2 - Valuation of Annuities

Question 1

What is an annuity?

Answer 1

The generic term used to describe a series of periodic payments.

Question 2

What is an annuity in a life insurance context?

Answer 2

In a life insurance context, an annuity is a “life-contingent” series of payments that are contingent on the survival of a specific individual or group of individuals.

Question 3

What is an annuity-certain?

Answer 3

The more precise term for a series of payments that are not contingent on the occurrence of any specified events (an annuity whose payments will definitely by made)

Question 4

What will we mean in this book when talking about annuities?

Answer 4

Annuity-certains.

Question 5

What is a key algebraic relationship used in valuing a series of payments?

Answer 5

The geometric series summation formula
1 + x + x^2 + … + x^k =
= (1-x^(k+1)) / (1-x) =
= (x^(k+1)-1) / (x-1)

Question 6

Let’s consider a series of n payments (or deposits) of amount 1 each, made at equally spaced intervals of time, and for which interest is at compound rate i per payment period, with interest credited on payment dates. The accumulated value of the series off payments, valued at the time of (an including) the final payment , can be represented as the sum of the accumulated values of the individual payments. Which expression captures this?

Answer 6

(1+i)^(n-1) + (1+i)^(n-2) + … + (1+i) + 1 =
= ((1+i)^(n) -1) / i
Since the valuation point is the time that the nth deposit is made, this is actually n-1 periods after the first deposit. Therefore the first deposit has grown with compound interest for n-1 periods.

Question 7

What is standard actuarial notation and terminology associated with these annuities?

Answer 7

The symbol Sn|i (where n|i is really meant to be a symbol where the n is enclosed on top and to the right of it like a half box) denotes the accumulated value, at the time of (and including) the final payment of a series of n payments of 1 each made at equally spaced intervals of time, where the rate of interest per payment period is i.

Question 8

What is the equation for the accumulated value of an n-payment annuity-immediate of 1 per period?

Answer 8

Sn|i = (1+i)^(n-1) + … + (1+i) + 1 =

= Sig t=0 to (n-1) of (1+i)^t = ((1+i)^n - 1) / i

Question 9

What is the number of payments in the series called?

Answer 9

The term of an annuity.

Question 10

What is the time between successive payments called?

Answer 10

The payment period, or frequency.

Question 11

What should we note about any interest rate i?

Answer 11

Note that for any interest rate i, S1|i = 1, but if i>0 and n>1, then Sn|i > n because of interest on earlier deposits.

Question 12

It should be emphasized that the Sn|i notation can be used to express the accumulated value of an annuity provided which 3 conditions are met?

Answer 12

1. The payments are of equal amount.
2. The payments are made at equal intervals of time, with the same frequency as the interest rate is compounded.
3. The accumulated value is found at the time of and including the final payment.

Question 13

It should be emphasized that the Sn|i notation can be used to express the accumulated value of an annuity provided 3 conditions are met:
1. The payments are of equal amount.
2. The payments are made at equal intervals of time, with the same frequency as the interest rate is compounded.
3. The accumulated value is found at the time of and including the final payment.
How is this series of payments referred to in actuarial terminology?

Answer 13

As an accumulated annuity-immediate.

Question 14

We often see a series of payments described with phrases “‘payments occur at the end of each year (or month),” with a valuation made at the end of n years. What is the conventional interpretation of this phrase?

Answer 14

The conventional interpretation of this phrase is to regard the valuation as an accumulated annuity-immediate.

Question 15

How can we re-write the equation
Sn|i = (1+i)^(n-1)+…+(1+i)+1 =
= Σt=0 to (n-1) of (1+i)^t =
= ((1+i)^n -1) / i

Answer 15

(1+i)^n = iSn|i + 1 = i[(1+i)^(n-1) + … + (1+i) + 1] + 1

Question 16

How can we interpret the expression (1+i)^n = i*Sn|i + 1 =

=i*[(1+i)^(n-1)+…+(1+i)+1] +1?

