{ "deckName": "Exam FM — Annuities (Level)", "examCode": "Exam FM", "cards": [ { "front": "What does the symbol $a_{\\overline{n|}}$ represent, and what is its formula at interest rate $i$ per period?", "back": "$a_{\\overline{n|}}$ is the present value, one period before the first payment, of an **annuity-immediate**: $n$ level payments of $1$ made at the *end* of each period. The formula is $a_{\\overline{n|}}=\\dfrac{1-v^{n}}{i}$, where $v=(1+i)^{-1}$.", "tag": "annuity-immediate" }, { "front": "What does $\\ddot{a}_{\\overline{n|}}$ represent, and what is its formula?", "back": "$\\ddot{a}_{\\overline{n|}}$ is the present value of an **annuity-due**: $n$ level payments of $1$ made at the *beginning* of each period. The formula is $\\ddot{a}_{\\overline{n|}}=\\dfrac{1-v^{n}}{d}$, where $d=\\dfrac{i}{1+i}=iv$ is the effective rate of discount.", "tag": "annuity-due" }, { "front": "Give the formula for the accumulated value $s_{\\overline{n|}}$ of an annuity-immediate of $1$ per period, valued at the time of the last payment.", "back": "$s_{\\overline{n|}}=\\dfrac{(1+i)^{n}-1}{i}$. Equivalently $s_{\\overline{n|}}=a_{\\overline{n|}}\\,(1+i)^{n}$, since accumulating the present value forward $n$ periods gives the value at time $n$.", "tag": "accumulated-value" }, { "front": "Give the formula for $\\ddot{s}_{\\overline{n|}}$, the accumulated value of an annuity-due, and say at what time it is valued.", "back": "$\\ddot{s}_{\\overline{n|}}=\\dfrac{(1+i)^{n}-1}{d}=s_{\\overline{n|}}\\,(1+i)$. It is valued one period *after* the last payment (at time $n$, while the last due-payment is at time $n-1$).", "tag": "accumulated-value" }, { "front": "State the conversion relationship between an annuity-due and an annuity-immediate of the same term and rate.", "back": "$\\ddot{a}_{\\overline{n|}}=(1+i)\\,a_{\\overline{n|}}$ and likewise $\\ddot{s}_{\\overline{n|}}=(1+i)\\,s_{\\overline{n|}}$. Each due payment occurs one period earlier than the corresponding immediate payment, so its value is multiplied by $(1+i)$. **Multiply** immediate by $(1+i)$ to get due; do not divide.", "tag": "relationships" }, { "front": "Prove/state the identity $\\ddot{a}_{\\overline{n|}}=1+a_{\\overline{n-1|}}$.", "back": "An $n$-payment annuity-due pays $1$ now plus $1$ at the start of each of the next $n-1$ periods; those later payments form an annuity-immediate of term $n-1$. Hence $\\ddot{a}_{\\overline{n|}}=1+a_{\\overline{n-1|}}$. Numeric check at $i=0.06,\\ n=10$: $\\ddot{a}_{\\overline{10|}}=7.801692=1+a_{\\overline{9|}}=1+6.801692$.", "tag": "relationships" }, { "front": "Calculate the present value of an annuity-immediate paying $\\$1{,}000$ at the end of each year for $10$ years, at $i=6\\%$.", "back": "$a_{\\overline{10|}}=\\dfrac{1-(1.06)^{-10}}{0.06}=\\dfrac{1-0.558395}{0.06}=7.360087$.\nPV $=1000\\times 7.360087=\\$7{,}360.09$.", "tag": "annuity-immediate" }, { "front": "Calculate the present value of an annuity-due paying $\\$1{,}000$ at the start of each year for $10$ years, at $i=6\\%$.", "back": "$\\ddot{a}_{\\overline{10|}}=(1+i)\\,a_{\\overline{10|}}=1.06\\times 7.360087=7.801692$.