S n a aR aR 2 ... aR n

(1)

where R is called the common ratio (between consecutive terms) of the GP sequence.

The reason why we want to derive a closed-form expression for S n is for the sake of calculating the summation, or otherwise we need to add all terms one-by-one together, which does not make a sense if the number of terms is huge, say a million terms!

Most importantly, we based on the closed-form expression to derive the PV and FV expressions for both ordinary annuity and annuity due.

Steps:

1. Multiply the both sides of equation (1) by the common ratio, R , to have S n R aR aR 2 aR 3 ... aR n1 (2)

2. Then subtract equation (1) by equation (2), (or vice versa; it doesn’t matter which subtracts which as the result will be the same.), i.e., S n S n R (a aR aR 2 ... aR n ) aR aR 2 aR 3 ... aR n1 (3) 3. Notice that all terms on the right hand side except for the first and last term, a, aR n1 , are cancelled. So, equation (3) becomes,

(4)

S n (1 R) a aR n1

4. Remember our objective is to calculate S n . From equation (4), S n is obvious to equal to,

a aR n 1 a(1 R n1 )

Sn

(1 R)

(1 R)

(*)

5. We are done. Equation (*) is the closed-form expression of that we want to obtain.

Applications:

1. The PV for ordinary annuity, e.g. constant end-of-period cash flows, C, for t periods.

PV

C

C

C

...

2

1 r (1 r )

(1 r ) t

(A1)

Comparing equation (A1) with equation (1), we can see that the PV is a summation of GP sequence, with a C and the common ratio R

1

. Therefore,

1 r

the closed-form expression, i.e., also the summation equals to

C

1

1

C

C

C 1

1

t

t 1

1 r (1 r )

(1 r ) t C

1 r (1 r )

1

PV

1

1

r

r

r (1 r ) t

1

1 r

1 r

(A2)

which is exactly the question in lecture note.

2. How about the FV (assuming the FV at the end-of-period of period t) of the same ordinary annuity? I.e.,

FV C (1 r ) t 1 C (1 r ) t 2 ... C (1 r ) 0

(A3)

Recall what we previously discussed about the relation between PV and FV, FV in equation (A3) can be calculated by

FV PV (1 r ) t

(A4)

provided that the interest rate (discount rate) remains constant. In other words, by equation (A2),

FV C (1 r ) t 1 C (1 r ) t 2 ... C (1 r ) 0 PV (1 r ) t

C

1

(1 r ) t

1

t

r (1 r )

C

(1 r ) t 1

r

Again, the result is exactly the equation in our lecture note.

(A5)

Remarks:

1. In general, we can apply the sum of GP formula (i.e. equation (*)) to obtain the closed form expressions for other cash flow patterns, .e.g. the constant cash flow annuity due, the growing annuity and so on.

2. In the examination, you are likely provided with a number of PV and FV formulas. But make sure you have identified the correct cash flow pattern before applying any one of those general formulas or otherwise you will easily get the answer wrong because different questions can have very different (or tricky) cash flow patterns and hence the given formulas may not or cannot be directly applied. You may need to do some modifications onto the general formulas to make it applicable to the specific questions. In normal time (or even during the examination), in case you are not sure whether a formula is directly applicable or not, it is always make good sense to derive the results from stretch. This is also the reason I write down this supplementary note for your reference.