Answer 16

Suppose that a single amount of 1 is invested at time 0 at periodic interest rate i, so that an interest payment of i is generated at the end of each period. Suppose further that each interest payment is reinvested and continues to earn interest at rate i. This is allowed to continue for n periods. Then the accumulation of the reinvested interest, along with the return of the initial amount 1 invested (the right hang side of the equation above), must be equal to the compound accumulation of 1 at rate i per period invested for n periods.

Question 17

We have seen that the value at the time of the nth deposit of a series of n deposits of amount 1 each is Sn|i = ((1+i)^n - 1) / i. If there are no further deposits, but the balance continues to grow with compound interest, then the accumulated value k periods after the nth deposit is (1+i)^(n-1) +(1+i)^(n-2)+…+(1+i) +1^k =
= (1+i)^(n+k-1) +(1+i)^(n+k-2)+…+(1+i)^(k+1)+(1+i)^k
= (((1+i)^n -1)/i))*((1+i)^k)
= Sn| * (1+i)^k =
= Value at time n * growth factor from time n to time n+k. (see page 78 of textbook)
How else can this also be represented?

Answer 17

Sn| * (1+i)^k =
= (((1+i)^n-1)/i)*(1+i)^k =
= ((1+i)^(n+k) - (1+i)^k)/i =
= [(1+i)^(n+k) - 1]/i  - [((1+i)^k)-1]/i =
= S(n+k)| - Sk|

Question 18

How can we explain
Sn| * (1+i)^k =
= (((1+i)^n-1)/i)*(1+i)^k =
= ((1+i)^(n+k) - (1+i)^k)/i =
= [(1+i)^(n+k) - 1]/i  - [((1+i)^k)-1]/i =
= S(n+k)| - Sk|?

Answer 18

If the annuity payments had continued to time n+k, which is the time of valuation, the accumulated value would be S(n+k)|. Since there are not any payments actually made for the final k payment periods, S(n+k)| must be reduced by Sk|, the accumulated value of k payments of 1 each ending at time n+k.

Question 19

How can we reformulate the relationship
Sn| * (1+i)^k =
= … =
= S(n+k)| - Sk|?

Answer 19

S(n+k)| =
= Sn| * (1+i)^k + Sk| =
= Sk| * (1+i)^n + Sn|

Question 20

What does the following equation show?
S(n+k)| =
= Sn| * (1+i)^k + Sk| =
= Sk| * (1+i)^n + Sn|

Answer 20

That a series of payments can be separated into components, and the accumulated value of the entire series at a valuation point can be represented as the sum of the accumulated values (at that valuation point) of the separate component series.

Question 21

The concept of dividing a series of payments into subgroups and valuing each subgroup separately can be applied to find the accumulated value of an annuity when the periodic interest rate changes during the term of the annuity.

Answer 21

In a situation in which the interest rate is at one level for a period of time and changes to another level for a subsequent period of time, it is necessary to separate the full term into separate time intervals over which the interest rate is constant.

Question 22

We can generalize the concept presented in Ex.2.4. Suppose that we consider an n+k payment annuity with equally spaced payments of 1 per period up to the time of the nth payment, followed by an interest rate of i2 per payment period from the time of the nth payment onward. How can the accumulated value of the annuity at the time of the final payment can be found?

Answer 22

a) The accumulated value of the first n payments valued at the time of the nth payment is sn|i1.
b)The accumulated value found in part a) grows with compound interest for an additional k periods at compound period interest rate i2, to a value of
sn|i1 * (1+i2)^k as of time n+k
c) The accumulated value of the final k payments is s|i2.
d) The total accumulated value at time n+k is the sum of b) and c), and equals
sn|i1 * (1+i2)^k + sk|i2

Question 23

a) The accumulated value of the first n payments valued at the time of the nth payment is sn|i1.
b)The accumulated value found in part a) grows with compound interest for an additional k periods at compound period interest rate i2, to a value of
sn|i1 * (1+i2)^k as of time n+k
c) The accumulated value of the final k payments is s|i2.
d) The total accumulated value at time n+k is the sum of b) and c), and equals
sn|i1 * (1+i2)^k + sk|i2
What is the use of this?

