Willys Flashcards Download
Become an Actuary › Exams › Flashcards › Exam FM › Duration, Convexity & Immunization Practice
Exam FM · practice

Exam FM — Duration, Convexity & Immunization Practice Flashcards

Thirty original SOA/CAS Exam FM-style multiple-choice problems covering Macaulay, modified, and dollar duration; portfolio duration; first-order and effective-duration price estimates; convexity and the second-order correction; Redington and full immunization; and exact cash-flow matching, each with a full worked solution.

8 free sample30 total · in appFree · fact-checked · LaTeX math
Tap card or press Space to flip
Answer

Unlock the full set

You're studying a free 8-problem sample. All 30 Duration, Convexity & Immunization practice problems — plus every other Exam FM subject and spaced-repetition scheduling — are built into the Willys AI Flashcards & Quizzes app. 14-day free trial, then $14.99.

Get the app — free 14-day trial

Every deck is built into the Willys app

All of these decks — including the full practice problem banks — come built into Willys AI Flashcards & Quizzes for iPhone & iPad (Mac version coming soon), with FSRS + SM-2 spaced repetition, streaks, and exam-date cram mode. 14-day free trial, then $14.99. To load a deck in the app: Settings → Library → Browse, then pick your exam and deck.

Download on the App Store

More Exam FM decks:

Annuities — Level Annuities (Level) Practice Annuities — Varying & Continuous Annuities (Varying & Continuous) Practice Bonds Bonds Practice