\nPV $=1000\\times 7.801692=\\$7{,}801.69$ (i.e. $\\$441.60$ more than the immediate version because every payment lands one year earlier).", "tag": "annuity-due" }, { "front": "Calculate the accumulated value at time $10$ of $\\$500$ deposited at the end of each year for $10$ years, at $i=6\\%$.", "back": "$s_{\\overline{10|}}=\\dfrac{(1.06)^{10}-1}{0.06}=\\dfrac{1.790847-1}{0.06}=13.180795$.\nAV $=500\\times 13.180795=\\$6{,}590.40$.", "tag": "accumulated-value" }, { "front": "A fund receives $\\$500$ at the *beginning* of each year for $10$ years at $i=6\\%$. What is its value at time $10$ (one year after the last deposit)?", "back": "$\\ddot{s}_{\\overline{10|}}=(1+i)\\,s_{\\overline{10|}}=1.06\\times 13.180795=13.971643$.\nAV $=500\\times 13.971643=\\$6{,}985.82$.", "tag": "accumulated-value" }, { "front": "What is the present value of a perpetuity-immediate of $1$ per period at rate $i$, and why?", "back": "$a_{\\overline{\\infty|}}=\\dfrac{1}{i}$. Taking the limit of $\\dfrac{1-v^{n}}{i}$ as $n\\to\\infty$ sends $v^{n}\\to 0$, leaving $\\dfrac{1}{i}$. Intuitively, interest $i$ on a principal of $1/i$ exactly funds a payment of $1$ each period forever.", "tag": "perpetuity" }, { "front": "What is the present value of a perpetuity-due of $1$ per period at rate $i$?", "back": "$\\ddot{a}_{\\overline{\\infty|}}=\\dfrac{1}{d}=\\dfrac{1+i}{i}=\\dfrac{1}{i}+1$. It exceeds the perpetuity-immediate by exactly $1$ because the first payment is made immediately rather than one period later.", "tag": "perpetuity" }, { "front": "Calculate the price of a perpetuity-immediate paying $\\$2{,}000$ per year if $i=8\\%$. Then give the price of the same payments as a perpetuity-due.", "back": "Immediate: PV $=\\dfrac{2000}{0.08}=\\$25{,}000$.\nDue: PV $=\\dfrac{2000}{d}$ with $d=\\dfrac{0.08}{1.08}=0.074074$, so PV $=\\dfrac{2000}{0.074074}=\\$27{,}000$. Equivalently $25{,}000\\times 1.08=\\$27{,}000$.", "tag": "perpetuity" }, { "front": "A perpetuity-due pays $\\$500$ now and at the start of every subsequent year forever. At $i=5\\%$, what is its present value?", "back": "$d=\\dfrac{0.05}{1.05}=0.047619$.\nPV $=\\dfrac{500}{d}=\\dfrac{500}{0.047619}=\\$10{,}500$. (Check: $\\dfrac{500}{0.05}+500=10000+500=10500$.)", "tag": "perpetuity" }, { "front": "Define a deferred annuity and give the present-value formula for an annuity-immediate of $1$ per period for $n$ periods, deferred $k$ periods.", "back": "A **deferred annuity** has its payments begin later than usual. The present value today of an annuity-immediate deferred $k$ periods (first payment at time $k+1$) is $v^{k}\\,a_{\\overline{n|}}={}_{k|}a_{\\overline{n|}}=a_{\\overline{k+n|}}-a_{\\overline{k|}}$.", "tag": "deferred" }, { "front": "Find the present value today of an annuity that pays $\\$1{,}000$ at the end of each year for $15$ years, with the first payment at the end of year $6$. Use $i=5\\%$.", "back": "This is a $15$-year annuity-immediate deferred $5$ years. $a_{\\overline{15|}}=\\dfrac{1-(1.05)^{-15}}{0.05}=10.379658$.