Answer 23

This method can be extended to situations in which the interest rate changes more than once during the term of the annuity.

Question 24

This previous method can be extended to situations in which the interest rate changes more than once during the term of the annuity. What else can it also be extended to?

Answer 24

To find the accumulated value of an annuity for which the payment amount changes during the course of the annuity.

Question 25

The previous discussion has been concerned with formulating and calculating the accumulated value of a series of payments. We now consider the present value of an annuity, which is a valuation of a series of payments some time before they begin. What can be seen from Ex.2.6 where someone wishes to open a bank account with a single deposit today that her grandchild can withdraw $1000 each year for four years starting in a year?

Answer 25

That the deposit amount needed is the combined present value of the four withdrawals that will be made. The present value of an annuity of payments is the value of the payments at the time, or some time before, the payments begin.

Question 26

Consider again a series of n payments of amount 1 each, made at equally spaced intervals for which interest is at effective interest rate i per payment period. How can the present value of the series of payments, valued one period before the first payment, be represented as the sum of the present values of the individual payments?

Answer 26

1/(1+i) + 1/(1+i)^2+...+ 1/(1+i)^(n-1) + 1/(1+i)^n =
= v + v^2+...+ v^(n-1) + v^n =
= v [1+v+...+v^(n-1)] =
= v*[(1-v^n)/(1-v)] =
= (1-v^n) / [(1+i)*(1-(1/(1+i)))] =
= (1-v^n) / (1+i-1) =
= (1-v^n) / i

Question 27

It is often the case that, as in Ex.6, a valuation of a series of payments is done one period before the first payment. There is a specific actuarial symbol that represents the present value of such an annuity. What is it?

Answer 27

an|i

where the n|i is the same half box symbol used in the Sn|i notation

Question 28

What is Definition 2.2 - Present Value of an n-Payment Annuity-Immediate of 1 Per Period?

Answer 28

The symbol an|i is specifically used to denote the present value of a series of equally spaced payments of amount 1 each when the valuation point is one payment period before the payments begin. 
an|i = v+v^2+...+v^n =
= Σfrom i=1 to n of v^t =
= Σv^t =
= (1-v^n) / i

Question 29

Similar to the use of the notation sn|, the symbol an| can be used to express the present value of an annuity provided which conditions are met?

Answer 29

1. There are n payments of equal amount.
2. The payments are made at equal intervals of time, with the same frequency as the frequency of interest compounding.
3. The valuation point is one payment period before the first payment is made.

Question 30

What is a typical situation in which the present value of an annuity-immediate arises?

Answer 30

The repayment of a loan.

Question 31

A typical situation in which the present value of an annuity-immediate arises is the repayment of a loan. Explain.

Answer 31

In financial practice, a loan being repaid with a series of payments is structured so that the original loan amount advanced to the borrower is equal to the present value of the loan payments to be made by the borrower, and the first loan repayment is made one period after the loan is made. The present value is calculated using the loan interest rate?

Question 32

What is the relationship between the present value of an annuity-immediate and the level of interest?

Answer 32

As interest rates increase, present value decreases.

Question 33

What is the relationship between the present value of an annuity-immediate and the number of payments?

Answer 33

The present value of a payment made far in the future is small.

Question 34

What is a caveat to remember when finding present values of a series of payments??

Answer 34

As in the case of an accumulated annuity in which the value was found some time after the final payment, it may be necessary to find the present value of a series of payments some time before the first payment is made.

Question 35

In example 2.8 where a car purchaser is allowed to start paying back the loan on the 9th month, since the payments are deferred for 8 months as compared to the situation in Ex.7, it should not be surprising that the new payment is equal to the old payment multiplied by an 8-month accumulation factor. Suppose an n-payment annuity of 1 per period is to be valued k+1 payment periods before the first payment is made. How can the present value be expressed?

Answer 35

As 
v^(k+1) + v^(k+2) +...+ v^(k+n)
which can be reformulated as
v^k * [v+v^2+...+v^n] =
= v^k * an|
Since an| represents the present value of the annuity one period before the first payment, the value k periods before that (for a total of k+1 periods before the first payment) should be v^k * an|.

