Yield-Based Bond Convexity and Portfolio Properties: formula sheet

Key formulas

Macaulay duration

MacDur = Σ [t × PV(CF_t)] ÷ Σ PV(CF_t)

t is in periods. Divide by periods per year to express in years. The denominator is the bond's price.

Macaulay duration, closed form for a fixed-rate bond

MacDur = (1 + r) ÷ r − [(1 + r) + N × (c − r)] ÷ [c × ((1 + r)^N − 1) + r]

r is the yield per period, c is the coupon rate per period, N is the number of periods. For a bond priced at par (c = r), this reduces to (1 + r) ÷ r × [1 − 1 ÷ (1 + r)^N]. The result is in periods.

Modified duration

ModDur = MacDur ÷ (1 + YTM ÷ m)

MacDur is in years. YTM is the annual rate, and m is the number of periods per year.

Approximate modified duration

ApproxModDur = (PV− − PV+) ÷ (2 × ΔYTM × PV0)

PV− is the price when YTM falls by ΔYTM. PV+ is the price when YTM rises by ΔYTM. PV0 is the starting price. Cash flows stay fixed.

Effective duration

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

ΔCurve is the parallel shift in the benchmark yield curve. Prices come from a model that lets cash flows change, such as an option-adjusted valuation.

Price change estimate

%ΔPrice ≈ −ModDur × ΔYield

Use EffDur × ΔCurve for bonds with embedded options. For larger changes, add the convexity term: + ½ × Convexity × (ΔYield)².

Money duration

MoneyDur = ModDur × Full price (PV^Full)

Use the full price, with accrued interest, and the position size you are asked about (for example, price per 100 par times par amount ÷ 100).

Price value of a basis point

PVBP = MoneyDur × 0.0001

Same as approximate currency change for a 1 bp yield move. Quoted as a positive number.

Exact PVBP

PVBP = (PV₋ − PV₊) ÷ 2

PV₋ is the price with yield down 1 bp, PV₊ with yield up 1 bp.

Estimated currency price change

ΔPV ≈ −MoneyDur × Δyield

Δyield in decimals (0.0050 for 50 bps). Add a convexity term for large moves.

Modified duration link

ModDur = MacDur ÷ (1 + y/m)

y is the annual yield, m is the periods per year. Needed when the question gives Macaulay duration.

Approximate modified duration

ApproxModDur = (PV− − PV+) ÷ (2 × PV0 × Δyield)

PV− is the price when yield falls by Δyield. PV+ is the price when yield rises by Δyield. Use Δyield as a decimal.

Approximate convexity

ApproxCon = (PV− + PV+ − 2 × PV0) ÷ (Δyield² × PV0)

Uses the bond's own yield-to-maturity change. Fits option-free bonds.

Effective convexity

EffCon = (PV− + PV+ − 2 × PV0) ÷ (ΔCurve² × PV0)

ΔCurve is a parallel shift in the benchmark yield curve. Use it for bonds with embedded options.

Price change from duration and convexity

%ΔPV ≈ −ModDur × Δy + ½ × Convexity × (Δy)²

The first term is the duration effect. The second term is the convexity adjustment. Δy is a decimal.

Money convexity version

ΔPV ≈ −MoneyDur × Δy + ½ × MoneyConvexity × (Δy)²

Money convexity = Convexity × full price. The result is a currency amount, not a percentage.

Weight of bond i

wᵢ = Market value of bond i ÷ Total portfolio market value

Use full price including accrued interest, not par value.

Portfolio duration (weighted average)

D_p = Σ wᵢ × Dᵢ = w₁D₁ + w₂D₂ + … + wₙDₙ

Use one type of duration throughout. Valid as an estimate for parallel yield shifts.

Portfolio convexity (weighted average)

C_p = Σ wᵢ × Cᵢ

Same market-value weights as for duration.

Approximate percentage price change

%ΔPV ≈ −D_p × Δy + ½ × C_p × (Δy)²

Enter Δy as a decimal. 50 bps = 0.005. The convexity term is always positive for an option-free bond portfolio.

Portfolio money duration or BPV

Portfolio BPV = Σ BPVᵢ

Money measures are additive across bonds because they are in currency units.

Cash flow yield approach

Find the IRR of the pooled portfolio cash flows, then compute duration at that yield

Better theoretically than the weighted average, but still assumes a single yield change.

Callable bond value

Callable bond price = Option-free bond price − Call option value

The issuer owns the call, so the investor's bond is worth less. The gap is biggest when yields are low.

