Variant: Cost of Debt using Matrix Pricing
Abstract
Find traded bonds with the same credit rating, compute their YTMs, average by maturity, then linear-interpolate to your target maturity. Average before interpolating — that is the main trap.
Brevis Solutions is a technology provider. Sunil Tilak, CFA is estimating the cost of debt, which represents 30% of Brevis' capital structure. The 6-year, BB-rated debt is thinly traded. Tilak collects data on similar BB-rated securities with liquid markets:
- Silva: 4-year, 5% coupon, price $99.45
- Deso: 4-year, 6% coupon, price $101.75
- Manfried: 7-year, 7% coupon, price $110.00
- Listor: 7-year, 8% coupon, price $114.00
Estimate the cost of debt using the matrix method.
What is the question really asking?
We need Brevis's cost of debt — the YTM a 6-year BB-rated bond should have. But Brevis's bond is thinly traded, so there is no reliable market price. The key question is: what yield does the market assign to BB-rated bonds near 6 years?
Note
Matrix pricing = borrow the yield from comparable traded bonds when your bond doesn't trade.
1. Calculate the YTM of each comparable bond
Each comparable is a standard TVM problem: today's price is the PV, coupons are PMT, par is FV, solve for I/Y.
Intuition: Price above par means yield below coupon; price below par means yield above coupon. Silva at $99.45 is almost at par, so its 5.16% YTM is just above its 5% coupon. Deso at $101.75 is a premium bond, so its 5.50% YTM is below its 6% coupon.
2. Construct the matrix and average yields by maturity
Arrange the YTMs into a grid of maturity vs coupon, then average each maturity row:
| Maturity | 5% | 6% | 7% | 8% | Average YTM |
|---|---|---|---|---|---|
| 4 yr | 5.16% | 5.50% | — | — | 5.33% |
| 7 yr | — | — | 5.26% | 5.53% | 5.39% |
Note
Average each maturity row first — this removes the coupon effect and isolates the maturity effect on yield.
3. Linear-interpolate to the target maturity (6 years)
We have the 4-year average (5.33%) and the 7-year average (5.39%). Brevis needs the 6-year yield, which sits between them.
Intuition: The yield curve from 4 to 7 years slopes gently upward (5.33% → 5.39%). Six years is two-thirds of the way from 4 to 7, so we take the 4-year yield and add two-thirds of the 0.06% spread.
Note
Linear interpolation: \(\text{Yield}_{target} = \text{Yield}_{short} + \frac{\Delta \text{Yield}}{\Delta \text{Mat}} \times (\text{target} - \text{short})\).
Variant: Cost of Debt from a Capital Lease (RIIL)
Abstract
The lease hides a loan. PV is what the lessor spends (fair value + direct costs), PMT is the lease payment, FV is the residual. Solve for I/Y — that is the hidden cost of debt.
Company A has signed a 15-year lease with annual payments of $10 million at the end of each year. The lease residual value is $30 million. The fair value of the asset is $120 million, and the lessor incurs a cost of $5 million at lease initiation.
Calculate the RIIL (rate implicit in the lease).
What is the question really asking?
We need the hidden interest rate baked into this lease — the rate that makes the lessor's investment worthwhile. The key question is: what is the lessor actually investing, and what are they getting back?
Note
RIIL is the IRR of the lease from the lessor's perspective: PV of what they pay out = PV of what they receive back.
1. Identify the lessor's cash flows
The lessor gives up the asset (fair value $120m) and pays extra costs ($5m) at inception. In return, they receive 15 annual payments of $10m plus the residual of $30m at the end.
Intuition: The lessor spends $125m today to buy and deliver the asset. They get $10m/year for 15 years, then the asset back (worth $30m). The IRR of that stream is the hidden borrowing rate.
2. Solve for the rate (TVM)
Why? The lessor lends $125m, receives $10m/year plus $30m at the end. The rate that balances those flows is 4.28% — that is the cost of debt implied by the lease.
Note
RIIL uses the lessor's perspective: \(PV = -(\text{fair value} + \text{direct cost})\), \(PMT = \text{lease payment}\), \(FV = \text{residual}\).
Variant: Equity Risk Premium using the Grinold-Kroner Model
Abstract
Expected equity return = dividend yield + repricing + inflation + real growth − dilution. Get inflation from the Fisher equation (Treasury vs TIPS, not simple subtraction). Subtract the risk-free rate for the ERP.
Patrick McGill is estimating the equity risk premium for the U.S. market. He uses the S&P 500 as the market proxy.
