Chapter 5: Level annuities

Question 1

immediate annuity

Answer 1

a series of n payments with the first payment at year 1 and the final at year n.
Payment occurs at the end of each period (aka in arrears).
an|=(1-v^n)/i

Question 2

an|

Answer 2

PV of an annnuity of 1 for n years
=(1-v^n)/i
=v^n Sn|

Question 3

sn|

Answer 3

AV of an annuity of 1 for n years
=(((1+i)^n)-1)/i
=(1+i)^n an|

Question 4

annuity-due

Answer 4

first payment is at time 0, last at time n-1.
Payments occur at the start of each time period (in advance)

Question 5

än|

Answer 5

Annuity-due
=(1-v^n)/d
=1+a(n-1)|

Question 6

s̈n|

Answer 6

AV of annuity-due
=((1+i)^n - 1)/d
valued at n years (1 year after the last payment is made)

Question 7

deferred annuity

Answer 7

An annuity that starts at some point after the first time period

Question 8

m|an|

Answer 8

PV of a deferred immediate annuity
n year immediate annuity that starts in m years (first payment occurs at m+1 up to m+n)
=v^m an|
=(m+1)än|

Question 9

m|än|

Answer 9

PV of a deferred annuity-due.
first payment at m, last at time m+n-1
=v^m än|

Question 10

continuously paid annuity

Answer 10

an annuity with payments made continuously over the year

Question 11

ā

Answer 11

PV of a continuously paid annuity of £1 per year recieved continuously from time 0 to n
=(1-v^n)/δ (if δ is constant)

Question 12

s̄

Answer 12

AV of a continuously paid annuity
=((1+i)n -1) /δ

Question 13

perpetuity

Answer 13

An annuity with payments that continue forever (e.g. where n =∞).
Is immediate if payment one is in year 1
Is due if they start at 0

Question 14

a∞|

Answer 14

PV of an immediate perpetuity
=1/i

Question 15

ä∞|

Answer 15

PV of a perpetuity-due
=1/d
=1+a∞|

Question 16

ā∞|

Answer 16

PV of a continuously payable perpetuity (rate of 1 per year forever
=1/δ