Financial Markets & Real Interest Rates

The Risk Premium

coreExam · highTA · PS2-Q1, TA4-Q1

Firms borrow at a rate that includes a risk premium x above the risk-free rate i. With default probability p and no recovery, the no-arbitrage condition (1+i) = (1−p)(1+i+x) gives x = (1+i)·p/(1−p). The IS relation with risk premium becomes Y depending on r+x: a shock to x (credit crunch) shifts IS left even at unchanged CB rate.

Derivation

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Risk-Neutral Pricing

A risky one-period bond promises $(1 + i + x)$ with probability $(1 - p)$ and zero with probability $p$ . Indifference with a risk-free bond returning $(1+i)$ requires equal expected payoffs:

(1 + i) = (1 - p)(1 + i + x)

Solving:

x = \frac{(1 + i) \cdot p}{1 - p}

For small $p$ , $x \approx p$ .

The Full Borrowing Rate

Firms borrow at $r + x$ , not just $r$ . The IS relation becomes $Y = C(Y-T) + I(Y, r+x) + G$ . A rise in $x$ has exactly the same output effect as a rise in $r$ .

Risk Aversion Wedge

Observed spreads exceed the risk-neutral formula because investors dislike variance. The extra wedge is time-varying and spikes during crises (flight-to-quality).

Policy Implications

| Shock | Direction of $x$ | CB response | |-------|------------------|-------------| | Banking crisis | $\uparrow$ sharply | Cut $i^T$ ; QE if at ZLB | | Recovery | $\downarrow$ | Let $i^T$ drift up | | Sovereign crisis (EU) | $\uparrow$ on periphery | OMT (Draghi) to cap $x$ |

Historical Episodes

2008 GFC: corporate spreads blew out → credit supply collapsed → Fed QE1/2/3 aimed directly at $x$ .
2011–12 euro crisis: Italian/Spanish sovereign $x$ surged → ECB OMT announcement (2012) compressed them.
COVID-19: rapid spread widening in March 2020 → Fed emergency facilities targeting corporate, muni, and consumer-credit $x$ .

Worked Example

i = 2%, initial p = 0.01. (a) Compute fair-value x. (b) A recession raises p to 0.08. Find new x. (c) The CB wants to keep r+x constant. By how much must it cut iᵀ?

(a) x = (1.02)(0.01)/(0.99) ≈ 0.0103 = 1.03 pp.
(b) x = (1.02)(0.08)/(0.92) ≈ 0.0887 = 8.87 pp. Δx ≈ +7.8 pp.
(c) To keep r + x constant, Δr ≈ −Δx ≈ −7.8 pp. With πe unchanged, ΔiᵀΔ ≈ −7.8 pp — an enormous cut, usually infeasible before ZLB.

Fair-value x rises from 1% to 8.9%. Offsetting fully would require a ~7.8 pp rate cut — typically impossible near zero, which is why CBs turn to QE during crises.

Common Mistakes

—Using x ≈ p without the (1+i)/(1-p) factor — fine as an approximation for small p, not for large p.
—Applying x only to investment instead of recognising that r + x is the firm's real borrowing cost.
—Ignoring risk aversion: actual x exceeds the fair-value formula, especially in stress.
—Forgetting that a rise in x has the same output effect as a rise in r — both shift IS left.

Exam Cues

→Formula: x = (1+i)p/(1−p). For small p, x ≈ p.
→IS with risk premium: Y depends on r + x. Appears in TA2 and Mock questions.
→Credit crisis: ↑x shifts IS left even at unchanged iᵀ. CB must cut by Δx to offset.
→Euro crisis 2011–12: sovereign risk premia rose sharply — OMT programme (Draghi) brought them down.