An Annuity is Continuously Varying and Payable for 23 Years
Problem 1
Find the accumulated value 18 years after the first payment is made of an annuity on which there are 8 payments of $\$ 2000$ each made at two-year intervals. The nominal rate of interest convertible semiannually is $7 \%$. Answer to the nearest dollar.
Problem 2
Find the present value of a ten-year annuity which pays $\$ 400$ at the beginning of each quarter for the first 5 years, increasing to $\$ 600$ per quarter thereafter. The annual effective rate of interest is $12 \%$. Answer to the nearest dollar.
Problem 3
A sum of $\$ 100$ is placed into a fund at the beginning of every other year for eight years. If the fund balance at the end of eight years is $\$ 520,$ find the rate of simple interest earned by the fund.
Problem 4
Rework Exercise 1 using the approach developed in Section 4.3.
Problem 5
Give an expression in terms of functions assuming a rate of interest per month for the present value, 3 years before the first payment is made, of an annuity on which there are payments of $\$ 200$ every 4 months for 12 years:
a) Expressed as an annuity-immediate.
b) Expressed as an annuity-due.
Problem 6
Show that the present value at time 0 of 1 payable at times $7,11,15,19,23,$ and 27 is
Problem 7
A perpetuity of $\$ 750$ payable at the end of every year and a perpetuity of $\$ 750$ payable at the end of every 20 years are to be replaced by an annuity of $R$ payable at the end of every year for 30 years. If $f^{(2)}=.04,$ show that
where all functions are evaluated at $2 \%$ interest.
Problem 8
Find an expression for the present value of an annuity-due of $\$ 600$ per annum payable semiannually for 10 years if $d^{(12)}=.09$.
Problem 9
The present value of a perpetuity paying 1 at the end of every three years is $125 / 91$ Find $l$.
Problem 10
Find an expression for the present value of an annuity on which payments are $\$ 100$ per quarter for five years, just before the first payment is made, if $\delta=.08$.
Problem 11
A perpetuity paying 1 at the beginning of each year has a present value of $20 .$ If this perpetuity is exchanged for another perpetuity paying $R$ at the beginning of every two years, find $R$ so that the values of the two perpetuities are equal.
Problem 12
Find an expression for the present value of an annuity on which payments are 1 at the beginning of each 4 -month period for 12 years, assuming a rate of interest per 3-month period.
Problem 13
Rework Exercise 2 using the approach developed in Section 4.4.
Problem 15
Derive the following formulas analogous to formulas (3.6) and (3.12)
$$\text { a) } \quad \frac{1}{a_{n}^{(m)}}=\frac{1}{s_{n}^{(m)}}+i^{(m)}$$
$$\text { b) } \quad \frac{1}{\dot{a}_{\bar{n}}^{(m)}}=\frac{1}{\dot{s}_{\bar{n}}^{(m)}}+d^{(m)}$$
Problem 16
Derive the following formulas analogous to formulas (3.13) and (3.14)
a) $\quad \vec{a}_{\vec{n} |}^{(m)}=a_{\vec{n} |}^{(m)}(1+i)^{1 / m}$
b) $\quad \dot{s}_{\bar{n}}^{(m)}=s_{\bar{n} |}^{(m)}(1+i)^{1 / m}$
Problem 17
Derive the following formulas analogous to formulas (3.15) and (3.16)
a) $\quad \vec{a}_{\vec{n} |}^{(m)}=1 / m+a \frac{(m)}{n-1)}$
b) $\quad \ddot{s}_{\bar{n} |}^{(m)}=s \frac{(m)}{n+1 / m}-1 / m$
Problem 18
Express $\ddot{a}_{n |}^{(12)}$ in terms of $a_{\vec{n} |}^{(2)}$ with an adjustment factor.
Problem 19
a) Show that $a_{n}^{(m)}=\frac{1}{m} \sum_{t=1}^{m} v^{t / m} \dot{a}_{\vec{n}}$
b) Verbally interpret the result obtained in $(a)$
Problem 20
A sum of $\$ 10,000$ is used to buy a deferred perpetuity-due paying $\$ 500$ every six months forever. Find an expression for the deferred period expressed as a function of $d$.
Problem 21
$$\text { If } 3 a_{\vec{n}}^{(2)}=2 a_{2 n}^{(2)}=45 s_{\mathrm{T}}^{(2)}, \text { find } i$$
Problem 22
Find an expression for the present value of an annuity which pays 1 at the beginning of each 3 -month period for 12 years, assuming a rate of interest per 4 -month period.
