Understand how rates affect equities and bonds

2.7.9 · Stock-Market › Economic Moats & Macro

Discounting Mechanism: Future Cash Flows Value Kyun Kho Dete Hain

Core Principle

Har asset apne future cash flows ke present value ke barabar hota hai. Tum jo discount rate use karte ho, woh decide karta hai ki woh flows aaj kitne worth hain.

Fundamental present value formula:

$P V = \frac{C F _{1}}{( 1 + r ) ^{1}} + \frac{C F _{2}}{( 1 + r ) ^{2}} + \frac{C F _{3}}{( 1 + r ) ^{3}} + \dots$

Jahan:

$P V$ = present value (aaj ki value abhi)
$C F_{t}$ = year $t$ mein cash flow
$r$ = discount rate (interest rate + risk premium)

YEH formula KYUN? Agar tum $r$ % risk-free kama sakte ho, toh agla saal $100 milna aaj $\frac{100}{1 + r}$ rakhne ke barabar hai—kyunki tum woh chhoti amount rate $r$ par invest kar sakte ho aur woh $100 tak badh jaayegi.

First principles se derivation:

Tumhare paas aaj $1 hai. Ek saal mein rate $r$ par yeh ban jaata hai: $1 \times (1 + r)$
Isko ulta karo: agla saal ka $1 aaj $\frac{1}{1 + r}$ worth hai
Year $t$ mein $1 ke liye $t$ compounding periods chahiye, isliye: $\frac{1}{( 1 + r ) ^{t}}$
Koi bhi cash flow $C F_{t}$ linearly scale hota hai: $\frac{C F _{t}}{( 1 + r ) ^{t}}$

Modified Duration: $D_{mod} = - \frac{1}{P} \frac{d P}{d r}$

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

KYUN? PV formula ka $r$ ke saath derivative lene par pata chalta hai ki door ke cash flows ke denominator mein bade $(1 + r)^{t}$ terms hote hain, isliye jab $r$ badhta hai toh woh zyada value kho dete hain. Ek 30-year bond ki duration ~20-30 saal hoti hai, jabki 1-year bill ki ~1 saal.

Bonds Rate Changes par Kaise React Karte Hain

Worked Example 1: 10-Year Treasury Bond

Setup: Tumhare paas ek 10-year Treasury bond hai, face value $1{,}000, \text{coupon rate} 4% \text{ (pays }$40/year). Market rates 4% hain.

Initial price (at par): $P = \frac{40}{1.04} + \frac{40}{1.0 4 ^{2}} + \dots + \frac{40}{1.0 4 ^{10}} + \frac{1000}{1.0 4 ^{10}}$

Coupons ke liye annuity formula plus final principal use karke: $P = 40 \times \frac{1 - ( 1.04 ) ^{- 10}}{0.04} + \frac{1000}{1.0 4 ^{10}} = 40 \times 8.111 + 675.56 = $1, 000$

Yeh step kyun? Jab market rate = coupon rate, toh bond par (face value) par trade karta hai.

Scenario: Market rates 5% tak jump karte hain.

New price: $P_{new} = 40 \times \frac{1 - ( 1.05 ) ^{- 10}}{0.05} + \frac{1000}{1.0 5 ^{10}}$ $P_{new} = 40 \times 7.722 + 613.91 = 308.88 + 613.91 = $922.79$

Impact: Tumhara bond $77.21 (7.7% decline) kho gaya kyunki naaye buyers 5% kahin aur se pa sakte hain. Tum 4% coupons mein locked ho, isliye tumhara bond compete karne ke liye discount par trade karna padega.

Math kyun kaam karta hai: Zyada discount rate (5% vs 4%) har future $40 \text{ payment aur }$1,000 principal ko aaj kam worth banata hai. 10-year maturity ka matlab hai ki yeh kai periods mein compound hota hai.

Worked Example 2: Short vs. Long Duration

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

$P_{short} = \frac{40}{1.05} + \frac{1040}{1.0 5 ^{2}} = 38.10 + 943.31 = $981.41$

Loss: $18.59 (1.86%)

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

$P_{long} = 40 \times \frac{1 - ( 1.05 ) ^{- 30}}{0.05} + \frac{1000}{1.0 5 ^{30}} = 615.08 + 231.38 = $846.46$

Loss: $153.54 (15.3%!)

