Notes on Nominal vs Real Rates, Fisher Equation, Term Structure, and Dividend Discount Model

Nominal vs Real Rates

Nominal rate (R_nominal) is the rate before adjusting for inflation.
Real rate (r_real) is the rate after removing inflation; it reflects true purchasing power growth.
Inflation rate (π, sometimes denoted h in some texts) erodes purchasing power and is the reason real returns differ from nominal ones.
Example (pizza/currency illustration):
- Start with $100 at the beginning of the year.
- Nominal return: end-of-year amount is $115.50, which is a nominal return of 15.5%.
- Inflation example: suppose pizza costs rise from $10 at the start to $10.50 at year end (inflation = 5%).
- With $100, you could buy 10 pizzas at the start and 11 pizzas at the end (115.50 / 10 = 11.55, i.e., you could buy 11 pizzas if rounding to whole pizzas).
- Real return on pizza = (11/10) − 1 = 10%.
- Inflation has eaten 5.5 percentage points of the nominal return, leaving a real return of 10% in this example.
Key takeaway: nominal returns can be higher when inflation is positive, but the real return (purchasing power adjusted) may be much lower.

The Fisher Equation (Fisher Effect)

Fisher equation relates nominal rate, real rate, and inflation: $1 + r{nom} = (1 + r{real})(1 + \, \pi)$ where:
- $r_{nom}$ is the nominal rate
- $r_{real}$ is the real rate
- $\pi$ is the inflation rate
Rearranged (exact):
$r{nom} = (1 + r{real})(1 + \pi) - 1$
In some texts, inflation is denoted by π or sometimes h; in this course, h is used for inflation, but the concept is the same.
Practical rule of thumb (approximation):
$r{nom} \approx r{real} + \pi$
Real-world intuition: you can determine the real rate if you know any two of the three: nominal rate, real rate, or inflation.

Example: exact vs. approximate Fisher results

Example 1: nominal return 15%? Let’s use the slide’s values.
- If real rate is 10% and inflation is 5% (π = 0.05):
- Exact nominal rate: $r_{nom} = (1 + 0.10)(1 + 0.05) - 1 = 1.155 - 1 = 0.155 = 15.5\%.$
- This matches the initial nominal return example of 15.5%.
- Note: sometimes the slide discusses 15.5% nominal with 5% inflation yielding a 10% real return (through a separate illustration about how much you can buy).
Example 2 (alternative numbers used on the slide):
- If real rate is 10% and inflation is 8%:
- Exact nominal rate: $r_{nom} = (1 + 0.10)(1 + 0.08) - 1 = 1.188 - 1 = 0.188 = 18.8\%.$
- Approximation would give $r_{nom} \approx 10\% + 8\% = 18\%.$ (exact is 18.8%, close but not identical; the difference grows with larger r and π)
Takeaway: the exact Fisher equation is needed for precise calculations; the approximation is handy for quick checks.

Approximation vs. Exact (Fisher) details

Fisher equation shows that when you multiply (1 + r_real) and (1 + π), you capture the compound effect of real growth and inflation.
The approximation rnom ≈ rreal + π neglects the product term r_real*π, which becomes more important as rates or inflation rise.
In practice, both forms are used:
- Exact: $r{nom} = (1 + r{real})(1 + \pi) - 1$
- Approximate: $r{nom} \approx r{real} + \pi$

Present Value concepts: nominal vs. real terms (DCF mindset)

