Understand how rates affect equities and bonds

The Discounting Mechanism: Why Future Cash Flows Lose Value

The Core Principle

Every asset is worth the present value of its future cash flows. The discount rate you use determines what those flows are worth today.

The fundamental present value formula:

$PV = \frac{CF_1}{(1+r)^1} + \frac{CF_2}{(1+r)^2} + \frac{CF_3}{(1+r)^3} + \cdots$

Where:

• $PV$ = present value (what it's worth now)
• $CF_t$ = cash flow in year $t$
• $r$ = discount rate (interest rate + risk premium)

WHY this formula? If you can earn $r$% risk-free, receiving $100 next year is equivalent to having $\frac{100}{1+r}$ today—because you could invest that smaller amount at rate $r$and it would grow to$100.

Derivation from first principles:

1. You have $1 today. In one year at rate $r$, it becomes: $1 \times (1+r)$
2. Reverse this: $1 next year is worth $\frac{1}{1+r}$ today
3. $1 in year $t$ requires$compounding periods, so:$\frac{1}{(1+r)^t}$
4. Any cash flow $CF_t$ scales linearly: $\frac{CF_t}{(1+r)^t}$

Modified Duration: $D_{\text{mod}} = -\frac{1}{P} \frac{dP}{dr}$

Price change approximation: $\Delta P \approx -D_{\text{mod}} \times P \times \Delta r$

WHY? Taking the derivative of the PV formula with respect to $r$ shows that distant cash flows have larger $(1+r)^t$ terms in the denominator, so they lose more value when $r$ increases. A 30-year bond has ~20-30 years of duration, while a 1-year bill has ~1 year.

How Bonds React to Rate Changes

Worked Example 1: 10-Year Treasury Bond

Setup: You own a 10-year Treasury bond, face value $1{,}000, \text{coupon rate} 4% \text{ (pays }$40/year). Market rates are 4%.

Initial price (at par): $P = \frac{40}{1.04} + \frac{40}{1.04^2} + \cdots + \frac{40}{1.04^{10}} + \frac{1000}{1.04^{10}}$

Using the annuity formula for coupons plus final principal: $P = 40 \times \frac{1 - (1.04)^{-10}}{0.04} + \frac{1000}{1.04^{10}} = 40 \times 8.111 + 675.56 = \$1,000$

Why this step? When market rate = coupon rate, the bond trades at par (face value).

Scenario: Market rates jump to 5%.

New price: $P_{\text{new}} = 40 \times \frac{1 - (1.05)^{-10}}{0.05} + \frac{1000}{1.05^{10}}$ $P_{\text{new}} = 40 \times 7.722 + 613.91 = 308.88 + 613.91 = \$922.79$

Impact: Your bond lost $77.21 (7.7% decline) because new buyers can get 5% elsewhere. You're locked into4% coupons, so your bond must trade at a discount to compete.

Why the math works: The higher discount rate (5% vs 4%) makes every future $40 \text{ payment and the }$1,000 principal worth less today. The10-year maturity means this compounds across many periods.

Worked Example 2: Short vs. Long Duration

Short-duration bond: 2-year bond, 4% coupon, rates rise4% → 5%.

$P_{\text{short}} = \frac{40}{1.05} + \frac{1040}{1.05^2} = 38.10 + 943.31 = \$981.41$

Loss: $18.59 (1.86%)

Long-duration bond: 30-year bond, 4% coupon, same rate rise.

$P_{\text{long}} = 40 \times \frac{1-(1.05)^{-30}}{0.05} + \frac{1000}{1.05^{30}} = 615.08 + 231.38 = \$846.46$

Loss: $153.54 (15.3%!)

Why this massive difference? The30-year bond has cash flows stretching far into the future. Each year's $40 payment in year 25-30 is crushed by $(1.05)^{25-30}$ in the denominator. The 2-year bond gets most of its value back quickly, so rates matter less.