Yeh itna bada difference kyun? 30-year bond ke cash flows bahut door future mein jaate hain. Year 25-30 mein har saal ka $40 payment denominator mein $(1.05)^{25 - 30}$ se crush ho jaata hai. 2-year bond apni zyaadatar value jaldi wapas le leta hai, isliye rates kam matter karte hain.

Figure — Understand how rates affect equities and bonds

Equities Rate Changes par Kaise React Karte Hain

Gordon Growth Model (ek perpetuity ke liye jo rate $g$ par grow karti hai):

$P_{0} = \frac{D _{1}}{r - g}$

Jahan:

$P_{0}$ = aaj stock price
$D_{1}$ = agla saal ka dividend (ya free cash flow)
$r$ = required return (risk-free rate + equity risk premium)
$g$ = perpetual growth rate

Derivation:

Infinite sum se shuru karo: $P_{0} = \frac{D _{1}}{1 + r} + \frac{D _{1} ( 1 + g )}{( 1 + r ) ^{2}} + \frac{D _{1} ( 1 + g ) ^{2}}{( 1 + r ) ^{3}} + \dots$
$D_{1}$ factor out karo: $P_{0} = D_{1} [\frac{1}{1 + r} + \frac{1 + g}{( 1 + r ) ^{2}} + \frac{( 1 + g ) ^{2}}{( 1 + r ) ^{3}} + \dots]$
Yeh ek geometric series hai jisme first term $a = \frac{1}{1 + r}$ aur ratio $q = \frac{1 + g}{1 + r}$ hai
Infinite geometric series ka sum: $\frac{a}{1 - q} = \frac{\frac{1}{1 + r}}{1 - \frac{1 + g}{1 + r}} = \frac{1}{( 1 + r ) - ( 1 + g )} = \frac{1}{r - g}$
Is tarah: $P_{0} = \frac{D _{1}}{r - g}$

Worked Example 3: Tech Stock Revaluation

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

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

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

$P_{0} = \frac{10}{0.08 - 0.15}$

Ruko—yeh negative hai! Jab $g > r$ , toh formula toot jaata hai (company "discount rate se tez grow kar rahi hai" ka matlab infinite value hai). High-growth firms ke liye, multi-stage DCF use karo ya samjho ki growth eventually slow hogi.

Realistic version: Maano 5 saal ke liye 15% growth, phir hamesha ke liye 4%.

Phase 1 (high growth, years 1-5): $P V_{1} = \frac{10}{1.08} + \frac{10 \times 1.15}{1.0 8 ^{2}} + \frac{10 \times 1.1 5 ^{2}}{1.0 8 ^{3}} + \frac{10 \times 1.1 5 ^{3}}{1.0 8 ^{4}} + \frac{10 \times 1.1 5 ^{4}}{1.0 8 ^{5}}$ $P V_{1} = 9.26 + 9.85 + 10.48 + 11.15 + 11.86 = $52.60$

Yeh steps kyun? Har saal ka FCF 15% grow karta hai, phir hum 8% par wapas discount karte hain. Year 2 FCF = $10 \times 1.15 = 11.5$ , year 3 = $10 \times 1.1 5^{2}$ , wagera.

Phase 2 (terminal value at year 5): Year 6 FCF = $10 \times 1.1 5^{5} \times 1.04 = 20.92$

$T V = \frac{20.92}{0.08 - 0.04} = \frac{20.92}{0.04} = $523$

Present value of terminal: $\frac{523}{1.0 8 ^{5}} = 356$

Total: $52.60 + 356 = \$ 408.60/share$

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

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

Phase 1: $P V_{1} = \frac{10}{1.11} + \frac{11.5}{1.1 1 ^{2}} + \frac{13.23}{1.1 1 ^{3}} + \frac{15.21}{1.1 1 ^{4}} + \frac{17.49}{1.1 1 ^{5}} = $48.23$

Phase 2: Year 6 FCF = $20.92$ , 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 $408.60 → $225.70 (44.8% decline!) gir gaya jab rates 3% badhe.

Yeh kyun hota hai:

Zyada discount rate seedha har cash flow ka PV ghata deta hai
Terminal value mein denominator $(r - g)$ 0.04 → 0.07 ho gaya, terminal value ~40% cut ho gaya
Growth stocks ki zyaadatar value door future mein hoti hai, isliye woh zyada discount rates se crush ho jaate hain (high duration)

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

2022 Case Study: Fed ne rates 0% → 5% badhaye. Nasdaq (growth-heavy) 33% gira, jabki Dow (value-heavy) 9% gira. Kyun? Growth stocks ki implicit duration 2-3× zyaada hoti hai.