Problem 23
Find the value of $t, 0< t< 1,$ such that 1 paid at time $t$ is equivalent to 1 paid continuously between time 0 and 1.
Problem 24
Show algcbraically and verbally that $a_{\text {?? }}< a_{\vec{n} |}^{(m)}< \bar{a}_{\bar{n} |}< \ddot{a}_{\text {?? }}^{(m)}<\ddot{a}_{\text {?? }}$ where $m>1$
Problem 25
Find an expression for $\bar{a}_{\vec{n} |}$ if $\delta_{i}=\frac{1}{1+t}$
Problem 26
There is $\$ 40,000$ in a fund which is accumulating at $4 \%$ per annum convertible continuously. If money is withdrawn continuously at the rate of $\$ 2400$ per annum, how long will the fund last?
Problem 27
$$\text { If } \bar{a}_{\bar{n}}=4 \text { and } \bar{s}_{\bar{n} |}=12, \text { find } \delta$$.
Problem 28
$$\text { Show that } \frac{d}{d n} a_{\bar{n}}=v^{n / s}_{\mathrm{T}}$$
Problem 29
Verbally interpret the result obtained in Example 4.13.
Problem 30
$$\text { Simplify } \sum_{t=1}^{20}(t+5) v^{t}$$.
Problem 31
Show algebraically, and by means of a time diagram, the following relationship between $(I a)_{\vec{n}}$ and $(D a)_{\vec{n}}$
$$(D a)_{\vec{n} |}=(n+1) a_{\vec{n} |}-(I a)_{\vec{n} |}$$
Problem 32
The following payments are made under an annuity: 10 at the end of the fifth year, 9 at the end of the sixth year, decreasing by 1 each year until nothing is paid. Show that the present value is
$$\frac{10-a_{\overline{14}}+a_{4]}(1-10 i)}{i}$$
Problem 33
Find the present value of a perpetuity under which a payment of 1 is made at the end of the first year, 2 at the end of the second year, increasing until a payment of $n$ is made at the end of the $n$ th year, and thereafter payments are level at $n$ per year forever.
Problem 34
A perpetuity-immediate has annual payments of $1,3,5,7 \ldots$ If the present value of the sixth and seventh payments are equal, find the present value of the perpetuity.
Problem 35
If $X$ is the present value of a perpetuity of 1 per year with the first payment at the end of the second year and $20 X$ is the present value of a series of annual payments $1,2,3, \ldots$ with the first payment at the end of the third year, find $d$.
Problem 36
An annuity-immediate has semiannual payments of $800,750,700, \ldots, 350,$ at $f^{(2)}=.16 .$ If $a_{101.08}=A,$ find the present value of the annuity in terms of $A$.
Problem 37
Annual deposits are made into a fund at the beginning of each year for 10 years. The first 5 deposits are $\$ 1000$ each and deposits increase by $5 \%$ per year therafter. If the fund earns $8 \%$ effective, find the accumulated value at the end of 10 years. Answer to the nearest dollar.
Problem 38
Find the present value of a 20 -year annuity with annual payments which pays $\$ 600$ immediately and each subsequent payment is $5 \%$ greater than the preceding payment. The annual effective rate of interest is $10.25 \%$. Answer to the nearest dollar.
Problem 40
a) Find the sum of the payments in $(I a)_{2}^{(12)}$
b) Find the sum of the payments in $\left(I^{(12)} a\right)_{2}^{(12)}$
Problem 41
$$\text { Show that }\left(I^{(m)} a\right) \frac{(m)}{\infty}=\frac{1}{m\left(i^{(m)}-d^{(m)}\right)}$$.
Problem 42
Show that the present value of a perpetuity on which payments are 1 at the end of the 5 th and 6 th years, 2 at the end of the 7 th and 8 th years, 3 at the end of the 9 th and 10 th years is
$$\frac{v^{4}}{i-v d}$$
Problem 43
A perpetuity has payments at the end of each four-year period. The first payment at the end of four years is $1 .$ Each subsequent payment is 5 more than the previous payment. It is known that $v^{4}=0.75$. Calculate the present value of this perpetwity.
Problem 44
A 10 -year annuity has the following schedule of payments:
$$\begin{aligned}&\text { On each January } 1, \ldots \ldots \ldots \ldots \ldots .100\\
&\text { On each April 1 } \ldots \ldots \ldots \ldots \ldots \ldots .200\\
&\text { On each July } 1 \ldots \ldots . . . . . . . . . . . . \quad 300\\
&\text { On each October } 1, \ldots \ldots \ldots \ldots \ldots400\end{aligned}$$
Show that the present value of this annuity on January 1 just before the first payment is made is
$$1600 \ddot{a}_{\overline{10} |}\left(I^{(4)} \dot{a}\right)_{\overline{1}}^{(4)}$$
Problem 45
A perpetuity provides payments every six months starting today. The first payment is 1 and each payment is $3 \%$ greater than the immediately preceding payment. Find the present value of the perpetuity if the effective rate of interest is $8 \%$ per annum.
