{ "deckName": "Exam FM — Duration, Convexity & Immunization", "examCode": "Exam FM", "cards": [ { "front": "Define **Macaulay duration** $D_{mac}$ of a stream of cash flows and explain what it measures.", "back": "$D_{mac}=\\dfrac{\\sum_{t} t\\,v^{t}\\,CF_{t}}{\\sum_{t} v^{t}\\,CF_{t}}$, the present-value-weighted **average time** of the cash flows.\nEach flow's weight is the fraction of total price contributed by its discounted value $v^{t}CF_{t}$, so $D_{mac}$ is measured in the same time units (usually years) as $t$.", "tag": "Duration" }, { "front": "What is the Macaulay duration of a single payment of $C$ made at time $n$?", "back": "Exactly $n$. With only one cash flow, the PV-weighted average time is just its own payment date: $D_{mac}=\\dfrac{n\\,v^{n}C}{v^{n}C}=n$.\nThis is why a zero-coupon bond maturing in $n$ years has Macaulay duration $n$ — the longest duration available for a given maturity.", "tag": "Duration" }, { "front": "Define **modified duration** $D_{mod}$ and give its relationship to Macaulay duration.", "back": "$D_{mod}=-\\dfrac{1}{P}\\dfrac{dP}{di}$, the negative percentage sensitivity of price to a change in the periodic yield.\nIt connects to Macaulay duration by $D_{mod}=\\dfrac{D_{mac}}{1+i}$. (Under a force of interest / continuous compounding the two coincide: $D_{mod}=D_{mac}$.)", "tag": "Duration" }, { "front": "A liability has Macaulay duration $7.5$ years and the effective annual yield is $i=4\\%$. Find its modified duration.", "back": "$D_{mod}=\\dfrac{D_{mac}}{1+i}=\\dfrac{7.5}{1.04}\\approx 7.2115$ years.\nModified duration is always slightly **smaller** than Macaulay duration (you divide by $1+i>1$).", "tag": "Duration" }, { "front": "Define **dollar duration** and explain how it relates to modified duration.", "back": "Dollar duration $=-\\dfrac{dP}{di}=P\\cdot D_{mod}$ — the **absolute** (not percentage) change in price per unit change in yield.\nUnlike $D_{mac}$ and $D_{mod}$, which are unitless time/elasticity measures, dollar duration carries currency units and is additive across a portfolio in dollars.", "tag": "Duration" }, { "front": "A bond is priced at $P=5000$ with modified duration $D_{mod}=4$. Compute its dollar duration and estimate the price change for a $+25$ bp yield move.", "back": "Dollar duration $=P\\cdot D_{mod}=5000\\times 4=20000$.\nFor $\\Delta i=+0.0025$: $\\Delta P\\approx -(\\text{dollar duration})\\cdot\\Delta i=-20000\\times 0.0025=-50$.\nThe bond loses about $\\$50$, falling to roughly $\\$4950$.", "tag": "Duration" }, { "front": "A $3$-year bond has annual coupons of $80$ and redeems for $1000$ at $t=3$. At an effective annual yield of $6\\%$, find its price and Macaulay duration.", "back": "Price: $P=\\dfrac{80}{1.06}+\\dfrac{80}{1.06^{2}}+\\dfrac{1080}{1.06^{3}}\\approx 1053.46$.\nWeighted-time numerator: $\\dfrac{1\\cdot80}{1.06}+\\dfrac{2\\cdot80}{1.06^{2}}+\\dfrac{3\\cdot1080}{1.06^{3}}\\approx 2938.24$.\n$D_{mac}=\\dfrac{2938.24}{1053.46}\\approx 2.789$ years. (Note $D_{mac}<3=$ maturity, as always for a coupon bond.)", "tag": "Duration" }, { "front": "For the bond above ($P\\approx1053.46$, $D_{mac}\\approx2.789$, $i=6\\%$), find its modified duration.", "back": "$D_{mod}=\\dfrac{D_{mac}}{1+i}=\\dfrac{2.789}{1.06}\\approx 2.631$ years.