How Equities React to Rate Changes

Gordon Growth Model (for a perpetuity growing at rate $g$):

$P_0 = \frac{D_1}{r - g}$

Where:

• $P_0$ = stock price today
• $D_1$ = next year's dividend (or free cash flow)
• $r$ = required return (risk-free rate + equity risk premium)
• $g$ = perpetual growth rate

Derivation:

1. Start with infinite sum: $P_0 = \frac{D_1}{1+r} + \frac{D_1(1+g)}{(1+r)^2} + \frac{D_1(1+g)^2}{(1+r)^3} + \cdots$
2. Factor out $D_1$: $P_0 = D_1 \left[ \frac{1}{1+r} + \frac{1+g}{(1+r)^2} + \frac{(1+g)^2}{(1+r)^3} + \cdots \right]$
3. This is a geometric series with first term $a = \frac{1}{1+r}$ and ratio $q = \frac{1+g}{1+r}$
4. Sum of infinite geometric series: $\frac{a}{1-q} = \frac{\frac{1}{1+r}}{1 - \frac{1+g}{1+r}} = \frac{1}{(1+r) - (1+g)} = \frac{1}{r-g}$
5. Thus: $P_0 = \frac{D_1}{r-g}$

Worked Example 3: Tech Stock Revaluation

Company: High-growth SaaS company. Expected FCF next year = $10/share, growth = 15%, equity risk premium = 6%.

Scenario A: Low-rate environment (risk-free = 2%)

Required return: $r = 2\% + 6\% = 8\%$

$P_0 = \frac{10}{0.08 - 0.15}$

Wait—this is negative! When $g > r$, the formula breaks (company "growing faster than discount rate" implies infinite value). For high-growth firms, use multi-stage DCF or realize growth must eventually slow.

Realistic version: Assume 15% growth for 5 years, then 4% perpetually.

Phase 1 (high growth, years 1-5): $PV_1 = \frac{10}{1.08} + \frac{10 \times 1.15}{1.08^2} + \frac{10 \times 1.15^2}{1.08^3} + \frac{10 \times 1.15^3}{1.08^4} + \frac{10 \times 1.15^4}{1.08^5}$ $PV_1 = 9.26 + 9.85 + 10.48 + 11.15 + 11.86 = \$52.60$

Why these steps? Each year's FCF grows at 15%, then we discount back at 8%. Year 2 FCF = $10 \times 1.15 = 11.5$, year 3 = $10 \times 1.15^2$, etc.

Phase 2 (terminal value at year 5): Year 6FCF = $10 \times 1.15^5 \times 1.04 = 20.92$

$TV = \frac{20.92}{0.08 - 0.04} = \frac{20.92}{0.04} = \$523$

Present value of terminal: $\frac{523}{1.08^5} = 356$

Total: 52.60 + 356 = \408.60/share$

Scenario B: High-rate environment (risk-free = 5%)

Required return: $r = 5\% + 6\% = 11\%$

Phase 1: $PV_1 = \frac{10}{1.11} + \frac{11.5}{1.11^2} + \frac{13.23}{1.11^3} + \frac{15.21}{1.11^4} + \frac{17.49}{1.11^5} = \$48.23$

Phase 2: Year 6FCF = $20.92$, TV = \frac{20.92}{0.11-0.04} = \298.86$

PV of TV = \frac{298.86}{1.11^5} = \177.47$

Total: 48.23 + 177.47 = \225.70/share$

Impact: Stock fell from $408.60 → $225.70 (44.8% decline!) when rates rose 3%.

Why this happens:

1. Higher discount rate directly lowers PV of every cash flow
2. The denominator $(r-g)$ in terminal value shrank from 0.04 → 0.07, cutting terminal value by ~40%
3. Growth stocks have most value in distant future, so they're crushed by higher discount rates (high duration)

Growth stocks (tech, biotech): Most value in years 10-30. Duration ~15-25 years. Rate rise → severe impact.

2022 Case Study: Fed raised rates from 0% → 5%. Nasdaq (growth-heavy) fell 33%, while Dow (value-heavy) fell 9%. Why? Growth stocks have implicit duration2-3× longer.

The Opportunity Cost Channel

The math: Equity Risk Premium (ERP) is the extra return stocks must provide over bonds.

$\text{ERP} = \text{Expected Stock Return} - \text{Risk-Free Rate}$

Historically, ERP averages ~5-6%. When risk-free rates spike, either stock prices must fall (lowering $P_0$, raising expected return) or earnings must accelerate dramatically.

