Yield Curve Strategies: formula sheet

Key formulas

Effective duration

EffDur = (PV₋ − PV₊) ÷ (2 × PV₀ × ΔCurve)

PV₋ is the price after the curve falls by ΔCurve, PV₊ after it rises. ΔCurve is in decimals, e.g. 0.0025 for 25 bps. This is for a parallel shift of the benchmark curve.

Approximate price change from duration

%ΔPV ≈ −EffDur × ΔYield

Add the convexity term when the move is large: + ½ × Convexity × (ΔYield)².

Key rate duration

KRD(k) = (PV₋ − PV₊) ÷ (2 × PV₀ × Δy_k)

Only the yield at key maturity k is shifted. Other key rates stay unchanged.

Sum of key rate durations

Σ KRD(k) ≈ EffDur

Holds approximately for a bond or portfolio with fixed cash flows. Portfolio KRD at a point is the market-value-weighted average of the bond KRDs.

Money duration

Money duration = EffDur × Market value

Expressed in currency. Often shown per 100 of par for a bond.

Price value of a basis point

PVBP ≈ Money duration × 0.0001

The approximate money change for a 1 bp change in yield.

Portfolio duration

Dur_p = Σ w_i × Dur_i

Weights are market-value weights. This is correct for a parallel shift of the curve.

Buy-and-hold return

Return ≈ YTM at purchase

Holds if the bond is held to maturity, there is no default, and coupons are reinvested at the YTM.

Roll-down return

Roll-down return = (Ending price − Beginning price) ÷ Beginning price

The ending price is found at the horizon using the yield for the shorter remaining maturity, read from the unchanged curve.

Rolling yield

Rolling yield = (Coupon income ÷ Beginning price) + Roll-down return

Coupon income here is cash coupons received over the horizon. If the bond is sold between coupon dates, include accrued interest consistently.

Carry trade return on equity

Return on equity = [Assets × Rolling yield − Borrowed amount × Funding rate] ÷ Equity

Assets = Equity + Borrowed amount. Assumes an unchanged curve and a constant funding rate.

Roll-down sign rule

Upward-sloping curve: roll-down > 0. Flat: ≈ 0. Inverted: < 0

Strictly this holds for the part of the curve you ride, so check the yields at both maturities.

Portfolio duration

D_P = Σ (w_i × D_i)

Weights are market value weights. Use this to size a barbell to equal the bullet's duration.

Two-bond barbell weights

w_short = (D_long − D_target) ÷ (D_long − D_short); w_long = 1 − w_short

Gives a barbell with the same duration as the target bullet.

Duration-neutral butterfly (market value basis)

w_short × D_short + w_long × D_long = w_body × D_body

Here w_short and w_long are the long positions in the short and long wings, and w_body is the size of the short position in the body. The body's duration contribution must equal the combined wing contributions, so net duration is about zero. For a market-value-neutral trade, also set w_short + w_long = w_body.

Approximate price change

%ΔP ≈ −D × Δy + ½ × C × (Δy)²

For the same duration, higher convexity means a better result when yield changes are large.

Level, slope, curvature

Slope = y_long − y_short; Curvature = 2 × y_middle − (y_short + y_long)

A common way to define the butterfly spread. A rising value means the middle yield rises relative to the wings and the curve becomes more humped. This favours the long butterfly (long wings, short body). A falling value means the opposite.

Expected return decomposition

E(R) ≈ Yield income + Rolldown return + E(price change from yield and spread changes) − E(credit losses) + E(currency gains or losses)

Add only the lines the question gives. Keep each in the same period, usually the horizon return.

Price change from yield shift

%ΔP ≈ −Duration × ΔY + ½ × Convexity × (ΔY)²

ΔY in decimals, so 50 bps = 0.0050. Use modified or effective duration, not Macaulay duration. Convexity must match the units of the one given.

Convexity effect alone

Convexity effect = ½ × Convexity × (ΔY)²

Positive when convexity is positive, whether yields rise or fall.

Key rate duration approximation

%ΔP ≈ −Σ (KRD_i × ΔY_i) + ½ × Convexity × (ΔY)²

Use for non-parallel shifts. Each ΔY_i is the yield change at that maturity point.

Credit spread effect

%ΔP from spread ≈ −Spread duration × ΔSpread

Widening spreads reduce price. Do not also count the spread change as part of the benchmark yield change.

Rolldown return

Rolldown return ≈ (Price at horizon with unchanged curve − Price today) ÷ Price today, excluding coupon

Positive for an upward-sloping curve, zero for a flat curve, negative for an inverted curve.

Forward rate from spot rates

(1 + z_B)^B = (1 + z_A)^A × (1 + f(A,B−A))^(B−A)

z is the spot rate, A and B are maturities in years. f(A,B−A) is the forward rate starting at A for B−A years. Solve for f.

Expectations view

Forward rate = expected future spot rate (pure expectations)

With a risk premium, forward = expected spot + premium. Do not treat the forward as a pure forecast.

Active view decision rule

Expected future spot rate < forward rate → the longer bond outperforms rolling shorter bonds; expected spot rate > forward → the longer bond underperforms rolling shorter bonds

Compare your expected spot with the forward, not with today's spot. Example: if the one-year spot in one year is 4.5% < 5.01%, rolling one-year bonds earns less than the two-year bond.