\nInterpretation: a $1\\%$ ($100$ bp) rise in yield drops the price by roughly $2.631\\%$ to first order.", "tag": "Duration" }, { "front": "Why is the Macaulay duration of a coupon-paying bond always **less than** its time to maturity?", "back": "Maturity reflects only the final redemption date, but duration is a PV-weighted average of **all** payment dates. The intermediate coupons land earlier than maturity and pull the average time down.\nOnly a zero-coupon bond (no intermediate flows) has duration equal to its maturity.", "tag": "Duration" }, { "front": "Give the formula for the Macaulay duration of a **level perpetuity-immediate** and compute it at $i=8\\%$.", "back": "For a perpetuity paying $1$ at the end of each period forever, $D_{mac}=\\dfrac{1+i}{i}$.\nAt $i=0.08$: $D_{mac}=\\dfrac{1.08}{0.08}=13.5$ years. (This special case is frequently misremembered — memorize $\\frac{1+i}{i}$.)", "tag": "Duration" }, { "front": "Find the Macaulay duration of a $5$-year annuity-immediate paying $100$ per year at an effective annual rate of $7\\%$.", "back": "PV $=100\\sum_{t=1}^{5}1.07^{-t}\\approx 410.02$.\nWeighted-time numerator $=100\\sum_{t=1}^{5} t\\cdot1.07^{-t}\\approx 1174.69$.\n$D_{mac}=\\dfrac{1174.69}{410.02}\\approx 2.865$ years. (The duration sits a bit **before** the middle date $t=3$ because the earlier level payments carry more present-value weight.)", "tag": "Duration" }, { "front": "How do you compute the duration of a **portfolio** from the durations of its component assets?", "back": "Portfolio duration is the **PV-weighted average** of the component durations:\n$D_{P}=\\dfrac{\\sum_{k} P_{k}\\,D_{k}}{\\sum_{k} P_{k}}$, where $P_{k}$ is the present value (market value) of asset $k$ and $D_{k}$ its duration.\nWeight by market value, not by face amount or by count of bonds.", "tag": "Portfolio duration" }, { "front": "Asset A has PV $4000$ and modified duration $3$; Asset B has PV $6000$ and modified duration $8$. Find the modified duration of the combined portfolio.", "back": "$D_{P}=\\dfrac{4000\\times3+6000\\times8}{4000+6000}=\\dfrac{12000+48000}{10000}=\\dfrac{60000}{10000}=6$ years.\nThe answer leans toward Asset B because B holds the larger share of the portfolio's value.", "tag": "Portfolio duration" }, { "front": "You want a portfolio with modified duration $5$, built from a short bond ($D_{mod}=2$) and a long bond ($D_{mod}=10$). What value-weight $w$ in the long bond is required?", "back": "Set the weighted average to $5$: $2(1-w)+10w=5$.\n$2+8w=5\\Rightarrow 8w=3\\Rightarrow w=0.375$.\nSo $37.5\\%$ of portfolio value in the long bond and $62.5\\%$ in the short bond gives $D_{mod}=5$.", "tag": "Portfolio duration" }, { "front": "State the **first-order (modified-duration) approximation** for the change in price when the yield moves by $\\Delta i$.", "back": "$\\Delta P\\approx -P\\cdot D_{mod}\\cdot\\Delta i$, equivalently $P(i+\\Delta i)\\approx P(i)\\,[\\,1-D_{mod}\\,\\Delta i\\,]$.\nA rate **increase** ($\\Delta i>0$) gives a price **decrease** ($\\Delta P<0$). This is the tangent-line (linear) estimate of the true, convex price curve.", "tag": "Price approximation" }, { "front": "A bond is priced at $P=1053.46$ with $D_{mod}=2.631$. Use the first-order modified-duration rule to estimate the price after the yield rises $50$ bp.", "back": "$\\Delta i=+0.005$.\n$\\Delta P\\approx -P\\,D_{mod}\\,\\Delta i=-1053.46\\times2.631\\times0.005\\approx -13.86$.