Problem 46
Find the ratio of the total payments made under $(\bar{I} \bar{a})_{\mathrm{T} 0}$ during the second half of the term of the annuity to those made during the first half.
Problem 47
$$\text { Evaluate }(\bar{I} \bar{a})_{\infty} \text { if } \delta=.08$$.
Problem 48
Payments under a continuous perpetuity are made at the periodic rate of $(1+k)^{t}$ at time $t .$ The annual effective rate of interest is $l,$ where $0< k< i .$ Find the present value of the perpetuity.
Problem 49
a) Find an integral expression for $(\bar{D} \bar{a})_{\vec{n}}$
b) Find an expression not involving integrals for $(\bar{D} \bar{a})_{\vec{n} |}$
Problem 50
A perpetuity is payable continuously at the annual rate of $1+t^{2}$ at time $t$ If $\delta=.05,$ find the present value of the perpetuity.
Problem 51
A one-year deferred continuous varying annuity is payable for 13 years. The rate of payment at time $t$ is $t^{2}-1$ per annum, and the force of interest at time $t$ is $(1+t)^{-1}$. Find the present value of the annuity.
Problem 52
a) (1) Show that $\frac{d}{d i} a_{\text {n }}=-v(I a)_{n}$
(2) Find $\frac{d^{*}}{d i} a_{\pi]}$ evaluated at $i=0$
b) (1) Show that $\frac{d}{d i} \bar{a}_{\bar{n} |}=-v(\bar{I} \bar{a})_{\pi |}$
(2) Find $\frac{d}{d i} \bar{a}_{\bar{n}}$ evaluated at $l=0$
Problem 53
$$\text { a) Show that } \frac{i}{i^{(m)}}=\frac{1}{1-\frac{m-1}{2 m} i} \div 1+\frac{m-1}{2 m} i$$
b) Derive the following approximate equality:
$$a_{\vec{n} |}^{(m)} \leftrightharpoons a_{\vec{n}}+\frac{m-1}{2 m}\left(1-v^{n}\right)$$
Problem 54
For a given $n,$ it is known that $\bar{a}_{\bar{n} |}=n-4$ and $\delta=10 \% .$ Find $\int_{0}^{n} \bar{a}_{7} d t$
Problem 55
Find expressions for:
a) $\sum_{i=1}^{n}(l a)_{t}$
b) $\sum_{t=1}^{n}(D a)_{\bar{t}}$
Problem 56
A family wishes to provide an annuity of $\$ 100$ at the end of each month to their daughter now entering college. The annuity will be paid for only nine months each year for four years. Prove that the present value one month before the first payment is
$$1200 \ddot{a}_{4} \text { a } \frac{(12)}{9 / 12}$$
Problem 57
Show that
$$\sum_{r=1}^{h}\left(a_{\bar{r} R}+s_{\bar{r} \bar{k}\rceil}\right)=\frac{1}{i}\left(\frac{\left.s_{\hbar \bar{k}}\right\rceil}{\left.a_{\bar{k}}\right]}-\frac{\left.a_{\bar{h} \bar{k}}\right\rceil}{s_{\bar{k}}}\right)$$
Problem 58
There are two perpetuitics. The first has level payments of $p$ at the end of each year. The second is increasing such that the payments are $q, 2 q, 3 q, \ldots$ Find the rate of interest which will make the difference in present value between these perpetuities:
a) Zero.
b) A maximum.
Problem 59
Fence posts set in soil last 9 years and cost $\$ 2 .$ Posts set in concrete last 15 years and cost $\$ 2+X .$ The posts will be needed for 35 years. Show that the break-even value of $X,$ i.e. the value at which a buyer would be indifferent between the two types of posts is
$$2\left(\frac{\left.a_{36}\right|^{a_{15}}}{a_{91} a_{451}}-1\right)$$
Problem 60
$$\text { If } \bar{a}_{\bar{n}}=a \text { and } \bar{a}_{2 n}=b, \text { express }(\bar{I} \bar{a})_{\bar{n} |} \text { in terms of } a \text { and } b$$.
Source: https://www.numerade.com/books/chapter/more-general-annuities/
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