\nPV $=1000\\times v^{5}\\,a_{\\overline{15|}}=1000\\times (1.05)^{-5}\\times 10.379658=1000\\times 0.783526\\times 10.379658=\\$8{,}132.73$.", "tag": "deferred" }, { "front": "A perpetuity-immediate pays $\\$1{,}000$ per year, with the first payment at the end of year $11$. At $i=6\\%$, what is its present value today?", "back": "Value the perpetuity one period before its first payment (at time $10$): $\\dfrac{1000}{0.06}=16{,}666.67$. Discount $10$ years: PV $=16{,}666.67\\times(1.06)^{-10}=16{,}666.67\\times 0.558395=\\$9{,}306.58$.", "tag": "deferred" }, { "front": "How do you convert an annual effective rate $i$ into the appropriate per-payment rate for an annuity payable $m$ times per year, and what symbol multiplies the payments?", "back": "Define the nominal rate convertible $m$-thly by $i^{(m)}=m\\!\\left[(1+i)^{1/m}-1\\right]$; the per-period (per $\\frac{1}{m}$-year) effective rate is $i^{(m)}/m$. For a total of $1$ per year (i.e. $1/m$ each $m$-th), the present value is $a_{\\overline{n|}}^{(m)}=\\dfrac{1-v^{n}}{i^{(m)}}$.", "tag": "m-thly" }, { "front": "Give the present and accumulated value formulas for an annuity-immediate payable $m$-thly with total annual payment $1$, in terms of an annual effective rate $i$.", "back": "Present value: $a_{\\overline{n|}}^{(m)}=\\dfrac{1-v^{n}}{i^{(m)}}=a_{\\overline{n|}}\\cdot\\dfrac{i}{i^{(m)}}$.\nAccumulated value: $s_{\\overline{n|}}^{(m)}=\\dfrac{(1+i)^{n}-1}{i^{(m)}}=s_{\\overline{n|}}\\cdot\\dfrac{i}{i^{(m)}}$.\nHere $n$ counts *years* and $i^{(m)}=m[(1+i)^{1/m}-1]$.", "tag": "m-thly" }, { "front": "An annuity pays $\\$1{,}000$ at the end of *each month* for $5$ years. The annual effective rate is $i=8\\%$. Find the present value.", "back": "Total annual payment is $12{,}000$. $i^{(12)}=12[(1.08)^{1/12}-1]=12(0.006434)=0.077208$.\n$a_{\\overline{5|}}^{(12)}=\\dfrac{1-(1.08)^{-5}}{0.077208}=\\dfrac{1-0.680583}{0.077208}=4.137075$.\nPV $=12{,}000\\times 4.137075=\\$49{,}644.90$.", "tag": "m-thly" }, { "front": "What is a continuous annuity $\\bar{a}_{\\overline{n|}}$, and what is its formula?", "back": "$\\bar{a}_{\\overline{n|}}$ values a payment stream made *continuously* at a rate of $1$ per period for $n$ periods. Its present value is $\\bar{a}_{\\overline{n|}}=\\dfrac{1-v^{n}}{\\delta}$, where $\\delta=\\ln(1+i)$ is the force of interest.", "tag": "m-thly" }, { "front": "Money is paid continuously at an annual rate of $\\$1{,}000$ for $8$ years. At $i=6\\%$, find the present value.", "back": "$\\delta=\\ln(1.06)=0.058269$. $\\bar{a}_{\\overline{8|}}=\\dfrac{1-(1.06)^{-8}}{0.058269}=\\dfrac{1-0.627412}{0.058269}=6.394279$.\nPV $=1000\\times 6.394279=\\$6{,}394.28$.", "tag": "m-thly" }, { "front": "A deposit of $\\$X$ is made at the end of each year for $20$ years to accumulate to $\\$50{,}000$ at $i=7\\%$. Find $X$.", "back": "$s_{\\overline{20|}}=\\dfrac{(1.07)^{20}-1}{0.07}=\\dfrac{3.869684-1}{0.07}=40.995492$.\n$X=\\dfrac{50{,}000}{s_{\\overline{20|}}}=\\dfrac{50{,}000}{40.995492}=\\$1{,}219.65$.", "tag": "unknown-payment" }, { "front": "A loan of $\\$25{,}000$ is to be repaid by $X$ at the end of each year for $12$ years at $i=9\\%$. Find the level payment $X$.", "back": "$a_{\\overline{12|}}=\\dfrac{1-(1.09)^{-12}}{0.09}=\\dfrac{1-0.355535}{0.09}=7.160725$.