In valuation models you discount cash flows using the same type of rate as the cash flows:
- Nominal cash flows with nominal discount rates
- Real cash flows with real discount rates
If you do everything consistently (nominal with nominal, real with real), you will get the same present value regardless of whether you work in nominal or real terms.
Example context from the lecture:
- Withdrawals intended to maintain purchasing power of $25,000 in today’s dollars with 4% inflation.
- If you project nominal withdrawals for 3 years increasing by 4% per year, discount at a nominal rate (e.g., 10%), you get one present value.
- If you instead express real withdrawals (constant purchasing power, i.e., $25,000 in today’s dollars) and discount at the real rate (computed via Fisher), you should obtain the same PV (subject to rounding).
Details from the example:
- Inflation = 4% per year, nominal rate = 10% per year.
- Real rate: $r_{real} = (1 + 0.10) / (1 + 0.04) - 1 = 1.10/1.04 - 1 \approx 0.0577 = 5.77\%.$
- Nominal withdrawals for years 1–3 to maintain $25,000 purchasing power grow by 4% each year:
- Year 1: $25{,}000(1.04)^1$ , Year 2: $25{,}000(1.04)^2$ , Year 3: $25{,}000(1.04)^3$
- PV of nominal cash flows at 10%: $PV{nom} = \sum{t=1}^3 \frac{25{,}000(1.04)^t}{(1+0.10)^t}$
- Real cash flows: in real terms, withdrawals are constant at $25{,}000$ per year; discounted at the real rate $r{real} = 5.77\%$ using the annuity formula: $PV{real} = 25{,}000 \times \frac{1 - (1 + r{real})^{-3}}{r{real}}$
- The two PVs converge to the same value (modest rounding differences) when computed consistently.

Term structure of interest rates

Definition: The term structure (yield curve) shows the yield on zero-coupon risk-free (default-free) bonds across maturities.
Common shapes:
- Normal (upward-sloping): longer maturities have higher yields due to more risk and longer exposure to inflation.
- Inverted (downward-sloping): occurs when short-term rates are higher than long-term rates; often associated with expectations of falling rates during a recession.
- Flat: little difference between short and long-term yields.
- Other shapes (e.g., humps) can occur due to varying risk and liquidity factors.
What determines the shape?
- Real rates (short and long-term real rate expectations)
- Expected inflation (inflation premium for longer horizons)
- Interest-rate risk premium (compensation for risk of changing rates; grows with time, but often at a decreasing rate)
Key insights:
- Inflation expectations drive part of the inflation premium embedded in longer maturities.
- Real rates are generally flatter with maturity, while inflation risk and rate risk premiums push long-term yields upward.
- The term structure is commonly illustrated using zero-coupon yields, whereas many bond yields are coupon-based.
Practical takeaway for investors: the shape of the yield curve informs expectations about future interest rates and the relative attractiveness of short vs. long-duration investments.

Nominal vs Real: which is more important for the typical investor?

Real rates are often more informative about the true purchasing power you gain or lose over an investment horizon.
Nominal rates are the practical numbers used in many pricing and discounting tasks, especially when cash flows themselves are nominal.
Conclusion from the lecture: real rates matter for understanding true wealth growth; but when performing standard PV/DV calculations, consistency (nominal with nominal or real with real) is the key to getting the same answer.

Chapter 8: Stock valuation and the Dividend Discount Model (DDM)