- Dividend yield: 1.2%
- Real GDP growth rate (forecast): 3%
- Market is fairly valued (no expected P/E change)
- 10-year Treasury yield: 2.4%
- 10-year TIPS yield: 0.25%
- No net change in shares outstanding
- Risk-free rate: 0.50%
Calculate the equity risk premium.
What is the question really asking?
We need the ERP — the extra return investors demand for holding equities over the risk-free rate. The key question is: what are the five drivers of expected equity return, and which are zero here?
1. Understand the Grinold-Kroner decomposition
The model says the expected market return comes from five pieces:
Then:
Note
Grinold-Kroner: \(ERP = [DY + \Delta P/E + i + G - \Delta S] - r_f\). Each piece is an independent driver of equity return.
2. Knock out the zeros
Two pieces are given as zero:
- Market is fairly valued \(\Rightarrow\) no expected P/E expansion or contraction \(\Rightarrow \Delta P/E = 0\)
- No net change in shares outstanding \(\Rightarrow \Delta S = 0\)
That leaves three live pieces: \(DY\), \(i\), and \(G\).
3. Estimate expected inflation from Treasury vs TIPS
The question gives both the nominal 10-year Treasury yield (2.4%) and the TIPS yield (0.25%). The difference reveals the market's inflation expectation — but not by simple subtraction.
Use the Fisher equation (exact):
Note
Do not simply subtract \(2.4\% - 0.25\% = 2.15\%\). Use the Fisher equation: \(i = \frac{1+Y_{\text{nominal}}}{1+Y_{\text{TIPS}}} - 1\).
Intuition: TIPS already include inflation protection, so the real yield is 0.25%. The nominal yield is 2.4%. The inflation rate that connects the two is not their difference but their ratio (minus one), because real and nominal rates compound multiplicatively.
4. Assemble the ERP
Intuition: Equity investors expect to earn the dividend yield (1.2%), plus inflation (2.1%), plus real economic growth (3.0%). That totals 6.3%. Since the risk-free rate pays 0.5%, the extra risk premium for holding equities is 5.8%.
Note
If the market were undervalued, \(\Delta P/E > 0\) (expected upward repricing adds to ERP). If overvalued, \(\Delta P/E < 0\).
Variant: Cost of Equity using the DDM
Abstract
Cost of equity = dividend yield + growth. Constant growth → Gordon formula (\(D_1/P_0 + g\)). Varying growth → IRR with the terminal stock price bundled into the last cash flow. The trap is forgetting that terminal price.
Calculate the cost of equity for two companies:
- Cogenics, Inc.: Expected dividend $4 at end of Year 1, dividends grow at a constant rate of 4% per year, current stock price $100.
- Betagenics, Inc.: Expected dividends of $1.50, $2.00, $2.50, and $3.00 at the end of each of the next four years. Current stock price $50, expected stock price at end of Year 4 is $60.
What is the question really asking?
We need each company's required return on equity (\(r_e\)). The key question is: does the dividend stream have constant growth or varying growth? Cogenics is constant (use the Gordon formula); Betagenics is varying (use IRR).
Note
DDM cost of equity: \(r_e = DY + CGY = \frac{D_1}{P_0} + g\). For non-constant dividends, solve for IRR instead.
1. Cogenics — constant growth (Gordon)
Dividends grow at a steady 4% forever, so the Gordon Growth Model applies directly:
Intuition: You pay $100 today. You get a $4 dividend yield immediately, and the stock grows at 4% per year (capital gains). Total required return = 8%.
2. Betagenics — non-constant growth (IRR)
Dividends change every year, so there is no single \(g\) to plug in. Instead, set up the cash flows and solve for IRR — the rate that makes the PV of all future cash flows equal to today's price.
The cash flows are:
| Time | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| CF | −50 | 1.50 | 2.00 | 2.50 | 63.00 |
Why \(63\) at Year 4? The last cash flow bundles the final dividend ($3.00) with the terminal stock price ($60): \(3 + 60 = 63\).
Note
The terminal stock price enters at the same time as the final dividend — bundle them: \(CF_{last} = D_{last} + P_{terminal}\).
On the TI BA II Plus:
Intuition: You pay $50 now and receive a growing dividend stream plus $60 at the end. The IRR that balances those flows — 8.78% — is what equity holders require.
Variant: Cost of Equity using Bond Yield Plus Risk Premium
Abstract
Cost of equity = bond YTM + equity risk premium. Solve the YTM from the bond price first — it is not the coupon rate. Then add the premium for equity being riskier than debt.