Question 36

What’s another way to write v^k * an|?

Answer 36

v^k * an| =

= a(n+k)| - ak|

Question 37

What is v^k * an| =

= a(n+k)| - ak| called?

Answer 37

Such an annuity is called a deferred annuity. The annuity considered here is usually called a k-period deferred, n-payment annuity of 1 per period.

Question 38

What is another of writing
v^k * an| =
= a(n+k)| - ak|
in actuarial notation?

Answer 38

k|an|

where the first | is what it should be whereas the second one is the semi box notation used thus far.

Question 39

What should you note and remember about k-period deferred annuity-immediates?

Answer 39

The firt payment comes k+1 periods after the valuation date, not k periods after.

Question 40

What is another way that we can rewrite the expression
v^k * an| =
= a(n+k)| - ak|

Answer 40

a(n+k)| =
= ak| + v^k * an| =
= an| + v^n * ak|

Question 41

Just as Eq. 2.5 can be used for accumulated annuities, Eq. 2.9 can be applied to find the present value of an annuity for which the interest rate changes during the term of the annuity. If we consider an n+k -payment annuity with equally spaced payments, with an interest rate of i1 per period up to the time of the nth payment followed by a rate of i2 per period from the nth payment onward, how can the present value of the annuity one period before the first payment can be found?

Answer 41

a) The present value of the first n payments valued one period before the first payment is an|i.
b) The present value of the final k payments valued at time n (one period before the first of the final k payments) at rate i2 is ak|i2.
c) The value of b) at time 0 (one period before the first payment of the entire series) at interest rate i1 per period over the first n periods is
v^ni1 * ak|i2 (where the vi1 in the v^ni1 is a symbol).
d) The total present value at time 0 is the sum of a) and c), which is an|i + v^ni1 * ak|i2.

Question 42

[Regarding the relationship between an|i and sn|i) We now return to annuities with a level interest rate. The valuation point for an n-payment accumulated annuity-immediate is the time of the nth payment and the accumulated value at that time is sn|i. The valuation point for the present value of an n-payment annuity-immediate is one period before the first payment, and the present value is an|i. We see that the valuation point for the present value of the annuity is n periods earlier than the valuation point for the accumulated value. What follows from this?

Answer 42

That
sn|i = (1+i)^n * an|i
and
an|i = v^n * sn|i

Question 43

How can it be observed algebrically that
sn|i = (1+i)^n * an|i
and
an|i = v^n * sn|i?

Answer 43

v^n * sn|i =
= v^n * [((1+i)^n -1)/i] =
= (1-v^n)/i =
= an|i

Question 44

What should we be able to see from the algebraic deduction 
v^n * sn|i =
= v^n * [((1+i)^n -1)/i] =
= (1-v^n)/i =
= an|i?

Answer 44

Given v^n * sn|i =
= v^n * [((1+i)^n -1)/i] =
= (1-v^n)/i =
= an|i
For a particular value of i, both sn|i and an|i increase as n increases. Furthermore
sn|i = 1+(1+i)+…+(1+i)^(n-1) increases as i increases (a higher interest rate results in greater accumulated values of the payments). However, since
v = 1/(1+i) decreases as i increases, we can see that
an|i = v +…+ v^n decreases as i increases. This again illustrates the inverse relationship between the present value of an income stream and the interest rate used for valuation.

Question 45

It was pointed out in Ch.1 that if i>0, then v^n decreases as n increases, and, in fact, v^n→0 as n→∞. Furthermore it was noted earlier in this section that an|i increases as n increases. What can we see as n→∞?

Answer 45

As n→∞ it is easy to see that

lim_n→∞ (1-v^n)/i = 1/i

Question 46

How else can we see that As n→∞ it is easy to see that

lim_n→∞ (1-v^n)/i = 1/i?

Answer 46

This expression can also be derived by summing the infinite series of present values of payments v+v^2+v^3+… . The infinite series becomes
v+v^2+v^3+… =
= v * (1/(1-v) =
= 1/i.

