Monetary and Fiscal Policy: Safeguarding Stability and Trust: formula sheet

Key formulas

Dual mandate (US Federal Reserve)

Mandate = maximum employment + stable prices

Set by Congress. The Fed's longer-run inflation goal is 2% (PCE). Moderate long-term interest rates are also named in the statute.

Inflation targeting

Policy rate adjusted so that expected inflation → announced target

A framework, not a legal mandate. The target is typically a number such as 2%, and the bank is held accountable for it.

Taylor-type rule (illustration)

i = r* + π + 0.5 × (π − π*) + 0.5 × (output gap)

A guide to how a bank may respond to inflation and output gaps. It is a rule of thumb, not a mandate. Coefficients vary by version.

Real interest rate

Real rate ≈ nominal rate − expected inflation

Policy works through real rates. Anchored expectations make this easier to steer.

Independence and credibility link

Independence + clear goal + transparency → credibility → anchored expectations

This is the logic examiners test, not a numeric formula.

Interest rate corridor

Deposit facility rate ≤ overnight market rate ≤ Lending facility rate

Standing facilities set the floor and ceiling. The policy rate usually sits inside the corridor, often near the middle.

Corridor width

Width = Lending facility rate − Deposit facility rate

A narrower corridor gives tighter control of the overnight rate. Quote the width in basis points (1 bp = 0.01%).

Required reserves

Required reserves = Reserve ratio × Reservable deposits

Excess reserves = Actual reserves − Required reserves.

Simple deposit multiplier (textbook)

Multiplier = 1 ÷ Reserve ratio

An upper-bound illustration only. It assumes no cash leakage and no excess reserves, so real-world expansion is smaller.

OMO direction rule

Buy securities / repo → reserves up → short rate down. Sell securities / reverse repo → reserves down → short rate up

Reverse repo terms are from the central bank's side when it absorbs cash.

Long yield decomposition

Long-term yield ≈ average expected future short rates + term premium

QE mainly lowers the term premium. Forward guidance mainly lowers expected short rates.

Bond price change from yield move

ΔP ≈ −D × Δy × P (D is modified duration)

Use it to size mark-to-market effects of QE or QT on bond holders. Convexity matters for large moves.

Central bank balance sheet under QE

Assets (bonds) ↑ and Liabilities (bank reserves) ↑ by the same amount

QT reverses this. Reserves fall as bonds mature without reinvestment.

Cash-flow effect of negative rate on reserves

Annual cost = reserves × |rate|

For example, ₹ or USD reserves of 100 million at −0.50% cost 0.5 million a year.

Primary balance

Primary balance = Revenue − Non-interest spending

Overall balance = Primary balance − Interest payments.

Debt dynamics (approximate)

Δ(d) ≈ [(r − g) ÷ (1 + g)] × d(prev) − pb

d is debt-to-GDP, pb is primary balance as a share of GDP (surplus positive), r is the nominal interest rate on debt, g is nominal GDP growth.

Debt-stabilising primary balance

pb* ≈ [(r − g) ÷ (1 + g)] × d

Primary surplus needed to hold the debt ratio constant. If r < g, it is negative, so a small primary deficit is consistent with a stable ratio.

Debt-to-GDP ratio

d = Public debt ÷ Nominal GDP

Use nominal GDP with nominal rates, or real GDP with real rates. Do not mix them.

Yield on a risky sovereign bond (approximate)

Yield ≈ Risk-free yield + Credit spread

Spread ≈ PD × LGD as a rough guide for the default component, plus risk premia.

Government debt dynamics

Change in debt-to-GDP ≈ (r − g) × d − primary surplus ratio

r is the average real interest rate on debt, g is real GDP growth, d is the starting debt-to-GDP ratio. Primary surplus is the balance before interest. If r > g, debt rises unless the primary surplus is large enough.

Debt-stabilising primary surplus

Primary surplus ratio = (r − g) × d

Approximation. It is the primary surplus (as a share of GDP) that holds debt-to-GDP constant. A higher r raises it, so rate rises worsen sustainability under high debt.

Fisher relation

Nominal rate ≈ real rate + expected inflation

Approximation. Used to link inflation expectations to nominal funding costs for the government.