\nEstimated new price $\\approx 1053.46-13.86=1039.60$. (The exact reprice is $1039.73$; the linear estimate slightly **understates** the price because it ignores convexity.)", "tag": "Price approximation" }, { "front": "State the **Macaulay-duration form** of the first-order price approximation and contrast it with the modified-duration form.", "back": "Macaulay form: $P(i+\\Delta i)\\approx P(i)\\left[\\dfrac{1+i}{1+i+\\Delta i}\\right]^{D_{mac}}$.\nModified form: $P(i+\\Delta i)\\approx P(i)\\,[1-D_{mod}\\,\\Delta i]$.\nBoth are first-order; the Macaulay form is multiplicative and uses $D_{mac}$ directly, the modified form is additive and uses $D_{mod}=D_{mac}/(1+i)$. **Do not** divide by $(1+i)$ twice by mixing them.", "tag": "Price approximation" }, { "front": "Using the **Macaulay-duration** form, estimate the new price of a bond with $P=1053.46$, $D_{mac}=2.789$, at $i=6\\%$ if the yield rises to $6.5\\%$.", "back": "$P(0.065)\\approx 1053.46\\left[\\dfrac{1.06}{1.065}\\right]^{2.789}$.\n$\\dfrac{1.06}{1.065}\\approx 0.995305$; raised to $2.789$: $\\approx 0.98696$.\n$P\\approx 1053.46\\times0.98696\\approx 1039.72$. (Essentially the exact reprice of $1039.73$ — the Macaulay form tracks the curve more closely than the linear modified form here.)", "tag": "Price approximation" }, { "front": "A liability portfolio has $P=2{,}000{,}000$ and modified duration $9$. By approximately how much does its value change if the yield falls $30$ bp?", "back": "$\\Delta i=-0.003$.\n$\\Delta P\\approx -P\\,D_{mod}\\,\\Delta i=-2{,}000{,}000\\times9\\times(-0.003)=+54{,}000$.\nA yield **drop** raises the present value of the liabilities by about $\\$54{,}000$ — longer-duration liabilities are more exposed to falling rates.", "tag": "Price approximation" }, { "front": "Define **convexity** $C$ for a set of cash flows (modified-convexity form) and state its role.", "back": "$C=\\dfrac{1}{P}\\dfrac{d^{2}P}{di^{2}}=\\dfrac{\\sum_{t} t(t+1)\\,v^{t+2}\\,CF_{t}}{P}$.\nIt is the **second-order** term capturing the curvature of the price-yield relationship. Because the price curve is convex, $C>0$ for ordinary bonds, so the duration-only estimate always understates the true price for both up and down moves.", "tag": "Convexity" }, { "front": "State the **second-order** price-change approximation using both duration and convexity.", "back": "$\\dfrac{\\Delta P}{P}\\approx -D_{mod}\\,\\Delta i+\\tfrac{1}{2}\\,C\\,(\\Delta i)^{2}$,\nso $P(i+\\Delta i)\\approx P\\left[1-D_{mod}\\,\\Delta i+\\tfrac{1}{2}C(\\Delta i)^{2}\\right]$.\nThe convexity term is always **positive** (it adds back the curvature the linear term misses) and matters most for large $\\Delta i$.", "tag": "Convexity" }, { "front": "For the $3$-year, $80$-coupon, $1000$-redemption bond at $6\\%$ ($P=1053.46$, $D_{mod}=2.631$, $C=9.681$), estimate the price after a $+50$ bp move using the **duration + convexity** rule.", "back": "$\\Delta i=0.005$.\nDuration term: $-D_{mod}\\,\\Delta i=-2.631\\times0.005=-0.013155$.\nConvexity term: $\\tfrac{1}{2}C(\\Delta i)^{2}=0.5\\times9.681\\times0.005^{2}=0.000121$.\n$\\dfrac{\\Delta P}{P}\\approx -0.013034\\Rightarrow P\\approx 1053.46(1-0.013034)\\approx 1039.73$ — matching the exact reprice to the cent.", "tag": "Convexity" }, { "front": "Compute the (modified) convexity of a $4$-year zero-coupon bond at an effective annual yield of $5\\%$.", "back": "For a single payment at time $n$, the modified convexity is $\\dfrac{n(n+1)}{(1+i)^{2}}$.\nAt $n=4$, $i=0.05$: $\\dfrac{4\\times5}{1.05^{2}}=\\dfrac{20}{1.1025}\\approx 18.14$.