Question 47

What is the infinite period annuity that results as n→∞ called and what notation do we use for it?

Answer 47

The infinite period annuity that results as n→∞ is called a perpetuity, and a notation that may be used to represent the present value of this perpetuity is
a∞|i. Since the valuation of the perpetuity occurs one period before the first payment, it would be referred to as a perpetuity-immediate.

Question 48

How can the notation of perpetuity a∞|i be considered from another point of view?

Answer 48

Suppose that X is the amount that must be invested at interest rate i per period in order to generate a perpetuity of 1 per period. In order to generate a payment of 1 without taking anything away from the existing principal amount X, the payment of 1 must be generated by interest alone. Therefore X*i = 1, or equivalently, X = 1/i.

Question 49

What should be noted about perpetuities?

Answer 49

That it is not possible to formulate the accumulated value of a perpetuity.

Question 50

A perpetuity-immediate pays X per year. Brian receives the first n payments. Colleen receives the next n payments, and Jeff receives the remaining payments. Brian’s share of the present value of the original perpetuity is 40%, and Jeff’s share is K. Calculate K.
What is the present value of a perpetuity?

Answer 50

X*a|i = X/i

Question 51

A perpetuity-immediate pays X per year. Brian receives the first n payments. Colleen receives the next n payments, and Jeff receives the remaining payments. Brian’s share of the present value of the original perpetuity is 40%, and Jeff’s share is K. Calculate K.
What is the present value of Brian’s portion of the perpetuity?

Answer 51

X*an|i = X[(1-v^n)/i], which we are told is 40% of X/i. Therefore, 
1-v^n = .4, so that v^n = .6.

Question 52

A perpetuity-immediate pays X per year. Brian receives the first n payments. Colleen receives the next n payments, and Jeff receives the remaining payments. Brian’s share of the present value of the original perpetuity is 40%, and Jeff’s share is K. Calculate K.
What is the present value of Colleen’s and Jeff’s portions of the perpetuity?

Answer 52

Xv^(n)an|i = .6Xan|i =
= (.6)(.4)(X/i), which is 24% of the value of the original perpetuity.
Therefore, Jeff’s share of the perpetuity is (100-40-24% = 36%.

Question 53

A perpetuity-immediate pays X per year. Brian receives the first n payments. Colleen receives the next n payments, and Jeff receives the remaining payments. Brian’s share of the present value of the original perpetuity is 40%, and Jeff’s share is K. Calculate K.
We’ve shown one way to solve this, what is the other?

Answer 53

Xv^(2n)a∞|i =
= (.6)^2Xa∞|i =
= (.36)*(X/i)

Question 54

How is it possible to interpret a level n-payment annuity as the “difference” between two perpetuities?

Answer 54

The present value of a perpetuity-immediate of 1 per year is
a∞|i =1/i.
An “n-year deferred” perpetuity-immediate of 1 per year would have payments starting in n+1 years, and would have present value
v^na∞|i = v^n/i.
Subtracting the second present value from the first, we get
(a∞|i) - (v^na∞|i) =
= (1-v^n)/i = an|i
Subtracting the deferred perpetuity cancels all payments after the nth payment, leaving an n-payment annuity-immediate.

Question 55

Which information is required to make a valuation of a level series of equally spaced payments?

Answer 55

1. The number of the payments.
2. The valuation point.
3. The interest rate per period.

Question 56

Other than an annuity-immediate, what is another standard annuity form?

Answer 56

That of the annuity-due.

Question 57

This form (annuity-due) occurs most frequently in the which context?

Answer 57

That of life annuities, but can also be defined in the case of annuities-certain.

Question 58

What does an annuity-due refer to in the case of present value?

Answer 58

In the case of present value, an annuity-due refers to the valuation of the annuity at the time of (and including) the first payment.

Question 59

What does an annuity-due refer to in the case of accumulated value?

Answer 59

In the case of accumulated value, an annuity-due refers to the valuation of the annuity one payment period after the final payment.

Question 60

What would be implied if an annuity is described as having payments occurring at the beginning of each period?