Opportunity Cost Channel

Math: Equity Risk Premium (ERP) woh extra return hai jo stocks ko bonds ke upar provide karna padta hai.

$ERP = Expected Stock Return - Risk-Free Rate$

Historically, ERP average ~5-6% hota hai. Jab risk-free rates spike karte hain, ya toh stock prices girni chahiye (lowering $P_{0}$ , expected return badhate hue) ya earnings dramatically accelerate honi chahiye.

Investors kyun shift karte hain: Risk-adjusted returns matter karte hain. Agar bonds 5% dete hain zero volatility ke saath aur stocks 7% dete hain 20% volatility ke saath, toh stocks ka Sharpe ratio deteriorate ho jaata hai. Rational investors kam stock allocation demand karte hain.

Common Feedback Loops

Isse reflexivity banti hai—rate changes valuations ko affect karte hain, jo real economy ko affect karte hain, jo earnings ko affect karte hain, jo valuation change ko validate karte hain.

Kyun sahi lagta hai: 30+ saaloon mein, equities historically bonds ko 4-6% annually outperform karte hain. Diversified portfolios ke liye sach.

Galti: Entry valuation ab bhi bahut matter karta hai. Agar tum stocks 30× earnings par khareedte ho jab rates 1% hain, aur rates 5% tak badhte hain (fair P/E → 15×), toh tum 50% drawdown face karoge. Chahe stocks eventually recover ho jaayein, tumhare returns agale decade ke liye near-zero ho sakte hain.

Stelman defense: "Time in the market beats timing the market." Dollar-cost averaging ke liye valid. Peak valuations par lump-sum investing ke liye invalid.

Fix: Maano ki rate regimes forward returns ke liye starting point set karte hain. Low rates + high valuations → agle 10 saaloon ke liye kam expected returns, chahe year 30+ theek lage. Allocation accordingly adjust karo (value, international, ya bonds ki taraf tilt karo jab domestic equities expensive hoon).

Kyun sahi lagta hai: Bonds ek bankrupt stock ki tarah zero nahi jaayenge. Agar tum hold karo toh maturity par par value milti hai.

Galti: Long-duration bonds rapid rate-hiking cycle mein 20-40% kho sakte hain (2022: 20-year Treasuries 30% gire). Agar tumhe maturity se pehle liquidity chahiye, toh tum loss mein sell kar rahe ho. Saath hi, opportunity cost—agar rates 2% → 6% badhte hain, toh tum decades ke liye 2% coupons mein locked ho jabki naaye bonds 6% de rahe hain.

Fix: Bond duration ko apne time horizon se match karo. 2 saal mein cash chahiye? 2-year bonds khareeedo (minimal rate risk). 20 saal ke liye invest kar rahe ho? Rate-hiking cycles ke dauran bond ladders ya short-duration funds use karo, duration tabhi extend karo jab rates peak karein.

Connections

Present-Value-and-Time-Value-of-Money – Saare DCF calculations ki foundation
Equity-Risk-Premium-and-Expected-Returns – Stocks ko rate risk ke liye kyun compensate karna padta hai
Fed-Policy-and-the-Business-Cycle – Rate changes pehli jagah mein kya drive karta hai
Duration-and-Convexity – Bond price sensitivity mein deep dive
Gordon-Growth-Model-and-Terminal-Value – Equity valuation mechanics
Value-vs-Growth-Investing – Rate regimes alag styles ko kyun favor karte hain
Portfolio-ConstructionAcross-Rate-Environments – Rates shift hone par kaise allocate karein

Active Recall Flashcards

#flashcards/stock-market

Interest rates aur bond prices ke beech mathematical relationship kya hai? :: Bond prices interest rates ke saath inversely chalte hain. Jab badhte hain, fixed coupon payments aur principal ka present value girta hai kyunki denominator mein discount rate $(1 + r)^{t}$ badh jaata hai. Formula: $P = \sum \frac{C F _{t}}{( 1 + r ) ^{t}}$ .

Rates badhne par long-duration bonds short-duration bonds se zyaada kyun girte hain?

Long-duration bonds ke cash flows door future mein hote hain (years 20-30). Yeh door ke cash flows

(1 + r)^{20 - 30}

se divide hote hain, isliye

r

mein thodi si bhi badhaav bahut bade denominator mein compound ho jaati hai, present value crush ho jaata hai. Short bonds apna principal jaldi wapas paate hain, rate sensitivity limit ho jaati hai.