\n(Longer-dated zeros have rapidly growing convexity because the $n(n+1)$ factor dominates.)", "tag": "Convexity" }, { "front": "Why does the duration-only price estimate always **understate** the true price of an option-free bond, regardless of the direction of the rate move?", "back": "The price-yield curve is **convex** (bows above its tangent line). Duration is the slope of the tangent at the current yield, so the tangent line lies **below** the actual curve on both sides.\nFor a yield rise the true price falls less than predicted; for a yield drop it rises more — in both cases the actual price exceeds the linear estimate. The positive convexity correction repairs this.", "tag": "Convexity" }, { "front": "Between two bonds with the **same duration**, why is higher convexity preferable to an investor?", "back": "With equal duration, the two react identically to small parallel moves, but the higher-convexity bond outperforms for **large** moves in either direction: it falls less when rates rise and gains more when rates fall.\nThis asymmetric benefit means convexity is a desirable property — investors will generally pay (accept a slightly lower yield) for it.", "tag": "Convexity" }, { "front": "State the three **Redington immunization** conditions for protecting a surplus against small interest-rate movements.", "back": "At the current yield: (1) **PV match** — $PV_{A}=PV_{L}$. (2) **Duration match** — $D_{mod,A}=D_{mod,L}$ (equivalently $\\dfrac{dP_{A}}{di}=\\dfrac{dP_{L}}{di}$). (3) **Convexity** — $C_{A}>C_{L}$ (asset cash flows more spread out than liability cash flows).\nTogether these make surplus $PV_{A}-PV_{L}$ a local minimum at the current rate, so any small shift leaves surplus $\\geq 0$.", "tag": "Immunization" }, { "front": "Why is matching present value and duration **not enough** for Redington immunization — what does the third condition add?", "back": "PV-and-duration matching makes the surplus function's first derivative zero at the current rate (a stationary point), but that point could be a maximum or a saddle.\nThe convexity condition $C_{A}>C_{L}$ forces the second derivative of surplus to be **positive**, guaranteeing a local **minimum** of zero surplus — so small moves in either direction can only increase surplus. Omitting condition (3) is the classic Redington mistake.", "tag": "Immunization" }, { "front": "A single liability of $1000$ is due at $t=2$. The effective annual rate is $10\\%$. What is its present value and (Macaulay) duration?", "back": "$PV_{L}=\\dfrac{1000}{1.10^{2}}=\\dfrac{1000}{1.21}\\approx 826.45$.\nBeing a single cash flow at $t=2$, its Macaulay duration is exactly $2$ years. Any immunizing asset portfolio must match $PV\\approx826.45$ and duration $2$.", "tag": "Immunization" }, { "front": "Immunize the $1000$-at-$t=2$ liability ($i=10\\%$, $PV_L\\approx826.45$) using two zero-coupon bonds maturing at $t=1$ and $t=3$. Find each face amount so PV and duration match.", "back": "Let the PVs of the two assets be $x_{1}$ (at $t=1$) and $x_{3}$ (at $t=3$). PV match: $x_{1}+x_{3}=826.45$. Duration match: $\\dfrac{1\\cdot x_{1}+3\\cdot x_{3}}{826.45}=2\\Rightarrow x_{1}+3x_{3}=1652.89$.\nSubtract: $2x_{3}=826.45\\Rightarrow x_{3}=413.22=x_{1}$.\nFaces: $A_{1}=413.22\\times1.10\\approx 454.55$; $A_{3}=413.22\\times1.10^{3}\\approx 550.00$.", "tag": "Immunization" }, { "front": "How does **full immunization** differ from Redington immunization, and what protection does it provide?", "back": "Redington protects only against **small** rate shifts (it is a local result). Full immunization protects against a single parallel shift of **any size**.