Answer 60

That annuity-due valuation is intended. There would be payments at time 0 (beginning of the 1st period), time 1 (beginning of the 2nd period),…, time n-1 (beginning of the nth period). The present value of the annuity would be found at time 0 and the accumulated value would be found at time n.

Question 61

What is Definition 2.3 - Annuity-Due

Answer 61

For n-payment annuities with payments of amount 1 each, the annuity-due present value is at the time of the first payment,
än|i = 1+v+v^2+...+v^(n-1) =
= (1-v^n)/(1-v) =
= (1-v^n)/d
and the accumulated value is one period after the final payment,
s̈n|i = (1+i)+(1+i)^2+...+(1+i)^n =
= (1+i)*[ ((1+i)^n -1) / i] =
= ((1+i)^n -1) / d

Question 62

Note that for both the present value and accumulated value of the annuity-due, the valuation point is one period after the valuation point for the corresponding annuity-immediate. This leads to which relationships?

Answer 62

än|i = (1+i)an|i
and
s̈n|i = (1+i)sn|i

Question 63

What is the present value of a perpetuity-due of 1 per year?

Answer 63

ä∞|i = 1+v+v^2+… =
= 1/(1-v) = 1/d =
= (1+i)/i = (1+i)*a∞|i =
= 1 + a∞|i

Question 64

A series of payments can be valued at any time. Annuity-immediate and annuity-due, accumulated and present value, are based on the most frequently used valuation points. We have seen that annuities can be valued some time after they end or some time before they begin. Valuations can also be done within the term of the annuity, so that we would find the accumulated value of payments already made combined with the present value of payments yet to be made. What is the general term to refer to the value of an annuity at any point in time?

Answer 64

Current value.

Question 65

In the annuities considered in Section 2.1, it had been assumed that the quoted compounding interest period is the same as the annuity payment period. It may often be the case that the quoted interest rate has a compounding period that does not coincide with the annuity payment period. How do we solve this problem?

Answer 65

For the purpose of a numerical evaluation of the annuity, we focus on the annuity payment period and determine and use the interest rate per payment period that is equivalent to the quoted interest rate.
What is meant by equivalence here is “compounding equivalence” as defined in Ch.1.

Question 66

When can the actuarial concept of the m^thly (mthly from now on) payable annuity be applied?

Answer 66

When the quoted interest rate is an effective annual rate of interest and the payments are made more frequently than once per year.

Question 67

What is the general form of an mthly payable annuity-immediate?

Answer 67

Ksn|i^(m)
where it really looks like the sn|i but with a constant K in front and a superscript (m).
The superscript “(m)” indicates that this total of K is split into m payments of K/m each to be made at the end of each quarter.

Question 68

Interpret the actuarial notation Ksn|i^(m) used to describe mthly payable annuity-immediates.

Answer 68

Given Ksn|i^(m)
The effective annual interest rate is i and payments of amount K/m each occur at the end of every 1/m -year period (total amount paid per year is K).
Ksn|i^(m) denotes the accumulated value of this series of payments at the end of n years of payments; there would be a total of m*n payments, and the valuation point is the time of the final payment.

Question 69

As seen, there exists actuarial notation Ksn|i^(m) to describe mthly payable annuity-immediates. Which similar actuarial notation describes the present value of an mthly payable annuity?

Answer 69

For the same set of payments previously described,

Kan|i^(m) denotes the present value of the series one payment period (or 1/m -year) before the first payment.

Question 70

The actuarial notation that describes the present value of an mthly payable annuity is Kan|i^(m). What is an|i^(m) also equal to?

Answer 70

an|i^(m) = (1-v^n) / (i^(m)) =

= an|i * i/i^(m)

Question 71

When does the mthly payable annuity notation arise?

Answer 71

In a life-annuity context, where it is more likely to be used.

Question 72

The annuities considered up to now all have specified individual payments at specified points in time. They are discrete annuities and frequently occur in practical situations. When is it sometimes useful to consider continuous annuities, those which have payments made continuously over a period of time?

Answer 72

For theoretical purposes and for modeling complex situations.

