The Investment Function in Financial Services Management: formula sheet

Key formulas

Net interest margin (NIM)

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

Securities income feeds NIM. Shifting to liquid, low-yield assets lowers NIM, all else equal.

Liquidity Coverage Ratio (LCR)

LCR = Stock of HQLA ÷ Total net cash outflows over the next 30 calendar days ≥ 100%

Basel III rule. The securities portfolio is a main source of HQLA.

Approximate price change from rates

ΔP ÷ P ≈ −Modified duration × Δy

Approximation for small yield changes. Longer duration means more price risk for extra yield.

Portfolio yield

Portfolio yield = Σ (weight × yield of each holding)

Use to see the income cost of holding more liquid securities.

Hierarchy of the framework

Risk capacity ≥ Risk appetite ≥ Risk limits

Capacity is the maximum risk the institution could bear. Appetite is what the board chooses to take. Limits are the day-to-day controls set within appetite.

Limit utilisation

Utilisation = Current exposure ÷ Limit × 100%

Used to monitor limits. A trigger is often set below 100%, for example at 80%, to give early warning.

Concentration limit check

Exposure to one issuer ÷ Reference base ≤ Limit %

The reference base is whatever the policy names, such as the portfolio, Tier 1 capital or total capital. Always use the base the policy states.

Portfolio duration

D(portfolio) = Σ wi × Di

Weights are market-value shares. Compare with the policy duration range and with the duration of liabilities.

Duration gap (simple form)

Duration gap = D(assets) − (Liabilities ÷ Assets) × D(liabilities)

A positive gap means asset value falls more than liability value when rates rise, so equity falls.

Price change from duration

ΔP ÷ P ≈ −D_mod × Δy + ½ × C × (Δy)²

Use for rate-risk questions on bonds. For MBS, effective duration and convexity (which may be negative) must reflect prepayments.

HQLA stock after haircut

Adjusted value = Market value × (1 − haircut)

Level 1: 0% haircut. Level 2A: 15%. Level 2B: higher haircuts (25% for RMBS, 50% for lower-rated corporate bonds and equities).

Level 2 caps in the LCR stock

Level 2 ≤ 40% of HQLA; Level 2B ≤ 15% of HQLA

Caps are applied after haircuts, using the Basel LCR formula. Check what the question gives you.

Liquidity Coverage Ratio

LCR = Stock of HQLA ÷ Total net cash outflows over 30 days ≥ 100%

Net outflows = outflows − min(inflows, 75% of outflows).

Accounting treatment summary

HTM: amortised cost | AFS: fair value, gains/losses in OCI | Trading: fair value, gains/losses in profit

Reclassification or sale of HTM can taint the category.

Portfolio duration

D_p = Σ wᵢ × Dᵢ

wᵢ are market-value weights. Duration of a portfolio is the weighted average of bond durations.

Price change from duration

ΔP ÷ P ≈ −D_mod × Δy + ½ × C × (Δy)²

D_mod is modified duration and C is convexity. Duration alone is a first-order estimate.

Modified duration

D_mod = D_Macaulay ÷ (1 + y ÷ m)

m is compounding periods per year.

Duration matching condition

D_A × A = D_L × L, i.e. D_A = D_L × L ÷ A

Immunizing equity against a small parallel shift requires matching dollar durations. This equals D_A = D_L only when A = L. If assets exceed liabilities, the required asset duration is lower than the liability duration.

Duration gap and equity change

ΔE ≈ −(D_A − D_L × L ÷ A) × A × Δy

Positive leveraged duration gap means equity falls when rates rise.

Barbell vs bullet

Same duration: Convexity(barbell) > Convexity(bullet)

Barbell usually has lower yield pickup but higher convexity.

Current yield

Current yield = Annual coupon ÷ Current price

Ignores price change and time to maturity.

Total return

Total return = (Income + Ending price − Beginning price) ÷ Beginning price

Include reinvestment income if the question gives it.

Modified duration

Modified duration = Macaulay duration ÷ (1 + y/m)

y is yield, m is compounding periods per year.

Price change with duration

ΔP/P ≈ −D_mod × Δy

First-order estimate. Works best for small yield changes.

Price change with convexity

ΔP/P ≈ −D_mod × Δy + ½ × C × (Δy)²

Use C in the same units as Δy, with Δy as a decimal.

DV01

DV01 ≈ D_mod × P × 0.0001

Value change for a 1 bp move in yield.

Sharpe ratio

Sharpe = (Rp − Rf) ÷ σp

Excess return per unit of total risk.

