Interest Rate Risk — Duration & Convexity

Interest-rate sensitivity measures how a bond's market value responds when required yield changes. Duration gives the slope of the price–yield curve; convexity measures its curvature. Together they convert a yield scenario into an estimated percentage and monetary price change.

1Interest-Rate Risk: The Starting Point

A fixed-rate bond promises cash flows in nominal currency. When the market requires a different yield, those cash flows do not change, but their present value does. This repricing is interest-rate risk.

Yield rises

Discount factors fall, so the bond price falls.

Yield falls

Discount factors rise, so the bond price rises.

Longer cash flows

More distant payments are generally more rate-sensitive.

Inverse relationship

Bond price and yield move in opposite directions. Duration attaches a number to that inverse slope; convexity explains why equal yield rises and falls do not produce equal price changes.

2Bond-Pricing Basics

For a bond with face value \(FV\), annual coupon rate \(c\), \(m\) payments per year, maturity \(T\), and nominal annual yield \(y\), the periodic coupon and yield are \(CPN=FV(c/m)\) and \(i=y/m\). With \(n=mT\):

\[P=\sum_{t=1}^{n}\frac{CF_t}{(1+i)^t}=\frac{CPN}{i}\left[1-(1+i)^{-n}\right]+FV(1+i)^{-n}\]

Relationship	Price implication	Reason
Coupon rate = yield	Price = par	Coupon compensates exactly for the market yield
Coupon rate > yield	Price > par	Above-market coupons create a premium
Coupon rate < yield	Price < par	Below-market coupons require a discount

Yield convention matters

A 7% nominal yield compounded semi-annually means 3.5% per half-year. Duration formulas must use a yield convention consistent with the bond price and coupon frequency.

3The Convex Price–Yield Relationship

The price–yield curve slopes downward and bows outward for an option-free bond. Its slope is steep when yields are low and flatter when yields are high.

\[\frac{\partial P}{\partial y}<0,\qquad \frac{\partial^2P}{\partial y^2}>0\]

Duration is the tangent

A straight-line estimate works well near the current yield.

Convexity is the bend

The actual curve lies above the duration tangent for an option-free bond.

Asymmetric response

A yield fall produces a larger gain than the loss from an equal yield rise.

Why convexity helps

Positive convexity is favourable: it adds to the estimated bond value whether yields rise or fall. It reduces the duration-estimated loss when yields rise and increases the duration-estimated gain when yields fall.

4What Determines Interest-Rate Sensitivity?

Bond feature	Usual duration effect	Intuition
Longer maturity	Higher	More value arrives later
Lower coupon	Higher	More weight remains in the final principal payment
Lower yield	Higher	Distant cash flows receive relatively greater present-value weight
More frequent coupon payments	Slightly lower	Cash is returned sooner
Embedded call	Can reduce duration and convexity	Falling yields make early redemption more likely

A zero-coupon bond has Macaulay duration equal to maturity because its only cash flow arrives at maturity. A coupon bond's duration is shorter than maturity because coupons return value earlier.

5Macaulay Duration

Macaulay duration is the present-value-weighted average time to receive the bond's cash flows. First calculate each cash flow's weight:

\[w_t=\frac{PV(CF_t)}{P},\qquad \sum_{t=1}^{n}w_t=1\]

\[D_{Mac}=\sum_{t=1}^{n}\tau_t w_t=\frac{\sum_{t=1}^{n}\tau_t\,PV(CF_t)}{P}\]

Here \(\tau_t=t/m\) is time in years. Macaulay duration is a timing measure, expressed in years, rather than a direct percentage sensitivity.

Economic interpretation

Macaulay duration is the bond's cash-flow centre of gravity. It is also the classical investment horizon at which price risk and coupon-reinvestment risk approximately offset for a small parallel rate change.

6Modified Duration

Modified duration converts Macaulay duration into first-order price sensitivity:

\[D_{Mod}=\frac{D_{Mac}}{1+y/m}\]

\[\frac{\Delta P}{P}\approx-D_{Mod}\Delta y\]

If modified duration is 7.30, a 100-basis-point rise in yield \((\Delta y=+0.01)\) implies an approximate price decline of 7.30%. A 50-basis-point fall implies an approximate gain of 3.65%.

Use decimal yield changes

One basis point is 0.0001, 50 bp is 0.005, and 100 bp is 0.01. Multiplying duration by “100” instead of 0.01 is a common and serious error.

Why the minus sign?

The minus sign records the inverse price–yield relationship. For a positive yield shock, the duration contribution is negative. For a negative yield shock, it is positive.

7Dollar Duration, PVBP and DV01

Portfolio managers often need currency P&L rather than a percentage. Dollar duration for a unit yield change is \(D_{Mod}P\). The price value of one basis point is:

\[DV01=PVBP\approx D_{Mod}\times P\times0.0001\]

DV01 is normally reported as a positive risk magnitude even though a one-basis-point yield rise creates an approximate price change of \(-DV01\).

Reference-bond DV01

With price £928.94 and modified duration 7.3037, DV01 is approximately £0.6785 per £1,000 face value. A 10 bp rise therefore creates an estimated loss of about £6.78 before convexity.

8Convexity

Convexity is the scaled second derivative of price with respect to yield:

\[\mathcal C=\frac{1}{P}\frac{\partial^2P}{\partial y^2}\]

A practical three-price estimate shocks the yield up and down by the same decimal amount \(\Delta y\):

\[\mathcal C\approx\frac{P_-+P_+-2P_0}{P_0(\Delta y)^2}\]

For fixed cash flows and discrete compounding, analytical convexity can also be obtained from the discounted cash-flow schedule, provided periods and annualisation are handled consistently.