\n$X=\\dfrac{25{,}000}{a_{\\overline{12|}}}=\\dfrac{25{,}000}{7.160725}=\\$3{,}491.27$.", "tag": "unknown-payment" }, { "front": "An annuity-immediate of $\\$200$ per year has a present value of exactly $\\$2{,}000$ at $i=6\\%$. Solve for the number of payments $n$.", "back": "$\\dfrac{1-v^{n}}{i}=\\dfrac{2000}{200}=10\\ \\Rightarrow\\ 1-v^{n}=10(0.06)=0.6\\ \\Rightarrow\\ v^{n}=0.4$.\nSince $v^{n}=(1.06)^{-n}$, take logs: $n=-\\dfrac{\\ln 0.4}{\\ln 1.06}=-\\dfrac{-0.916291}{0.058269}=15.73$. So a bit under $16$ payments are needed; the exact equation is solved by a non-integer $n\\approx 15.73$.", "tag": "unknown-term" }, { "front": "Explain what a balloon payment and a drop payment are when the solved term $n$ is not an integer.", "back": "When $n$ is non-integer, you make the largest whole number of level payments, then settle the small remaining balance. A **drop (smaller) payment** at the next regular date settles the leftover (final payment smaller than the level amount). A **balloon (larger) payment** instead adds the leftover to the last full payment (final payment larger). Both are valued so the equation of value balances exactly.", "tag": "unknown-term" }, { "front": "A loan of $\\$10{,}000$ at $i=8\\%$ is repaid by level annual payments of $\\$1{,}500$ plus a final smaller (drop) payment. Find the time and amount of the drop payment.", "back": "$a_{\\overline{n|}}=\\dfrac{10{,}000}{1500}=6.6667\\Rightarrow v^{n}=1-i a_{\\overline{n|}}=1-0.53333=0.46667\\Rightarrow n=9.90$, so $9$ full payments then a drop at $t=10$.\nBalance at $t=9$: $10{,}000(1.08)^{9}-1500\\,s_{\\overline{9|}}=19{,}990.05-18{,}731.34=1{,}258.71$.\nDrop at $t=10$: $1{,}258.71\\times 1.08=\\$1{,}359.41$.", "tag": "unknown-term" }, { "front": "What calculator/algebraic approaches solve for an unknown interest rate $i$ in a level annuity equation such as $P\\cdot a_{\\overline{n|}}=L$?", "back": "There is no closed-form solution for $i$. Use the BA II Plus **TVM keys** (enter $N$, $PV$, $PMT$, $FV$ and CPT $I/Y$) or the **CF/IRR worksheet**. By hand, use linear interpolation between two trial rates, or Newton's method on $f(i)=P\\,a_{\\overline{n|}}-L$.", "tag": "unknown-rate" }, { "front": "Set up (do not fully solve) the equation of value to find the yield rate $i$ if a $\\$50{,}000$ loan is repaid by $\\$6{,}000$ at the end of each year for $12$ years.", "back": "Solve $6000\\,a_{\\overline{12|}}=50{,}000$, i.e. $a_{\\overline{12|}}=\\dfrac{50{,}000}{6000}=8.3333$. Since $a_{\\overline{12|}}=\\dfrac{1-(1+i)^{-12}}{i}=8.3333$, iterate: the solution is $i\\approx 6.11\\%$ (BA II Plus: $N=12,\\ PV=-50000,\\ PMT=6000\\Rightarrow I/Y\\approx 6.11$).", "tag": "unknown-rate" }, { "front": "An annuity-immediate pays $\\$100$ per year for $10$ years at $i=5\\%$. Find its *current value* at the end of year $6$ (just after the 6th payment).", "back": "Accumulate the first $6$ payments and discount the last $4$ to time $6$:\nCV $=100\\,(s_{\\overline{6|}}+a_{\\overline{4|}})=100(6.801913+3.545951)=\\$1{,}034.79$.