Question 73

(Regarding an annuity payable continuously) Suppose an annuity has a level rate of continuous payment of 1 per period, and an effective rate of interest of i per period. Then the amount paid during the interval from time t1 to time t2 (measured using the period as the unit of time) is equal to t2-t1. Suppose the payment continues for n periods, measured from time 0 to time n. In order to find the accumulated value of the n period of payments, it is not possible to add up the accumulated values of individual payments as was done for the discrete annuities considered earlier. How can determine the accumulated value at time n of the infinitesimal amount paid between time t1 and time t2 using differential calculus?

Answer 73

The accumulated value as of time n of this amount is
(1+i)^(n-t1) dt.
These accumulated amounts are “added up” by means of an integral, so that the accumulated value of the continuous annuity, paid at rate 1 per period for n periods, denoted ?Sn|i is given by
?Sn|i = ∫(1+i)^(n-t1) dt from [0,n]
where the symbol ?S is really an overlined S.

Question 74

How else can we define ?Sn|i = ∫(1+i)^(n-t1) dt from [0,n]?

Answer 74

?Sn|i = [-(1+i)^(n-t)] / ln(1+1) from [0,n] =
= [(1+i)^n -1] / δ =
= (e^(nδ) -1) / δ =
= [((1+i)^n -1) / i]*(i/δ) =
= (i/δ)*sn|i

In short
?Sn|i = (i/δ)*sn|i

Question 75

Note also that 
?Sn|i =  [(1+i)^n -1] / δ =
= lim as m→∞ of [(1+i)^n -1] / i^(m) =
= lim as m→∞ of sn|i^(m)
What is the interpretation of this relationship?

Answer 75

That as m gets larger, the annuity payment is more frequent and is spread more evenly throughout the year, with the limit being a continuous distribution of payment throughout the year.

Question 76

What is the present value, at the time payment begins, of a continuous annuity paying a total of 1 per period at effective periodic interest rate i?

Answer 76

?an|i = ∫v^t dt from[0,n] =
= (1-v^n)/(ln(1+i) =
= (1-v^n)/δ = 
= (1-e^(-nδ))/δ =
= (i/δ)*an|i = 
= lim as m→∞ of an|i^(m)

Question 77

Suppose a general accumulation function is in effect, where a(t1,t2) is the accumulated value at time t2 of an amount 1 invested at time t1. Then what do ∫a(t,te)dt from [t0,te] and ∫1/(a(t0,t)dt from [t0,te] represent?

Answer 77

The accumulated value at time te and the present value at time t0, respectively, of a continuous annuity of 1 per unit time, payable from time t0 to time te.

Question 78

Suppose a general accumulation function is in effect, where a(t1,t2) is the accumulated value at time t2 of an amount 1 invested at time t1. Then ∫a(t,te)dt from [t0,te] and ∫1/(a(t0,t)dt from [t0,te] represent the accumulated value at time te and the present value at time t0, respectively, of a continuous annuity of 1 per unit time, payable from time t0 to time te. If accumulation is based on the force of interest δr, what then follows?

Answer 78

Then

a(t1,t2) = exp[ ∫δrdr [t1,t2] ]

Question 79

Suppose a general accumulation function is in effect, where a(t1,t2) is the accumulated value at time t2 of an amount 1 invested at time t1. Then ∫a(t,te)dt from [t0,te] and ∫1/(a(t0,t)dt from [t0,te] represent the accumulated value at time te and the present value at time t0, respectively, of a continuous annuity of 1 per unit time, payable from time t0 to time te. If accumulation is based on the force of interest δr, then
a(t1,t2) = exp[ ∫δrdr [t1,t2] ]
and we have present and accumulated annuity values at time 0 and time n, respectively given by which equations?

Answer 79

?an|δr = ∫e^(-∫δrdr [0,t])dt [0,n]
and
?sn|δr = ∫e^(∫δrdr [t,n])dt [0,n]

Question 80

Which relationships relates
?an|δr = ∫e^(-∫δrdr [0,t])dt [0,n]
to
?sn|δr = ∫e^(∫δrdr [t,n])dt [0,n]?

