 5 — Convexity (Asymmetry)

For equal yield changes up and down, price appreciation exceeds price depreciation. This asymmetry (convexity) benefits bondholders — the price-yield curve is convex, not linear.

Rule 6 — Yield Level Effect

Price sensitivity to a given yield change is greater at lower initial yields than at higher initial yields. A 1% fall from 2% has a bigger impact than a 1% fall from 8%.

Yield Spread

Definition: YTM(corporate) − YTM(government bond of same maturity). Compensates investors for default risk above the risk-free rate.

Market Environment	Spread Direction	Reason
Recession / pessimism	Widens ↑	Higher perceived default risk; flight to quality (govt bonds rally, corps fall)
Expansion / optimism	Narrows ↓	Lower perceived risk; credit demand for corporate bonds compresses spreads
Investment Grade (BBB)	Narrowest IG spread	Still considered safe; traded actively by pension funds
High Yield (BB and below)	Wide	Significant default risk premium required

4

Duration & Immunization

Duration — Definition & Calculation

Duration is the weighted average time to receipt of a bond's cash flows, where the weights are the present values of each cash flow as a proportion of total bond price.

Macaulay Duration D = Σ [t × PV(CF_t)] / Bond Price Where: t = time period, PV(CF_t) = PV of cash flow at time t

      Worked Example: 8-year bond, 9% annual coupon, YTM = 10%

      Calculate PV of each coupon and face value, multiply by year, divide by price.

      Result: Duration = 5.97 years

Duration is also the price sensitivity measure — a 1% rise in yields causes roughly a Duration% fall in price (Modified Duration).

Duration Properties

Property	Effect on Duration
Zero coupon bond	Duration = Maturity (all cash flow at end)
Higher coupon rate	Shorter duration (more weight on earlier payments)
Higher YTM	Shorter duration (future flows discounted more heavily)
Longer maturity	Longer duration (but less than proportionally)
Perpetuity	Duration = (1 + y) / y

Duration Immunization Strategies

Strategy 1: Target Date Immunization

Goal: Ensure a specific sum is available at a target date, regardless of interest rate changes
Method: Match the bond's duration to the investor's holding period (not maturity)
Works because: Price risk and reinvestment risk offset each other when duration = horizon

Target Date Example 8% coupon, 4-yr bond, YTM=10% Duration = 3.48 years Investor horizon = 3.5 years Whether rates go to 8%, 10%, or 12%: Total accumulated value ≈ $131.6M (price risk and reinvestment risk cancel)

Strategy 2: Net Worth Immunization

Goal: Protect the surplus (assets − liabilities) against rate changes
Method: Set asset portfolio duration = liability duration
Used by: Insurance companies, pension funds
If asset duration = liability duration, any rate change affects both sides equally → surplus preserved

Net Worth Immunization Condition Duration(Assets) = Duration(Liabilities) If rates rise: Asset value falls, but liability PV also falls by same amount → Net worth unchanged ✓

Exam point: Immunization is not a one-time fix. As time passes and rates change, the portfolio must be rebalanced to maintain the duration match. Duration is a moving target.

5

Dividend Discount Model (DDM)

DDM Framework

The intrinsic value of a stock equals the present value of all future dividends, discounted at the investor's required return (cost of equity, k_e).

General DDM P_0 = D_1/(1+ke) + D_2/(1+ke)^2 + D_3/(1+ke)^3 + ... = Σ D_t / (1+ke)^t ke = cost of equity (required return / discount rate)

      Key assumption: Dividends, not earnings, are the cash flows shareholders actually receive. The model assumes the firm will pay dividends indefinitely.
    

Gordon Growth Model (Constant Growth DDM)

When dividends grow at a constant rate g forever, the infinite sum collapses to the growing perpetuity formula.

Gordon Growth Model P_0 = D_1 / (ke − g) Where: D_1 = next year's dividend (= D_0 × (1+g)) ke = required return on equity g = constant perpetual dividend growth rate REQUIRES: ke > g

Example 1 D_1 = $1.30, ke = 10%, g = 5% P_0 = 1.30 / (0.10 − 0.05) = $26.00

Class Question (Answer C) D_0 = $1.00 (just paid), g = 5%, ke = 10% D_1 = $1.00 × 1.05 = $1.05 P_0 = 1.05 / (0.10 − 0.05) = $21.00

Common mistake: Using D_0 in the numerator. The formula uses D_1 (next period's dividend). Always grow D_0 by (1+g) to get D_1 first.

Multi-Stage DDM

Real companies often have a high-growth phase followed by stable long-run growth. Value each period separately, then apply Gordon Growth for the terminal value.

Multi-Stage Framework P_0 = D_1/(1+ke) + D_2/(1+ke)^2 + ... + (D_n/(ke-g))/(1+ke)^(n-1) Terminal Value at end of stage 1 = D_(n+1) / (ke − g_stable) Discount that terminal value back to today

Worked Example: Multi-Stage

Given: D_0 = $2.00, ke = 15%, g1 = 10% for 2 years, then g2 = 5% forever

D_1 = $2.00 × 1.10 = $2.20 D_2 = $2.20 × 1.10 = $2.42 D_3 = $2.42 × 1.05 = $2.541 (first stable-growth dividend) Terminal Value at end of Year 2: P_2 = D_3 / (ke − g_2) = $2.541 / (0.15 − 0.05) = $25.41 P_0 = D_1/(1.15) + (D_2 + P_2)/(1.15)^2 = 2.20/1.15 + (2.42 + 25.41)/1.1525^2 = $1.91 + $21.04 = $22.95

DDM Sensitivity

The Gordon Growth formula is highly sensitive to assumptions. Small changes in ke or g have a large impact on valuation.

Sensitivity Example D_1 = $0.83; ke = 6.20%; g = 3.70% P_0 = 0.83 / (0.0620 − 0.0370) = $33.20 If ke rises to 7.20%: P_0 = 0.83/0.035 = $23.71 (−29%) If g falls to 2.70%: P_0 = 0.83/0.035 = $23.71 (−29%)

      Key insight: The denominator (ke − g) is the key driver. A tiny change in this spread dramatically changes value. This is why equity valuation requires conservative, defensible assumptions.
    

★

Formula Reference — Session 2

Complete Formula Sheet

Zero Coupon Bond Price PV = FV / (1 + r)^n Zero Coupon Bond Yield r = (FV / PV)^(1/T) − 1 Coupon Bond Price (Annual) B = C × [1 − (1+YTM)^-n] / YTM + F / (1+YTM)^n Semi-Annual Adjustments C_period = Annual Coupon / 2 n_periods = Years × 2 r_period = Annual YTM / 2 Macaulay Duration D = Σ [t × PV(CF_t)] / Bond Price Perpetuity Duration D = (1 + y) / y Gordon Growth Model (Constant-Growth DDM) P_0 = D_1 / (ke − g) [requires ke > g] D_1 = D_0 × (1 + g) Yield Spread Spread = YTM(Corporate) − YTM(Government) Excel for YTM =RATE(nper, pmt, pv, fv, type) =RATE(20, 50, -925.61, 1000, 0) → 3% per period

Six Rules — Quick Reference

#	Rule	Direction
1	Yield ↑ → Price ↓ (Inverse relationship)	Always
2	Longer maturity → Greater price sensitivity	Always
3	Sensitivity increases at a decreasing rate as maturity lengthens	Always
4	Lower coupon → Greater price sensitivity	Always
5	Price gain from yield fall > Price loss from equivalent yield rise (convexity)	Benefit to holders
6	Lower initial yield → Greater sensitivity to a given change	Always

Session 2: Bonds & Dividend Discount Model