Positive and negative convexity

Option-free bonds generally have positive convexity. Callable bonds and mortgage-backed securities can develop negative convexity when falling yields trigger calls or prepayments and cap price appreciation.

9Total Price Change: Duration Plus Convexity

The second-order approximation combines slope and curvature:

\[\frac{\Delta P}{P}\approx\underbrace{-D_{Mod}\Delta y}_{\text{duration effect}}+\underbrace{\frac12\mathcal C(\Delta y)^2}_{\text{convexity effect}}\]

\[\Delta P\approx P\left[-D_{Mod}\Delta y+\frac12\mathcal C(\Delta y)^2\right]\]

Yield movement	Duration term	Convexity term	Total
Yield rises	Negative	Positive for an option-free bond	Loss is smaller than duration alone
Yield falls	Positive	Positive for an option-free bond	Gain is larger than duration alone

Accuracy rule

Duration alone is usually adequate for small changes. As the yield shock grows, the squared convexity term becomes material. Full repricing remains the benchmark whenever accuracy matters.

10Worked Example

Consider the workbook's reference bond: £1,000 face value, 6% annual coupon, semi-annual payments, 10-year maturity and 7% nominal annual yield.

Measure	Result	Interpretation
Bond price	£928.94	Coupon is below yield, so price is below par
Macaulay duration	7.5593 years	PV-weighted receipt time
Modified duration	7.3037	Approximate % sensitivity per 100 bp
Effective duration from ±50 bp	7.3065	Three-price estimate
Convexity	66.9210	Positive curvature
DV01	£0.6785	Approximate loss for a 1 bp rise

Estimate a 100 bp yield rise

\[\text{Duration effect}=-7.3037(0.01)=-7.3037\%\]

\[\text{Convexity effect}=\tfrac12(66.9210)(0.01)^2=+0.3346\%\]

\[\text{Estimated total change}=-6.9691\%,\qquad \widehat P\approx £864.20\]

Duration alone would estimate £861.10. The positive-convexity correction adds about £3.11 and moves the estimate toward the fully repriced value.

11Effective Duration and Effective Convexity

Modified duration assumes fixed cash flows. When cash flows change with rates, revalue the security under an option-adjusted model:

\[D_{Eff}=\frac{P_--P_+}{2P_0\Delta y}\]

\[\mathcal C_{Eff}=\frac{P_-+P_+-2P_0}{P_0(\Delta y)^2}\]

Measure	Best suited to	Main assumption
Modified duration	Option-free fixed-cash-flow bonds	Cash flows remain unchanged
Effective duration	Callable, putable and mortgage-backed securities	Cash flows are regenerated under each rate scenario

Do not mix measures casually

The shocked prices for effective measures should come from a consistent term-structure and option model. Merely changing a single YTM while holding option-dependent cash flows fixed defeats the purpose.

12Portfolio Duration and Key-Rate Risk

For market-value weights \(w_i=MV_i/\sum MV_i\), portfolio duration and convexity are approximately:

\[D_P=\sum_i w_iD_i,\qquad \mathcal C_P=\sum_iw_i\mathcal C_i\]

These measures describe a parallel yield-curve shift. Actual curves can steepen, flatten, twist or develop local movements. Key-rate duration measures exposure at selected maturities:

\[KRD_k\approx-\frac{P_k-P_0}{P_0\Delta y_k}\]

From one number to a risk vector

A portfolio can have the desired total duration and still be exposed to a curve twist. Key-rate durations show where on the curve the risk sits. Their sum is approximately effective duration for a parallel shift.

13Duration Matching and Immunization

Immunization aims to fund a liability despite small interest-rate changes. Classical conditions at inception are:

\[PV(A)=PV(L),\qquad D_A=D_L\]

If asset convexity exceeds liability convexity, the asset portfolio has a favourable second-order cushion for small parallel shifts.

Price risk

Rising yields reduce the current value of bonds.

Reinvestment risk

Falling yields reduce the rate earned on coupons.

Duration match

At the target horizon, the two effects approximately offset.

Immunization is not permanent

Duration changes as time passes, yields move and cash flows are received. Portfolios require rebalancing. A simple duration match does not protect against non-parallel curve shifts, spread changes, defaults or model error.

14Interactive Duration–Convexity Lab

Adjust the reference bond and yield shock. The lab calculates exact price, Macaulay and modified duration, convexity, duration-only change, convexity correction and the full-repricing result.

Bond interest-rate sensitivity calculator

Annual coupon rate 6.0% Maturity 10 years Current yield 7.0% Yield shock +100 bp

Current price

Macaulay duration

Modified duration

Convexity

Duration effect

Convexity effect

Total estimate

Exact price change

Estimate error

15Practical Measurement Workflow

Confirm cash flows, settlement date, day-count convention and yield-compounding convention.
Price the bond from its cash flows and current term structure or YTM.
Use Macaulay duration for timing and modified duration for fixed-cash-flow price sensitivity.
Report DV01 for monetary exposure and aggregate it using market values.
Add convexity for material rate moves; use full repricing as the accuracy benchmark.
Use effective measures when cash flows depend on rates.
Use key-rate duration for non-parallel yield-curve risk.
Backtest estimates against actual repricing and rebalance hedges or immunized portfolios.

The complete sensitivity equation

Yield shock → duration effect + convexity correction → estimated percentage change → estimated monetary P&L. The estimate is local; full repricing captures all higher-order effects.

Prepared for The Mountain Path Academy · Reference example adapted from Module 5: Fixed Income Securities & Analysis