Monetary financing identity

Deficit funded by central bank = increase in central bank holdings of government debt = money created

Conceptual link. The central bank buys bonds and credits reserves, which expands its balance sheet.

Approximate bond price change

ΔP ÷ P ≈ −D_mod × Δy + ½ × C × (Δy)²

D_mod is modified duration, C is convexity, Δy is the yield change in decimals. For small moves, use only the duration term.

Approximate dollar loss on a portfolio

Loss ≈ Market value × D_mod × Δy

Gives a quick estimate of unrealised loss from a parallel rate rise. Convexity makes the true loss slightly smaller.

Net interest margin

NIM = (Interest income − Interest expense) ÷ Average earning assets

Falls when deposit costs reprice faster than asset yields.

Loss relative to capital

Unrealised loss ÷ Equity capital

Shows whether fair-value losses would threaten solvency if the assets had to be sold.

Uninsured deposit share

Uninsured deposits ÷ Total deposits

Higher share means higher run risk. It is a funding stability indicator.

Quick revision

• Central banks are judged against their mandate, usually price stability and often financial stability or employment as well.
• Raising the policy rate tightens financial conditions; cutting it eases them.
• Conventional tools include policy rates, reserve requirements, open market operations and standing facilities.
• QE means large-scale central bank asset purchases, which aims to lower longer-term yields.
• Forward guidance shapes expectations about the future path of rates.
• Unconventional tools matter most when policy rates are near the lower bound.
• Sovereign risk rises with high debt, high interest cost and weak growth, and is worse with foreign currency debt.
• Fiscal dominance means debt pressure limits the central bank's ability to fight inflation.
• Sovereign stress can pass to banks through their holdings of government bonds.
• Policy trade-offs exist: tightening to curb inflation can strain financial stability and debt servicing.
• Credibility and trust anchor inflation expectations; losing them makes inflation harder to control.

Common mistakes

• Saying the Fed has a numeric employment target Fix: Remember that the Fed sets a numeric inflation goal but assesses employment broadly, without a fixed target.
• Treating inflation targeting and a dual mandate as opposites Fix: A dual mandate is the legal set of goals. Inflation targeting is a way to pursue price stability. A bank can have both.
• Confusing repo and reverse repo direction. Fix: Ask who receives cash. If the central bank lends cash against collateral, it is a repo for the central bank and adds liquidity. If it absorbs cash, it drains liquidity.
• Saying the central bank directly sets all market interest rates. Fix: State that it sets the policy rate and steers the overnight rate with OMOs and facilities. Longer rates reflect expectations and term premia too.
• Saying QE is the same as printing cash for the government. Fix: QE is the purchase of assets in exchange for newly created bank reserves. It is a balance sheet operation, not a direct transfer.
• Treating QT as simply the opposite with an exactly equal effect. Fix: QT effects are uncertain and depend on market liquidity and reserve scarcity. Do not state fixed yield impacts.
• Confusing primary balance with overall balance. Fix: Ask whether interest is included. Primary excludes interest. Overall includes it.
• Assuming a deficit always means debt-to-GDP rises. Fix: Debt can grow while the ratio falls if nominal GDP grows faster. Always check g.
• Treating all quantitative easing as debt monetisation. Fix: Separate purpose and design. QE aimed at the inflation target with a credible exit differs from purchases made to fund deficits or cap yields under government pressure.
• Saying large deficits always cause inflation. Fix: State the conditions: monetisation, demand above capacity or lost confidence in future fiscal backing. Without these, the effect may be weak.

Exam tips

• Read the institution's name first. The Fed, the ECB and inflation-targeting banks are framed differently.
• Watch for wording that treats a framework and a mandate as the same thing.
• Expect case-like stems linking a policy action to a trade-off between inflation, growth and financial stability.
• Be ready to explain how credibility and anchored expectations lower the cost of policy.
• Reject extreme options such as independence with no accountability or price stability meaning zero inflation.
• Always decide first whether an action adds or drains reserves. Most MCQs reduce to that direction.
• Read repo wording carefully. Check whether the question uses the central bank's view or the bank's view.
• If a question gives facility rates, test every option against the floor and ceiling.