\nIt is typically achieved by funding each liability with two asset cash flows that **straddle** the liability's date (one before, one after) while matching PV and duration. The earlier/later asset flows guarantee surplus $\\geq 0$ for every shift, not just infinitesimal ones.", "tag": "Immunization" }, { "front": "State the conditions used to **fully immunize** a single liability cash flow $L$ due at time $t_{L}$ with two asset cash flows.", "back": "Place asset cash flows at times $t_{1}C_{L}$. Is the position immunized, and against what?", "back": "Yes — all three Redington conditions hold (PV match, duration match, asset convexity exceeds liability convexity), so the surplus is protected against **small** parallel rate movements.\nIf, additionally, the asset cash flows straddle the single liability date, the position is **fully** immunized and protected against a parallel shift of any size.", "tag": "Immunization" }, { "front": "A duration-matched, PV-matched asset/liability position has **asset convexity less than liability convexity**. What can happen, and why?", "back": "Redington's third condition fails, so the surplus is at a local **maximum** rather than a minimum. A small move of the yield in **either** direction can push the surplus **negative** — the firm is exposed despite matching PV and duration.\nThe fix is to lengthen/spread the asset cash flows (raise $C_{A}$) so asset convexity exceeds liability convexity.", "tag": "Immunization" }, { "front": "Why does immunization (Redington or full) generally protect only against **parallel** shifts of the yield curve, and what real-world risk does that leave?", "back": "Duration and convexity summarize sensitivity to a single rate $i$ moving uniformly. The matching conditions are derived assuming the **whole** curve shifts by the same $\\Delta i$.\nIf the curve **twists** (short and long rates move by different amounts — a non-parallel shift), a matched portfolio can still lose surplus. This residual is **reinvestment / yield-curve-shape risk**, which only exact cash-flow matching fully eliminates.", "tag": "Immunization" }, { "front": "Summarize the practical hierarchy: **cash-flow matching vs full immunization vs Redington immunization** — strength of protection and cost.", "back": "**Cash-flow matching** is strongest (no rate risk at all, any curve move) but most costly/restrictive. **Full immunization** protects against any-size **parallel** shift using straddling assets with PV + duration match. **Redington** is weakest — only **small** parallel shifts — but needs just PV match, duration match, and $C_{A}>C_{L}$.\nProtection rises and flexibility falls as you move from Redington to full to exact matching.", "tag": "Immunization" }, { "front": "Estimate **effective (approximate) modified duration** by repricing: a bond is worth $1053.46$ at $i=6\\%$, $1067.45$ at $5.5\\%$, and $1039.73$ at $6.5\\%$. Compute it with the central-difference formula.", "back": "$D_{eff}=\\dfrac{P_{-}-P_{+}}{2\\,P_{0}\\,\\Delta i}$, using a $\\pm50$ bp shock ($\\Delta i=0.005$).\n$D_{eff}=\\dfrac{1067.45-1039.73}{2\\times1053.46\\times0.005}=\\dfrac{27.72}{10.5346}\\approx 2.631$ years.\nThis numerical estimate matches the analytic modified duration of $2.631$ — handy when a closed-form $D_{mod}$ is awkward.", "tag": "Price approximation" } ] }