Vasicek model

dr = a(b − r)dt + σ dz

a is speed of mean reversion, b the long-run mean, σ constant volatility. Rates can go negative. Affine, with a closed-form bond price.

CIR model

dr = a(b − r)dt + σ√r dz

Volatility rises with the rate level. Rates stay non-negative under standard conditions. Affine, with a closed-form bond price.

Nelson-Siegel components

Yield curve = level + slope + curvature components, with decay parameter λ

Three components (level, slope, curvature), governed by a decay parameter λ, fit the shape. The Svensson extension adds a second curvature term.

Basis point value of a bond or portfolio

BPV ≈ Modified duration × Market value × 0.0001

Gives the money change for a 1 bp yield move. Use the same yield-change convention for every leg.

Number of futures to reach target BPV

N = (BPV target − BPV current) ÷ BPV futures

Positive N means buy futures. Negative N means sell futures.

Futures BPV

BPV futures ≈ BPV of cheapest-to-deliver (CTD) ÷ Conversion factor

For bond futures, adjust the CTD bond's BPV by its conversion factor.

Hedge ratio for bond futures

N = −BPV portfolio ÷ (BPV CTD ÷ Conversion factor)

This is the target formula with a target BPV of zero, using futures BPV = BPV CTD ÷ Conversion factor. It is the same as −(BPV portfolio ÷ BPV CTD) × Conversion factor. The negative sign means the result is the number of contracts to sell.

Swap BPV

BPV swap ≈ BPV fixed leg − BPV floating leg

This is from the fixed receiver's perspective, so the receiver of fixed has positive BPV. The fixed payer's BPV is the negative of this. The floating leg BPV is small.

Yield curve trade rule

Steepener: long short-end BPV, short long-end BPV. Flattener: the reverse.

Size legs so that net BPV is about zero if you want a pure shape trade.

Quick revision

• A curve move is described by level, slope and curvature changes.
• Effective duration measures exposure to a parallel shift; key rate duration measures exposure at specific maturities.
• Money duration is the duration times the position value, so use it to size positions and hedges.
• Buy-and-hold return comes from coupons, reinvestment and the price at the end of the horizon.
• Roll-down return exists when the curve is upward sloping and the bond is priced at a lower yield as maturity shortens, assuming an unchanged curve.
• Expected return = carry + roll-down + price effect of the expected yield change, adjusted for any other items given.
• A barbell has more convexity than a bullet of the same duration, but yields differ and the benefit depends on the curve change.
• For equal duration, a bullet tends to outperform a barbell when the curve flattens (long-end yields fall by less than short-end yields, or short-end yields rise), and a barbell tends to outperform a bullet when the curve steepens. The result depends on the actual key rate changes, so check them rather than relying on the label.
• Match duration first, then compare structures on curvature and slope exposure.
• Run several scenarios and compare expected returns; do not rely on one forecast.
• Futures and swaps adjust duration or key rate exposure without selling bonds.
• In essays, tie the recommendation to the client's objectives and constraints in the fewest words that answer the command word.

Common mistakes

• Using effective duration to estimate the effect of a steepening or flattening. Fix: Effective duration assumes a parallel shift. For a twist or butterfly, use key rate durations at each maturity.
• Mixing up what steepness and curvature mean. Fix: Steepness is long yield minus short yield. Curvature is about the middle of the curve versus the ends. Match the term to the maturities that move.
• Using the original yield to price the bond at the horizon. Fix: Price the bond at the horizon with the yield for its remaining maturity, taken from the unchanged curve.
• Leaving out coupon income when computing rolling yield. Fix: Always add coupon income to the price change and divide the total by the beginning price.
• Comparing a barbell and a bullet with different durations. Fix: Solve for weights that give equal duration before drawing any conclusion.
• Saying the barbell always outperforms because it has more convexity. Fix: The barbell usually gives up yield. It wins only if yield moves are big enough, or the curve changes in its favour.
• Entering ΔY as 0.5 or 50 instead of 0.0050 in the convexity term Fix: Convert bps to decimals before using either formula. Squaring 0.0050 gives 0.000025.
• Forgetting the ½ in the convexity term Fix: Write the formula as ½ × C × (ΔY)² at the top of your working every time.
• Comparing the expected spot rate with today's spot rate instead of the forward rate. Fix: The forward is already priced in. You gain or lose only against the forward.
• Treating forward rates as unbiased forecasts. Fix: Forwards can include a risk premium. Say so when asked about forecasting ability.

Exam tips

• For any non-parallel shift, go straight to key rate durations. Write each KRD times its yield change in a column, then add.
• Show every step of the calculation. A correct number typed alone earns full credit, but a worked line protects you if the final figure is wrong.
• Obey the command word. If asked to identify the curve move, name it (steepening, flattening, butterfly) and stop. If asked to justify, link the move to the maturity exposure in one sentence.
• Read the units: bps to decimals, per 100 of par versus total market value.
• In a recommendation, tie the duration positioning to the client's liability or objective, such as which key rate to match.
• In item sets, check the curve shape first. It tells you the sign of roll-down before you calculate.
• In essays, a command word such as calculate needs only the number. A command word such as explain or justify needs a short reason tied to the unchanged-curve assumption.
• State the assumption each time: curve unchanged, no default, coupons reinvested at the stated yield.