Answer 80

?sn|δr = ?an|δr*e^(∫δrdr [0,n])

Question 81

Suppose we consider the basic relationship for the accumulated value, M, of an annuity-immediate with n level payments of amount J each with an interest rate of i per payment period. What, then, is the relationship between M and J?

Answer 81

M = J[(1+i)^(n-1) + (1+i)^(n-2)+…+ (1+i) +1] =
= J[((1+i)^n) -1) / i] =
= Jsn|i

Question 82

M = J[(1+i)^(n-1) + (1+i)^(n-2)+…+ (1+i) +1] =
= J[((1+i)^n) -1) / i] =
= Jsn|i
We can regard M, J, i, and n as “variables” in this equation, and given any three of these variables it is possible to solve for the fourth. In examples considered so far, either we have been given J, i, and n and solved for M or we have been given M, i, and n, and solved for J.
How can we solve for the unknown time factor n algebraically?

Answer 82

M = J[((1+i)^n) -1) / i] →
→ (1+i)^n = 1 + (Mi)/J →
→
n = (ln(1+(M*i)/J)) / ln(1+i)

Question 83

M = J[(1+i)^(n-1) + (1+i)^(n-2)+…+ (1+i) +1] =
= J[((1+i)^n) -1) / i] =
= Jsn|i
We can regard M, J, i, and n as “variables” in this equation, and given any three of these variables it is possible to solve for the fourth. In examples considered so far, either we have been given J, i, and n and solved for M or we have been given M, i, and n, and solved for J.
We solve for the unknown time factor n algebraically with equation
n = (ln(1+(M*i)/J)) / ln(1+i)
In general, it will not be possible to solve algebraically for the unknown interest factor i. In either case, the solution would be done using appropriate functions on a financial calculator. Most financial calculators have functions that solve for the fourth variable if any three of M, J, I, and n are known. How does the same comments apply to the present value of a level payment annuity-immediate?

Answer 83

L = K[v+v^2+…+v^n] =
= K[(1-v^n)/i]
where the four variables are the present value L, the payment amount K, the number of payments n, and the interest rate i. Solving for n results in
n = (ln(1-(L*i)/K)) / ln(v)

Question 84

Solving for the unknown time will usually result in a value for n that is not an integer. The integer part will be the number of full periodic payments required. What then is the fractional part?

Answer 84

The fraction will be the additional part of a payment required to complete the annuity. This additional fractional payment may be made at the time of the final full payment (called a “balloon payment”) or may be made one period after the final full payment.

Question 85

The solution for n in the equation L = Kan|i was see to be
n = (ln(1-(Li)/K)) / ln(v). It is implicitly assumed that 1 - (Li)/K > 0, since otherwise it would be impossible to find the natural logarithm. What do we see upon closer inspection?

Answer 85

Upon closer inspection, if
1 - (Li)/K ≤ 0, then K ≤ Li, so the loan payment will at most cover the periodic interest due on the loan and will never repay any principal. Therefore, if the loan payment isn’t sufficient to cover the periodic interest due, the loan will never be repaid and n=∞.

Question 86

There are a few points to keep in mind when considering a situation involving an unknown rate of interest. Why can it be assumed that interest rates are greater or equal to -100%?

Answer 86

Since at a rate of -100% the accumulated value of 1 would be 0 at any future point except in unusual circumstances.

Question 87

There are a few points to keep in mind when considering a situation involving an unknown rate of interest. It can be assumed that interest rates are greater or equal to -100%, since at a rate of -100% the accumulated value of 1 would be 0 at any future point except in unusual circumstances. What is one such circumstance?

Answer 87

Where an investor has at risk more than the amount invested (such as with “leveraged” investments or when investing on “margin”).

Question 88

There are a few points to keep in mind when considering a situation involving an unknown rate of interest. It can be assumed that interest rates are greater or equal to -100%, since at a rate of -100% the accumulated value of 1 would be 0 at any future point except in unusual circumstances. What is a second such circumstance?

Answer 88

Where the investment consists of a series of varying cashflows, each one of which can be either positive and negative (i.e., disbursements and receipts).