\nCheck: $100\\,a_{\\overline{10|}}(1.05)^{6}=100(7.721735)(1.340096)=\\$1{,}034.79$.", "tag": "relationships" }, { "front": "Convert: if an annuity-immediate has present value $\\$8{,}000$ at $i=7\\%$, what is the present value of the corresponding annuity-due (same payments and term)?", "back": "Each payment moves one period earlier, so multiply by $(1+i)$: PV(due) $=8{,}000\\times 1.07=\\$8{,}560$. (This is exact regardless of $n$ — the whole stream simply shifts back one period.)", "tag": "relationships" }, { "front": "Define $i^{(m)}$ and $d^{(m)}$ and state the unifying equality that links $i$, $i^{(m)}$, $d$, $d^{(m)}$, and $\\delta$.", "back": "$i^{(m)}$ is the nominal rate of *interest* convertible $m$-thly and $d^{(m)}$ the nominal rate of *discount* convertible $m$-thly. The chain is $\\left(1+\\dfrac{i^{(m)}}{m}\\right)^{m}=1+i=(1-d)^{-1}=\\left(1-\\dfrac{d^{(m)}}{m}\\right)^{-m}=e^{\\delta}$.", "tag": "m-thly" }, { "front": "A retirement account needs to fund a perpetuity-immediate of $\\$40{,}000$ per year starting one year after retirement. If the fund earns $i=5\\%$, how much must be in the account at retirement, and what if payments instead start *immediately* (perpetuity-due)?", "back": "Immediate: $\\dfrac{40{,}000}{0.05}=\\$800{,}000$.\nDue: $\\dfrac{40{,}000}{d}$ with $d=\\dfrac{0.05}{1.05}=0.047619$, giving $\\dfrac{40{,}000}{0.047619}=\\$840{,}000$ (equivalently $800{,}000\\times 1.05$).", "tag": "perpetuity" }, { "front": "A $20$-year annuity-immediate of $\\$2{,}000$ per year is deferred so that the first payment is at the end of year $9$. At $i=6\\%$, find the present value today.", "back": "Deferral period is $8$ years (first payment at time $9$ is one period after time $8$). $a_{\\overline{20|}}=\\dfrac{1-(1.06)^{-20}}{0.06}=11.469921$.\nPV $=2000\\times v^{8}a_{\\overline{20|}}=2000\\times(1.06)^{-8}\\times 11.469921=2000\\times 0.627412\\times 11.469921=\\$14{,}392.73$.", "tag": "deferred" }, { "front": "An annuity-due of $\\$1{,}200$ per year for $15$ years is valued at $i=8\\%$. Find both its present value and its accumulated value at time $15$.", "back": "$a_{\\overline{15|}}=\\dfrac{1-(1.08)^{-15}}{0.08}=8.559479$, so $\\ddot{a}_{\\overline{15|}}=1.08\\times 8.559479=9.244238$; PV $=1200\\times 9.244238=\\$11{,}093.09$.\n$s_{\\overline{15|}}=\\dfrac{(1.08)^{15}-1}{0.08}=27.152114$, so $\\ddot{s}_{\\overline{15|}}=1.08\\times 27.152114=29.324283$; AV $=1200\\times 29.324283=\\$35{,}189.14$.", "tag": "annuity-due" }, { "front": "A sinking fund must accumulate $\\$100{,}000$ in $15$ years. Level deposits are made at the *end* of each year and the fund earns $i=5\\%$. Find the required deposit.", "back": "$s_{\\overline{15|}}=\\dfrac{(1.05)^{15}-1}{0.05}=\\dfrac{2.078928-1}{0.05}=21.578564$.\nDeposit $=\\dfrac{100{,}000}{21.578564}=\\$4{,}634.23$.", "tag": "unknown-payment" }, { "front": "Two annuities are equivalent in value at $i=6\\%$: Annuity A pays $\\$1{,}000$ at the end of each year for $10$ years; Annuity B pays a level $\\$P$ at the *start* of each year for $10$ years. Find $P$.", "back": "Equate present values: $1000\\,a_{\\overline{10|}}=P\\,\\ddot{a}_{\\overline{10|}}=P(1+i)\\,a_{\\overline{10|}}$.