Equity valuation ke liye Gordon Growth Model kya hai? :: $P_{0} = \frac{D _{1}}{r - g}$ , jahan $D_{1}$ agla saal ka dividend/FCF hai, $r$ required return hai, $g$ perpetual growth rate hai. Rate $r$ par discounted growing cash flows ki geometric series sum karke derive kiya jaata hai.

Interest rates badhne par growth stocks value stocks se zyaada kyun girte hain?

Growth stocks ke zyaadatar cash flows 10-30 saal baad aate hain (high duration ~15-25 saal). Zyada discount rates door ke cash flows crush kar dete hain. Value stocks cash abhi generate karte hain (duration ~5-8 saal), isliye near-term cash flows discounting se kam affect hote hain.

Equity Risk Premium (ERP) kya hai aur yeh rate sensitivity ke liye kyun matter karta hai?

ERP = Expected Stock Return - Risk-Free Rate. Jab risk-free rates badhte hain (e.g., bonds 5% vs 1% pay karte hain), toh stocks ko apne added risk compensate karne ke liye proportionally zyada returns offer karne padte hain. Agar earnings nahi badhti, toh stock prices girni chahiye expected returns boost karne ke liye.

Fed ka rates badhana valuation compression spiral kaise trigger karta hai?

(1) Rates up → discount rates up → DCF se stock valuations girte hain. (2) Girte stocks → wealth effect → consumer spending kam → earnings girte hain. (3) Kam earnings → valuations aur girte hain. (4) Credit spreads widen → borrowing costs badhte hain → growth slow hoti hai. Reflexive loop.

"Stocks hamesha bonds beat karte hain, isliye rates matter nahi karte" mein galti kya hai? :: Entry valuation matter karta hai. Agar tum stocks 30× P/E par khareedte ho jab rates low hain, aur rates badhte hain (fair P/E → 15×), toh tum 50% drawdown face karoge. Chahe stocks 30 saaloon mein outperform karein, tumhare agale 10-saal ke returns multiple compression ki wajah se near-zero ho sakte hain.

Bond investing mein duration-matching kyun important hai?

Agar tumhe 2 saal mein cash chahiye, toh 20-year bond tumhe massive rate risk expose karta hai (rates spike hone par 20-40% kho sakte ho). Tumhe loss mein sell karna pad sakta hai. Bond maturity ko apne time horizon se match karo—near-term needs ke liye short bonds, long bonds tabhi jab maturity tak hold kar sako.

Recall Feynman Technique: 12-Saal ke Bachche ko Explain Karo

Socho tumhare paas ek lemonade stand hai jo har garmiyoon mein $10 banata hai. Tumhara dost poochta hai, "Tumhara lemonade stand kitne ka hai?" Well, yeh depend karta hai ki tum apne paise ke saath aur kya kar sakte ho!

Agar bank 1% interest deta hai, tumhara lemonade stand (hamesha ke liye $10/saal banata hua) bahut ZYAADA worth hai—shayad \ $1,000! Kyun? Kyunki bank boring hai, isliye tumhara \$ 10 lemonade ka paisa comparison mein exciting lagta hai.

Lekin agar bank suddenly 10% interest dene lagta hai, toh ab tumhara lemonade stand special nahi raha. $10/saal banane ke liye tumhe bank mein $100 chahiye hoga. Toh tumhara lemonade stand ab sirf $100 worth hai. Same \$10/saal, lekin 10× kam worth hai kyunki alternative behtar ho gaya.

Stocks aur bonds ke saath yahi hota hai. Jab interest rates badhte hain, tumhara har future dollar aaj kam worth hota hai, kyunki tum bank mein paisa rakh ke risk-free zyaada kama sakte ho. Jab rates girte hain, toh woh future dollars zyaada worth hote hain kyunki bank phir boring ho jaata hai.

Bonds aur bhi simple hain: woh fixed payments promise karte hain. Agar tumne ek bond khareeeda jo $5/saal pay karta hai aur rates 10% tak jump karti hain, toh koi tumhara bond nahi chaahega—woh naaye bonds khareedenge jo $10 pay karte hain. Toh tumhe apna bond half-price par bechna padega use attractive banane ke liye. Isliye bond prices rates badhne par girte hain.

Yaad rakho: "High rates, light wallets" (assets tumhare portfolio mein kam worth hote hain).