Core idea: The value of a stock today equals the present value of all its future dividends.
- This is the Dividend Discount Model (DDM).
Why stocks can be harder to value than bonds:
- Bonds have fixed cash flows (coupons) and a known maturity; stock dividends can be variable, possibly infinite (perpetual) and may not exist for a period.
- The maturity of a stock is effectively infinite (perpetuity of dividends), whereas most bonds have finite maturity.
- Some stocks pay no dividends (e.g., some growth stocks); value then comes from expected future dividends or from expectations of capital gains via growth.
Intuition about dividends and growth:
- If a company never paid a dividend (charter forbids payouts forever), the stock would have zero value under the basic DDM framework.
- Growth companies often reinvest earnings and only later start paying dividends; mature firms tend to have higher dividend payouts. ETFs with high dividends tend to include mature companies; growth stock portfolios often have little/no current dividends.
Basic formulas:
- Constant (zero growth) dividends: if D is constant forever, the stock value is a perpetuity:
 $P_0 = \frac{D}{r}$
- Constant growth (Gordon Growth Model): dividends grow at rate g every year; the first dividend after today is $D1 = D0(1+g)$ , and the price is
 P0 = \frac{D1}{r - g} = \frac{D_0(1+g)}{r - g}\qquad (r > g)
- In general, when growth is constant after some year T, use a terminal (terminal value) calculation.
Example 1: constant growth case
- Suppose D0 = 2.30, r = 13%, g = 5%.
- D1 = 2.30 × (1 + 0.05) = 2.415
- P0 = D1 / (r − g) = 2.415 / (0.13 − 0.05) = 2.415 / 0.08 ≈ 30.19
Example 2: a growth event starts after year 5 (a “non-dividend” period for years 1–4)
- Scenario: no dividends for years 1–4; in year 5, a dividend of D5 = $0.50; after year 5, dividends grow at g = 10% per year; required return r = 20%
- First, compute the terminal value at year 4 (since growth starts in year 5):
- D5 is the first dividend of the perpetuity starting in year 5, so the perpetuity value at time 4 is
 $P{4} = \frac{D{5}}{r - g} = \frac{0.50}{0.20 - 0.10} = \frac{0.50}{0.10} = 5.00$
- Present value today (P0) involves discounting the four missing years plus the terminal value at year 4:
- PV = \frac{P_4}{(1+r)^4} = \frac{5}{(1.20)^4} ≈ \frac{5}{2.0736} ≈ 2.41
- So the stock price today is about $2.41 in this setup (the rest of the cash flows are zero for years 1–4).
Example 3: non-constant growth for a finite period, then constant growth after year 3
- Cash dividends: D1 = $1, D2 = $2, D3 = $2.50; after year 3, dividends grow at g = 5% per year; r = 10%
- Compute the terminal value at year 3 (start of constant growth 4):
- D4 = D3 × (1 + g) = 2.50 × 1.05 = 2.625
- P3 = D4 / (r − g) = 2.625 / (0.10 − 0.05) = 2.625 / 0.05 = 52.50
- Present value of all cash flows up to year 3 plus the terminal value:
- PV = D1/(1+r) + D2/(1+r)^2 + D3/(1+r)^3 + P3/(1+r)^3
- Numerically: 1/1.10 + 2/1.10^2 + 2.50/1.10^3 + 52.50/1.10^3
- ≈ 0.9091 + 1.6529 + 1.8777 + 39.452 ≈ 43.89
- The approximate value is about $44, which matches the intuition for this example.
Important conceptual point about non-constant growth: the terminal value should be computed at the moment growth becomes constant; before that, you sum the explicit dividends for the non-constant period, then add the terminal value for the constant-growth period.
Quick recap of the two ways you can model stock value:
- Explicit forecast of dividends for a finite horizon, plus a terminal value for the perpetual growth after that horizon.
- Use a constant-growth model if you expect perpetual, steady growth starting now (Gordon Growth Model).
Real-world interpretation: many “growth stocks” do not pay dividends now, but investors price them based on expected future dividends or on anticipated capital gains from growing earnings; mature firms tend to pay more regular dividends.

Practical implications and exam-minded notes

Always be clear whether you are using nominal or real values and rates; ensure consistency to avoid errors in PV calculations.
For stock valuation, remember the core formulas:
- Constant dividends (perpetuity): $P_0 = \frac{D}{r}$
- Constant growth: $P0 = \frac{D1}{r - g}, \quad D1 = D0(1+g)$
- Non-constant growth with a terminal value: compute explicit dividends for the non-constant years, then compute the terminal value at the switch year using $PT = \frac{D{T+1}}{r - g}$ , and discount it back to today.
The term structure section reminds you that bond pricing and rate expectations depend on real rates, expected inflation, and risk premia; longer horizons carry more inflation and rate risk, which helps explain why yields generally rise with maturity.
The discussion on Bitcoin and other assets highlights that assets generating no cash flow (like some cryptocurrencies) can still have value due to demand, expected future cash flows elsewhere, or store-of-value characteristics—these are outside the strict DDM framework.
Ethical/philosophical notes: The DDM assumes value derives from cash flows (dividends). Some assets (like certain crypto or tech platforms) may be valued for reasons beyond current cash flows (network effects, future monetization potential, etc.). This emphasizes the importance of understanding model assumptions and limitations in practice.