Understand credit ratings and default risk

2.1.6 · Stock-Market › Equity & Fixed Income

KYA measure kiya ja raha hai?

YEH KYU exist karta hai: Investors personally har company ko audit nahi kar sakte. Rating agencies ek baar analysis karti hain aur ek comparable grade publish karti hain, jisse poore market ka information cost kam hota hai.

Rating ladder (letters ka MATLAB kya hai)

KAISE padhein: Higher letter = lower default probability = lower spread = lower yield. "D" matlab already default mein hai.

Default risk price mein KAISE banta hai (spread derive karo)

Hum kyu derive karte hain ki ek risky bond zyada pay karta hai, first principles se — koi bhi formula dump nahi.

Setup. Aap ya toh:

1. Ek risk-free bond khareed sakte ho jo $r_f$ return karta hai certain taur par, YA
2. Ek risky bond khareed sakte ho jo yield $y$ promise karta hai, lekin default probability $p$ ke saath (us period ke liye), aur agar default ho toh aap fraction $R$ recover karte ho (yeh recovery rate hai) jo aapka tha.

Step 1 — Aapke liye bother karne ke liye kya true hona chahiye? Yeh step kyu? Ek rational investor tabhi default risk leta hai jab expected return kam se kam safe return match kare. Toh expected payoffs ko equal set karo.

$1investkaro.Risk-freedetahai$(1+r_f)$.Riskybond:probability$(1-p)$sefull$(1+y)$miltahai;probability$p$sesirf$R(1+y)$ recover hota hai.

Step 2 — Risky bond ka expected payoff likho. $\mathbb{E}[\text{payoff}]=(1-p)(1+y)+pR(1+y)=(1+y)[(1-p)+pR]$ Yeh step kyu? Hum har outcome ko uski probability se weight karte hain — yahi expectation ki definition hai.

Step 3 — Expectations equal set karo (no-arbitrage / risk-neutral pricing). $(1+y)[1-p(1-R)]=1+r_f$ Yeh step kyu? Agar risky bond ki expected value zyada hoti, toh sab usse khareedte, uska price badhta, uska yield girta jab tak equality na aa jaye. Markets is balance ki taraf push karte hain.

Step 4 — Yield $y$ solve karo. $1+y=\frac{1+r_f}{1-p(1-R)}\Rightarrow y=\frac{1+r_f}{1-p(1-R)}-1$

Step 5 — Spread read off karo. $\boxed{\text{spread}=y-r_f=\frac{(1+r_f)p(1-R)}{1-p(1-R)}}$ Yeh step kyu? Safe rate subtract karo. Spread badhta hai ==default probability $p$== ke saath aur loss given default $(1-R)$ ke saath. Agar $p=0$, spread $=0$ — ek certain bond risk-free jaisa price hota hai.

Worked examples

Common mistakes (Steel-manned)

Active recall

Connections

• Bond Yields and Prices — spread risk-free yield ke upar add hota hai.
• Yield Curve and Interest Rates — $r_f$ government yield curve se aata hai.
• Duration and Interest-Rate Risk — default risk se alag risk hai.
• Diversification and Portfolio Risk — default risk ko issuers mein spread karna.
• Risk and Return Tradeoff — spread risk bear karne ki price hai.

Concept Map