Portfolio duration

D_portfolio = Σ (wi × Di)

Weights are market-value weights.

Liquidity Coverage Ratio

LCR = Stock of HQLA ÷ Total net cash outflows over the next 30 calendar days ≥ 100%

Net outflows = outflows − min(inflows, 75% of outflows). Inflows are capped at 75% of outflows.

HQLA after haircuts

HQLA value = Level 1 + Level 2A × (1 − 15%) + Level 2B × (1 − haircut), subject to caps

Level 2B haircut is 25% for some assets (such as qualifying RMBS) and 50% for others (such as qualifying equities). Check the question.

Level 2 caps

Level 2 ≤ 2/3 × Level 1; Level 2B ≤ 15/85 × (Level 1 + Level 2A)

These are the working forms of the limits that Level 2 is at most 40% of HQLA and Level 2B at most 15% of HQLA. Use the amounts after haircuts, and after adjusting for secured transactions maturing within 30 days. Use the 2/3 and 15/85 forms to compute the cap.

Collateral required with haircut

Collateral market value = Loan amount ÷ (1 − haircut)

Haircut is a percentage of collateral value. With a 5% haircut, USD 95 of lending needs USD 100 of collateral (95 ÷ 0.95 = 100).

AFS fair value change

Unrealised gain or loss → OCI (equity); Trading change → net income; HTM → no fair value entry

Applies to the US-style three-bucket scheme.

Quick revision

• The investment function invests to meet liabilities and objectives within board-set limits, not to maximise return alone.
• Risk appetite sets the overall amount of risk; limits and policy put it into practice.
• Match assets to liabilities: long-dated liabilities need long-dated or duration-matched assets.
• Liability-driven investing focuses on the funding position, not on beating a market index.
• Passive strategies track an index; active strategies take risk against a benchmark to seek excess return.
• Tracking error measures how much active returns vary around the benchmark.
• Sharpe ratio = (portfolio return − risk-free rate) ÷ standard deviation of portfolio return.
• Duration measures price sensitivity to yield changes; higher duration means more interest rate risk.
• VaR gives a loss threshold at a confidence level and horizon; it says nothing about losses beyond it.
• Accounting classification can change how gains and losses reach earnings and capital, so check it in a case.
• Liquid, high-quality assets can be a regulatory requirement and may limit the yield you can chase.
• Always check concentration, liquidity and regulatory limits before choosing the answer.

Common mistakes

• Treating income as the only reason banks hold securities. Fix: Remember the order: liquidity and balance sheet management come first, income is the residual benefit.
• Assuming higher yield is always better. Fix: Ask what extra risk pays for the extra yield: longer duration, lower credit quality or thinner market.
• Treating risk appetite and investment policy as the same thing. Fix: Appetite is the board's statement of how much risk overall. Policy is the set of portfolio rules that carry it out.
• Choosing the highest-yielding option. Fix: For an institution, the first test is fit with liabilities, liquidity and appetite. Return comes after that.
• Treating all agency MBS as having no risk because credit risk is low. Fix: Separate credit risk from prepayment and extension risk. Agency MBS can still lose value from rate moves and negative convexity.
• Saying MBS shorten in duration when rates rise. Fix: Rates down means faster prepayments and shorter life. Rates up means slower prepayments and longer life, so losses are larger.
• Saying a barbell always outperforms a bullet. Fix: Barbell wins in large yield moves but usually gives up yield. For small moves or a steepening or flattening, results depend on the curve change.
• Believing duration matching removes all interest rate risk. Fix: It covers small parallel shifts only. Non-parallel shifts, convexity mismatch and drift over time still create risk, so rebalance.
• Treating yield as total return Fix: Always add price change to income when the question gives ending and beginning prices.
• Forgetting the sign of the price change Fix: For a plain bond, a yield rise gives a price fall. Write the minus sign in the formula.

Exam tips

• Start from the funding profile. It decides which objective wins in most case questions.
• Expect tradeoff wording such as 'most appropriate' or 'primary objective'. Pick the option that balances goals, not the extreme one.
• Link securities to the LCR and HQLA when a regulatory ratio is mentioned.
• Check small calculations for yield or income by weights. Write each holding's income first.
• Read what each limit is measured against. Capital, total assets and portfolio value give different answers.
• When asked who is responsible, the board approves appetite and policy, management implements, and an independent risk function monitors.
• Expect options that offer higher yield by breaking a limit or a liquidity need. Reject them.
• Questions often test the order: capacity, then appetite, then policy, then limits.