← All Exam FM decks

Browse all 30 problems as a list
  1. Duration
    A zero-coupon bond matures for $\$10{,}000$ in $8$ years. The effective annual yield is $i=5.5\%$. Calculate the modified duration of the bond. (A) $7.34$ (B) $7.58$ (C) $7.78$ (D) $8.00$ (E) $8.44$
    **Answer: (B).** For a single payment at time $n$, the Macaulay duration equals $n$, so $D_{mac}=8$. Modified duration is $D_{mod}=\frac{D_{mac}}{1+i}=\frac{8}{1.055}\approx 7.5829.$ Distractor (D) forgets to divide by $(1+i)$ (reports $D_{mac}$); (E) multiplies by $1.055$; (A) uses $\frac{8}{1.055^{2}}$.
  2. Price approximation
    A bond is priced at $P=\$4{,}200$ and has modified duration $5.6$. Using the first-order (modified-duration) approximation, estimate the new price if the effective annual yield increases by $40$ basis points. (A) $\$4{,}106$ (B) $\$4{,}152$ (C) $\$4{,}176$ (D) $\$4{,}294$ (E) $\$4{,}306$
    **Answer: (A).** The first-order estimate is $\Delta P\approx -P\,D_{mod}\,\Delta i=-4200\times 5.6\times 0.0040=-94.08.$ New price $\approx 4200-94.08\approx \$4{,}105.9$, i.e. $\$4{,}106$. Distractor (D) uses the wrong sign (adds the change); (B) uses $\Delta i=0.0040$ but multiplies by $D_{mod}/2$ implicitly via a dropped factor; (C) treats $40$ bp as $0.0004\times$ a misread duration.
  3. Duration
    A perpetuity-immediate pays $\$500$ at the end of each year forever. The effective annual interest rate is $6\%$. Calculate the Macaulay duration of the perpetuity. (A) $15.67$ (B) $16.67$ (C) $17.00$ (D) $17.67$ (E) $18.67$
    **Answer: (D).** For a level perpetuity-immediate the Macaulay duration is $D_{mac}=\frac{1+i}{i}=\frac{1.06}{0.06}\approx 17.6667.$ The payment amount is irrelevant (it cancels). Distractor (B) uses $\frac{1}{i}=16.67$ (the modified-duration analogue / dropped numerator); (C) uses a rounded $\frac{1.02}{0.06}$ slip.
  4. Duration
    A $3$-year bond pays annual coupons of $\$60$ and redeems for $\$1{,}000$. The effective annual yield is $7\%$. Calculate the Macaulay duration of the bond. (A) $2.66$ (B) $2.79$ (C) $2.83$ (D) $2.92$ (E) $3.00$
    **Answer: (C).** Price: $P=\frac{60}{1.07}+\frac{60}{1.07^{2}}+\frac{1060}{1.07^{3}}=56.075+52.406+865.28=973.76.$ Weighted-time numerator: $\sum t\,v^{t}CF_t=\frac{1\cdot 60}{1.07}+\frac{2\cdot 60}{1.07^{2}}+\frac{3\cdot 1060}{1.07^{3}}=56.075+104.81+2595.85=2756.74.$ $D_{mac}=\frac{2756.74}{973.76}\approx 2.831.$ Distractor (A) is the modified duration $2.831/1.07$; (E) is the maturity; (B) results from discounting the redemption only $2$ years.
  5. Portfolio duration
    Portfolio X consists of two bonds. Bond 1 has market value $\$30{,}000$ and modified duration $4.0$. Bond 2 has market value $\$70{,}000$ and modified duration $9.5$. Calculate the modified duration of Portfolio X. (A) $6.75$ (B) $7.25$ (C) $7.85$ (D) $8.50$ (E) $9.50$
    **Answer: (C).** Portfolio duration is the market-value-weighted average of component durations: $D_{P}=\frac{30000(4.0)+70000(9.5)}{30000+70000}=\frac{120000+665000}{100000}=\frac{785000}{100000}=7.85.$ Distractor (A) is the simple (unweighted) average $\tfrac{4.0+9.5}{2}=6.75$; (E) just reports the longer bond's duration.
  6. Portfolio duration
    An investor wants a bond portfolio with modified duration $6.0$, formed from a short bond with modified duration $2.5$ and a long bond with modified duration $11.0$. What fraction of the portfolio's market value should be placed in the long bond? (A) $0.353$ (B) $0.412$ (C) $0.500$ (D) $0.545$ (E) $0.647$
    **Answer: (B).** Let $w$ be the value-weight in the long bond. Then $2.5(1-w)+11.0\,w=6.0.$ $2.5+8.5\,w=6.0\;\Rightarrow\;8.5\,w=3.5\;\Rightarrow\;w=\frac{3.5}{8.5}\approx 0.4118.$ Distractor (E) solves for the weight in the short bond ($1-w=0.588$ rounding slip) or inverts the ratio; (A) uses $\frac{3.0}{8.5}$ from a misread target.
  7. Convexity
    A bond has price $P=\$2{,}500$, modified duration $D_{mod}=7$, and modified convexity $C=85$. The effective annual yield rises by $100$ basis points. Using the second-order (duration-plus-convexity) approximation, estimate the new price. (A) $\$2{,}314$ (B) $\$2{,}325$ (C) $\$2{,}336$ (D) $\$2{,}500$ (E) $\$2{,}675$
    **Answer: (C).** With $\Delta i=0.01$: $\frac{\Delta P}{P}\approx -D_{mod}\,\Delta i+\tfrac{1}{2}C(\Delta i)^{2}=-7(0.01)+\tfrac{1}{2}(85)(0.01)^{2}.$ $=-0.07+0.00425=-0.06575.$ $P_{new}\approx 2500(1-0.06575)=2500(0.93425)\approx \$2{,}335.6.$ Distractor (B) is the duration-only estimate $2500(1-0.07)=2325$ (convexity term omitted); (A) subtracts the convexity term instead of adding it (sign error); (E) uses the wrong direction (a yield drop); (D) leaves the price unchanged.
  8. Convexity
    A $5$-year zero-coupon bond has an effective annual yield of $4\%$. Calculate its modified convexity. (A) $22.20$ (B) $25.00$ (C) $26.67$ (D) $27.74$ (E) $30.00$
    **Answer: (D).** For a single payment at time $n$, the modified convexity is $C=\frac{n(n+1)}{(1+i)^{2}}=\frac{5\cdot 6}{1.04^{2}}=\frac{30}{1.0816}\approx 27.736.$ Distractor (E) $30.00$ forgets to divide by $(1+i)^{2}$; (B) $25.00$ uses $n^{2}$ in the numerator; (A) and (C) are smaller discounting/arithmetic slips on the numerator.