Putable bond value

Putable bond price = Option-free bond price + Put option value

The investor owns the put, so the bond is worth more. The gap is biggest when yields are high.

Effective duration

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

PV₋ and PV₊ are prices after the curve falls and rises by ΔCurve. Use for bonds with embedded options.

Effective convexity

EffCon = (PV₋ + PV₊ − 2 × PV₀) ÷ (PV₀ × ΔCurve²)

A negative result means negative convexity. ΔCurve is a decimal, such as 0.0025.

Price change estimate

%ΔPV ≈ (−EffDur × ΔYield) + (½ × EffCon × ΔYield²)

Use the same yield change in decimal form. Convexity adds a positive term if EffCon > 0 and a negative term if EffCon < 0.

Convexity by bond type

Option-free: positive. Putable: positive. Callable: positive at high yields, negative at low yields.

Callable bonds show negative convexity only when the call option is near or in the money.

Quick revision

• Bond prices and yields move in opposite directions, and the relationship is convex for option-free bonds.
• Modified duration = Macaulay duration ÷ (1 + YTM per period), with YTM per period = YTM ÷ periods per year.
• Effective duration uses price changes from a curve shift and suits bonds with embedded options.
• %ΔPrice ≈ −ModDur × ΔYield for the first-order estimate.
• Convexity adjustment = ½ × Convexity × (ΔYield)², and it is positive for positive convexity.
• Money duration = ModDur × full price of the position. It is the currency price change per 1.00 (100%) change in yield, so for a 1 percentage point (100 basis point) change use about money duration × 0.01.
• PVBP is the price change for a one basis point change in yield, about money duration × 0.0001.
• Portfolio duration is the market-value-weighted average of bond durations.
• A callable bond has negative convexity at low yields, when the call is likely, and its price rises less than an option-free bond's. At high yields it has positive convexity.
• A putable bond has positive convexity. At high yields the put moves into the money, so its price falls less than an option-free bond's. At low yields the put is far out of the money and the bond behaves like an option-free bond.
• Higher convexity is favourable: it gains more when yields fall and loses less when yields rise.

Common mistakes

• Using modified duration for a callable or putable bond. Fix: If the bond has an embedded option, cash flows can change when rates change. Use effective duration from the repriced bond.
• Dividing Macaulay duration by (1 + annual YTM) for a semiannual bond. Fix: Divide by (1 + YTM ÷ m). For a 6% semiannual bond, divide by 1.03, not 1.06.
• Using clean price instead of full price Fix: Add accrued interest whenever the question gives it. Money duration is based on the full price.
• Forgetting to multiply by 0.0001 for PVBP Fix: Always scale: PVBP = money duration × 0.0001.
• Leaving out the ½ in the convexity adjustment. Fix: Memorise the price-change formula as one line: −ModDur × Δy + ½ × Convexity × (Δy)².
• Using Δy in percent instead of decimal. Fix: Convert first: 1% = 0.01, 50 bp = 0.005. Then square the decimal.
• Weighting by par value or number of bonds instead of market value. Fix: Use market value including accrued interest. A discount or premium bond has a different weight than its par suggests.
• Leaving out the ½ in the convexity adjustment, or forgetting to square Δy. Fix: Write the full formula first: ½ × C × (Δy)². Convert Δy to decimals before squaring.
• Saying a callable bond always has negative convexity. Fix: Negative convexity appears only when yields are low enough for the call to matter. At high yields the callable bond looks like an option-free bond.
• Thinking a putable bond has negative convexity because it has an embedded option. Fix: The put gives the investor a price floor, so the putable bond keeps positive convexity. Ask who owns the option.

Exam tips

• When a stem mentions a call, put or other embedded option, expect effective duration and a table of prices after curve shifts.
• Check the units of the shift. Basis points must be converted to decimals in the formula, so 25 bps becomes 0.0025.
• Remember the ranking for a positive yield: Macaulay duration is above modified duration. This eliminates options quickly.
• Know the zero-coupon rule. Macaulay duration equals maturity, and the modified duration is that figure divided by (1 + yield per period).
• For price-change questions, give the sign. Yields up mean prices down, so the estimate is negative.
• Check whether the question gives modified or Macaulay duration before calculating.
• Watch the unit of price: per 100 par or the whole position.
• Check magnitude: PVBP should be roughly one ten-thousandth of price × duration, so reject options off by powers of ten.