Company LMN has bonds with 15 years to maturity, a coupon of 8.2%, and a price of 101.70. An analyst estimates that the additional risk of holding equity rather than bonds justifies a risk premium of 3.8%. Calculate the cost of equity using the bond-yield-plus-risk-premium approach.
What is the question really asking?
We need LMN's cost of equity using a build-up shortcut. The key question is: what is the YTM of LMN's bonds? That is the starting point, not the coupon rate.
1. Compute the bond's YTM
The bond price (101.70) differs from par (100), so YTM ≠ coupon. Solve a TVM problem:
Intuition: The bond trades at a slight premium, so its YTM (8.0%) is just below its coupon (8.2%). Bondholders demand 8.0%.
2. Add the equity risk premium
Equity sits below debt in the capital structure, so equity holders demand more. The analyst has quantified that "more" as 3.8%:
Intuition: Bondholders get 8.0%. Equity holders take all the residual risk, so they demand 3.8% more. Total: 11.8%.
Note
BYRPM: \(r_e = YTM_{\text{debt}} + \text{equity risk premium over debt}\). Use YTM, not the coupon rate.
Variant: Cost of Equity using the CAPM
Abstract
Cost of equity = risk-free rate + beta × equity risk premium. One factor, one multiplication. Beta scales the market premium — below 1 means less risk, above 1 means more.
The expected risk-free rate is 4%, and the equity risk premium is 3.9%. Calculate the required return on equity for a stock with a beta of 0.8.
What is the question really asking?
We need the required return on equity using the simplest single-factor model. The key question is: how much market risk does this stock carry? Beta = 0.8 means less than the market.
1. Apply the CAPM formula
Intuition: The risk-free floor is 4%. The market premium is 3.9%, but this stock only captures 80% of market risk (beta = 0.8), so it earns only 3.12% of the premium. Total: 7.12%.
Note
CAPM: \(r_e = r_f + \beta \times ERP\). Beta scales the market premium — a beta below 1 means less market risk, a beta above 1 means more.
Variant: Cost of Equity using the Fama–French Five-Factor Model
Abstract
Like CAPM but with five risk buckets. Multiply each beta by its premium (mind negative signs — they subtract), sum all contributions, add the risk-free rate.
Suppose the current risk-free rate is 2.1%. Calculate the cost of equity for Fulton Corp. using the Fama–French five-factor model:
- Market: beta 1.1, risk premium 3.2%
- Size (SMB): beta 0.2, risk premium 1.3%
- Value (HML): beta −0.3, risk premium 2.0%
- Profitability (RMW): beta 0.18, risk premium 4.2%
- Investment Style (CMA): beta 0.5, risk premium 2.4%
What is the question really asking?
We need Fulton's required return on equity (\(r_e\)). The key question is: how many risk buckets does this model use? The question fixes Fama–French five-factor, so there are five, not one.
Note
FF5 = CAPM with four extra risk buckets: \(r_e = r_f + \sum_i \beta_i \times RP_i\). Each factor chips in its own \(\beta \times\) premium.
1. Recognise the game: CAPM with five risk buckets
CAPM prices equity with a single market bucket. FF5 keeps that skeleton but adds four extra buckets — different flavours of risk a stock is exposed to:
- Size (SMB, Small Minus Big) — small firms tend to out-earn big firms.
- Value (HML, High Minus Low book-to-market) — value firms tend to out-earn growth firms.
- Profitability (RMW, Robust Minus Weak) — robust earners tend to win.
- Investment Style (CMA, Conservative Minus Aggressive) — conservative investors tend to win.
2. Multiply each beta by its risk premium
Each bucket's contribution is sensitivity × price per unit of sensitivity:
Note
Negative beta = subtraction. Fulton's value beta is \(-0.3\), so it behaves like a growth stock and the value premium is removed, not added.
3. Sum the buckets and add the risk-free rate
Intuition: Start at the risk-free floor of 2.1%. Fulton earns extra for market risk (+3.52%), a little for size (+0.26%), profitability (+0.76%) and investment style (+1.2%), but loses a bit because it is growth-leaning, not value (−0.60%). Net ≈ 7.2%.
Units check: beta is unitless; \(\beta \times RP\) yields percentage points; the running total is a rate in %. Consistent.
4. Interpret the result
\(\boxed{r_e \approx 7.2\%}\) is what Fulton's shareholders require. It is the discount rate for Fulton's equity cash flows and the hurdle rate for equity-funded projects — any project returning below 7.2% destroys equity value.