\nThe $a_{\\overline{10|}}$ cancels: $P=\\dfrac{1000}{1+i}=\\dfrac{1000}{1.06}=\\$943.40$. (Annuity-due payments can be smaller because they earn an extra period of interest.)", "tag": "relationships" }, { "front": "An annuity pays $\\$5{,}000$ at the end of each *half-year* for $8$ years. Interest is $i^{(2)}=8\\%$ convertible semiannually. Find the present value.", "back": "The per-period rate is $\\dfrac{i^{(2)}}{2}=0.04$ over $n=16$ half-years. $a_{\\overline{16|}}$ at $4\\%=\\dfrac{1-(1.04)^{-16}}{0.04}=\\dfrac{1-0.533908}{0.04}=11.652296$.\nPV $=5000\\times 11.652296=\\$58{,}261.48$.", "tag": "m-thly" }, { "front": "A perpetuity-immediate is purchased for $\\$60{,}000$ and pays a level amount forever at $i=4.5\\%$. What is the annual payment? What level payment would the same price buy as a perpetuity-due?", "back": "Immediate: payment $=60{,}000\\times i=60{,}000\\times 0.045=\\$2{,}700$ per year.\nDue: payment $=60{,}000\\times d=60{,}000\\times\\dfrac{0.045}{1.045}=60{,}000\\times 0.043062=\\$2{,}583.73$ per year (smaller, since due payments arrive earlier).", "tag": "perpetuity" }, { "front": "A $\\$30{,}000$ loan at $i=6\\%$ is repaid by $X$ at the *beginning* of each year for $10$ years (annuity-due). Find $X$.", "back": "$\\ddot{a}_{\\overline{10|}}=(1.06)\\,a_{\\overline{10|}}=1.06\\times 7.360087=7.801692$.\n$X=\\dfrac{30{,}000}{\\ddot{a}_{\\overline{10|}}}=\\dfrac{30{,}000}{7.801692}=\\$3{,}845.32$.", "tag": "unknown-payment" }, { "front": "An annuity-immediate of $\\$1{,}000$ per year for $n$ years has present value $\\$7{,}023.58$ at $i=7\\%$. Solve for $n$.", "back": "$a_{\\overline{n|}}=\\dfrac{7023.58}{1000}=7.02358\\Rightarrow 1-v^{n}=i\\,a_{\\overline{n|}}=0.07(7.02358)=0.491651\\Rightarrow v^{n}=0.508349$.\nSince $v^{n}=(1.07)^{-n}$, take logs: $n=-\\dfrac{\\ln 0.508349}{\\ln 1.07}=-\\dfrac{-0.676587}{0.067659}=10.00$. So $n=10$ years.", "tag": "unknown-term" }, { "front": "An annuity pays $\\$5{,}000$ at the end of each *year* for $10$ years, but interest is quoted as a nominal $6\\%$ convertible *monthly* ($i^{(12)}=6\\%$). Here the payment period (annual) is **less frequent** than the conversion period (monthly). Find the present value.", "back": "When the payment period is longer than the interest-conversion period, first collapse the interest basis to the *annual effective* rate that matches the once-a-year payments: $i=\\left(1+\\dfrac{i^{(12)}}{12}\\right)^{12}-1=(1.005)^{12}-1=0.061678$.\nThen value an ordinary $10$-year annuity-immediate at that effective $i$: $a_{\\overline{10|}}=\\dfrac{1-(1.061678)^{-10}}{0.061678}=\\dfrac{1-0.549633}{0.061678}=7.301933$.\nPV $=5000\\times 7.301933=\\$36{,}509.67$. (Do **not** use the nominal $6\\%$ directly — that would over-discount by ignoring monthly compounding between payments.)", "tag": "m-thly" }, { "front": "A continuously-paid annuity and an annuity-due both pay a total of $\\$12{,}000$ per year. At $i=6\\%$ over $10$ years, which has the larger present value, and what general ordering holds among $\\bar{a}$, $a^{(m)}$, $a$, and $\\ddot{a}$?", "back": "For a fixed total annual amount and term, present value increases as payments are made *earlier/more often*: $a_{\\overline{n|}}i^{(m)}>\\delta>d^{(m)}>d$.", "tag": "m-thly" } ] }