Exam FM — Annuities (Level) Flashcards

$a_{\overline{n|}}$ 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 $v=(1+i)^{-1}$.

$\ddot{a}_{\overline{n|}}$ 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.

$s_{\overline{n|}}=\dfrac{(1+i)^{n}-1}{i}$. Equivalently $s_{\overline{n|}}=a_{\overline{n|}}\,(1+i)^{n}$, since accumulating the present value forward $n$ periods gives the value at time $n$.

$\ddot{s}_{\overline{n|}}=\dfrac{(1+i)^{n}-1}{d}=s_{\overline{n|}}\,(1+i)$. It is valued one period *after* the last payment (at time $n$, while the last due-payment is at time $n-1$).

$\ddot{a}_{\overline{n|}}=(1+i)\,a_{\overline{n|}}$ 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.

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 $i=0.06,\ n=10$: $\ddot{a}_{\overline{10|}}=7.801692=1+a_{\overline{9|}}=1+6.801692$.

$a_{\overline{10|}}=\dfrac{1-(1.06)^{-10}}{0.06}=\dfrac{1-0.558395}{0.06}=7.360087$. PV $=1000\times 7.360087=\$7{,}360.09$.

$\ddot{a}_{\overline{10|}}=(1+i)\,a_{\overline{10|}}=1.06\times 7.360087=7.801692$. PV $=1000\times 7.801692=\$7{,}801.69$ (i.e. $\$441.60$ more than the immediate version because every payment lands one year earlier).

$s_{\overline{10|}}=\dfrac{(1.06)^{10}-1}{0.06}=\dfrac{1.790847-1}{0.06}=13.180795$. AV $=500\times 13.180795=\$6{,}590.40$.

$\ddot{s}_{\overline{10|}}=(1+i)\,s_{\overline{10|}}=1.06\times 13.180795=13.971643$. AV $=500\times 13.971643=\$6{,}985.82$.

$a_{\overline{\infty|}}=\dfrac{1}{i}$. 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.

$\ddot{a}_{\overline{\infty|}}=\dfrac{1}{d}=\dfrac{1+i}{i}=\dfrac{1}{i}+1$. It exceeds the perpetuity-immediate by exactly $1$ because the first payment is made immediately rather than one period later.

Immediate: PV $=\dfrac{2000}{0.08}=\$25{,}000$. Due: PV $=\dfrac{2000}{d}$ with $d=\dfrac{0.08}{1.08}=0.074074$, so PV $=\dfrac{2000}{0.074074}=\$27{,}000$. Equivalently $25{,}000\times 1.08=\$27{,}000$.

$d=\dfrac{0.05}{1.05}=0.047619$. PV $=\dfrac{500}{d}=\dfrac{500}{0.047619}=\$10{,}500$. (Check: $\dfrac{500}{0.05}+500=10000+500=10500$.)

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 $v^{k}\,a_{\overline{n|}}={}_{k|}a_{\overline{n|}}=a_{\overline{k+n|}}-a_{\overline{k|}}$.

This is a $15$-year annuity-immediate deferred $5$ years. $a_{\overline{15|}}=\dfrac{1-(1.05)^{-15}}{0.05}=10.379658$. PV $=1000\times v^{5}\,a_{\overline{15|}}=1000\times (1.05)^{-5}\times 10.379658=1000\times 0.783526\times 10.379658=\$8{,}132.73$.

Value the perpetuity one period before its first payment (at time $10$): $\dfrac{1000}{0.06}=16{,}666.67$. Discount $10$ years: PV $=16{,}666.67\times(1.06)^{-10}=16{,}666.67\times 0.558395=\$9{,}306.58$.

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 $a_{\overline{n|}}^{(m)}=\dfrac{1-v^{n}}{i^{(m)}}$.

Present value: $a_{\overline{n|}}^{(m)}=\dfrac{1-v^{n}}{i^{(m)}}=a_{\overline{n|}}\cdot\dfrac{i}{i^{(m)}}$. Accumulated value: $s_{\overline{n|}}^{(m)}=\dfrac{(1+i)^{n}-1}{i^{(m)}}=s_{\overline{n|}}\cdot\dfrac{i}{i^{(m)}}$. Here $n$ counts *years* and $i^{(m)}=m[(1+i)^{1/m}-1]$.

Total annual payment is $12{,}000$. $i^{(12)}=12[(1.08)^{1/12}-1]=12(0.006434)=0.077208$. $a_{\overline{5|}}^{(12)}=\dfrac{1-(1.08)^{-5}}{0.077208}=\dfrac{1-0.680583}{0.077208}=4.137075$. PV $=12{,}000\times 4.137075=\$49{,}644.90$.

$\bar{a}_{\overline{n|}}$ 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.

$\delta=\ln(1.06)=0.058269$. $\bar{a}_{\overline{8|}}=\dfrac{1-(1.06)^{-8}}{0.058269}=\dfrac{1-0.627412}{0.058269}=6.394279$. PV $=1000\times 6.394279=\$6{,}394.28$.

$s_{\overline{20|}}=\dfrac{(1.07)^{20}-1}{0.07}=\dfrac{3.869684-1}{0.07}=40.995492$. $X=\dfrac{50{,}000}{s_{\overline{20|}}}=\dfrac{50{,}000}{40.995492}=\$1{,}219.65$.

$a_{\overline{12|}}=\dfrac{1-(1.09)^{-12}}{0.09}=\dfrac{1-0.355535}{0.09}=7.160725$. $X=\dfrac{25{,}000}{a_{\overline{12|}}}=\dfrac{25{,}000}{7.160725}=\$3{,}491.27$.