Why investors shift: Risk-adjusted returns matter. If bonds pay 5% with zero volatility and stocks pay 7% with 20% volatility, the Sharpe ratio for stocks deteriorates. Rational investors demand a lower stock allocation.

Common Feedback Loops

This creates reflexivity—rate changes affect valuations, which affect the real economy, which affects earnings, which validates the valuation change.

Why it feels right: Over 30+ years, equities historically outperform bonds by 4-6% annually. True for diversified portfolios.

The error: Entry valuation still matters enormously. If you buy stocks at 30× earnings when rates are 1%, and rates rise to 5% (fair P/E → 15×), you'll endure a 50% drawdown. Even if stocks eventually recover, your returns for the next decade could be near-zero.

Stelman defense: "Time in the market beats timing the market." Valid for dollar-cost averaging. Invalid for lump-sum investing at peak valuations.

The fix: Acknowledge that rate regimes set the starting point for forward returns. Low rates + high valuations → lower expected returns for the next 10 years, even if year 30+ looks fine. Adjust allocation accordingly (tilt toward value, international, or bonds when domestic equities are expensive).

Why it feels right: Bonds won't go to zero like a bankrupt stock. You get par value at maturity if you hold.

The error: Long-duration bonds can lose 20-40% in a rapid rate-hiking cycle (2022: 20-year Treasuries fell 30%). If you need liquidity before maturity, you're selling at a loss. Also, opportunity cost—if rates rise from 2% → 6%, you're locked into 2% coupons for decades while new bonds pay 6%.

The fix: Match bond duration to your time horizon. Need cash in 2 years? Buy 2-year bonds (minimal rate risk). Investing for 20 years? Use bond ladders or short-duration funds during rate-hiking cycles, extend duration only when rates peak.

Connections

• Present-Value-and-Time-Value-of-Money – The foundation for all DCF calculations
• Equity-Risk-Premium-and-Expected-Returns – Why stocks must compensate for rate risk
• Fed-Policy-and-the-Business-Cycle – What drives rate changes in the first place
• Duration-and-Convexity – Deep dive into bond price sensitivity
• Gordon-Growth-Model-and-Terminal-Value – Equity valuation mechanics
• Value-vs-Growth-Investing – Why rate regimes favor different styles
• Portfolio-ConstructionAcross-Rate-Environments – How to allocate when rates shift

Active Recall Flashcards

#flashcards/stock-market

What is the mathematical relationship between interest rates and bond prices? :: Bond prices move inversely to interest rates. When rise, the present value of fixed coupon payments and principal declines because the discount rate $(1+r)^t$ in the denominator increases. Formula: $P = \sum\frac{CF_t}{(1+r)^t}$.

What is the Gordon Growth Model for equity valuation? :: $P_0 = \frac{D_1}{r-g}$, where $D_1$ is next year's dividend/FCF, $r$ is required return, $g$ is perpetual growth rate. Derived from suming a geometric series of growing cash flows discounted at rate $r$.

What is the mistake in "stocks always beat bonds, so rates don't matter"? :: Entry valuation matters. If you buy stocks at 30× P/E when rates are low, and rates rise (fair P/E → 15×), you'll face a 50% drawdown. Even though stocks may outperform over 30 years, your next 10-year returns could be near-zero due to multiple compression.

If the bank pays 1% interest, your lemonade stand (making $10/year forever) is worth a LOT—maybe \1,000! Why? Because the bank is boring, so your \10 lemonade money is exciting by comparison.

But if the bank suddenly pays 10% interest, now your lemonade stand isn't special anymore. You'd need $100 in the bank to make \10/year. So your lemonade stand is only worth \100 now. Same \backslash$10/year, but it's worth 10× less because the alternative got better.

That's what happens to stocks and bonds. When interest rates go up, every future dollar you'll earn is worth less today, because you could just put money in the bank and earn more risk-free. When rates go down, those future dollars are worth more because the bank is boring again.

Bonds are even simpler: they promise fixed payments. If you bought a bond paying $5/year and rates jump to 10%, nobody wants your bond—they'll buy new ones paying$10. So you have to sell yours for half-price to make it attractive. That's why bond prices fall when rates rise.

Remember: "High rates, light walets" (assets are worth less in your portfolio).

Concept Map

Hinglish (regional understanding)