$\dfrac{1-v^{n}}{i}=\dfrac{2000}{200}=10\ \Rightarrow\ 1-v^{n}=10(0.06)=0.6\ \Rightarrow\ v^{n}=0.4$. Since $v^{n}=(1.06)^{-n}$, 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 $n\approx 15.73$.

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.

$a_{\overline{n|}}=\dfrac{10{,}000}{1500}=6.6667\Rightarrow v^{n}=1-i a_{\overline{n|}}=1-0.53333=0.46667\Rightarrow n=9.90$, so $9$ full payments then a drop at $t=10$. Balance at $t=9$: $10{,}000(1.08)^{9}-1500\,s_{\overline{9|}}=19{,}990.05-18{,}731.34=1{,}258.71$. Drop at $t=10$: $1{,}258.71\times 1.08=\$1{,}359.41$.

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 $f(i)=P\,a_{\overline{n|}}-L$.

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: $N=12,\ PV=-50000,\ PMT=6000\Rightarrow I/Y\approx 6.11$).

Accumulate the first $6$ payments and discount the last $4$ to time $6$: CV $=100\,(s_{\overline{6|}}+a_{\overline{4|}})=100(6.801913+3.545951)=\$1{,}034.79$. Check: $100\,a_{\overline{10|}}(1.05)^{6}=100(7.721735)(1.340096)=\$1{,}034.79$.

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.)

$i^{(m)}$ is the nominal rate of *interest* convertible $m$-thly and $d^{(m)}$ the nominal rate of *discount* convertible $m$-thly. The chain is $\left(1+\dfrac{i^{(m)}}{m}\right)^{m}=1+i=(1-d)^{-1}=\left(1-\dfrac{d^{(m)}}{m}\right)^{-m}=e^{\delta}$.

Immediate: $\dfrac{40{,}000}{0.05}=\$800{,}000$. Due: $\dfrac{40{,}000}{d}$ with $d=\dfrac{0.05}{1.05}=0.047619$, giving $\dfrac{40{,}000}{0.047619}=\$840{,}000$ (equivalently $800{,}000\times 1.05$).

Deferral period is $8$ years (first payment at time $9$ is one period after time $8$). $a_{\overline{20|}}=\dfrac{1-(1.06)^{-20}}{0.06}=11.469921$. PV $=2000\times v^{8}a_{\overline{20|}}=2000\times(1.06)^{-8}\times 11.469921=2000\times 0.627412\times 11.469921=\$14{,}392.73$.

$a_{\overline{15|}}=\dfrac{1-(1.08)^{-15}}{0.08}=8.559479$, so $\ddot{a}_{\overline{15|}}=1.08\times 8.559479=9.244238$; PV $=1200\times 9.244238=\$11{,}093.09$. $s_{\overline{15|}}=\dfrac{(1.08)^{15}-1}{0.08}=27.152114$, so $\ddot{s}_{\overline{15|}}=1.08\times 27.152114=29.324283$; AV $=1200\times 29.324283=\$35{,}189.14$.

$s_{\overline{15|}}=\dfrac{(1.05)^{15}-1}{0.05}=\dfrac{2.078928-1}{0.05}=21.578564$. Deposit $=\dfrac{100{,}000}{21.578564}=\$4{,}634.23$.

Equate present values: $1000\,a_{\overline{10|}}=P\,\ddot{a}_{\overline{10|}}=P(1+i)\,a_{\overline{10|}}$. The $a_{\overline{10|}}$ 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.)

The per-period rate is $\dfrac{i^{(2)}}{2}=0.04$ over $n=16$ half-years. $a_{\overline{16|}}$ at $4\%=\dfrac{1-(1.04)^{-16}}{0.04}=\dfrac{1-0.533908}{0.04}=11.652296$. PV $=5000\times 11.652296=\$58{,}261.48$.

Immediate: payment $=60{,}000\times i=60{,}000\times 0.045=\$2{,}700$ per year. Due: 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).

$\ddot{a}_{\overline{10|}}=(1.06)\,a_{\overline{10|}}=1.06\times 7.360087=7.801692$. $X=\dfrac{30{,}000}{\ddot{a}_{\overline{10|}}}=\dfrac{30{,}000}{7.801692}=\$3{,}845.32$.

$a_{\overline{n|}}=\dfrac{7023.58}{1000}=7.02358\Rightarrow 1-v^{n}=i\,a_{\overline{n|}}=0.07(7.02358)=0.491651\Rightarrow v^{n}=0.508349$. Since $v^{n}=(1.07)^{-n}$, take logs: $n=-\dfrac{\ln 0.508349}{\ln 1.07}=-\dfrac{-0.676587}{0.067659}=10.00$. So $n=10$ years.

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: $i=\left(1+\dfrac{i^{(12)}}{12}\right)^{12}-1=(1.005)^{12}-1=0.061678$. Then value an ordinary $10$-year annuity-immediate at that effective $i$: $a_{\overline{10|}}=\dfrac{1-(1.061678)^{-10}}{0.061678}=\dfrac{1-0.549633}{0.061678}=7.301933$. PV $=5000\times 7.301933=\$36{,}509.67$. (Do **not** use the nominal $6\%$ directly — that would over-discount by ignoring monthly compounding between payments.)

For a fixed total annual amount and term, present value increases as payments are made *earlier/more often*: $a_{\overline{n|}}<a^{(m)}_{\overline{n|}}<\bar{a}_{\overline{n|}}<\ddot{a}_{\overline{n|}}$. The continuous annuity beats the immediate and $m$-thly versions but the annuity-*due* (all payments at period starts) has the largest PV of these. Equivalently the denominators satisfy $i>i^{(m)}>\delta>d^{(m)}>d$.