-
Compound Interest Part 1
-
Compound Interest Part 2
-
Present Value
-
Annuities
-
Perpetuities
-
Bonds
-
Dividend Discount Model
-
Risk
-
Capital Asset Pricing Model
Annuity Due
Annuity Due¶
While regular annuities pay off at the end of each year, there is a such thing as an annuity due. These instruments actually pay off in the beginning of the year instead. We can visualize a 5 year annuity due below.
plt.rcParams["figure.figsize"] = [12,4]
#There is also a such thing as an annuity due, which is an annuity where you get your payment one period earlier
#So on a timeline an annuity due for 5 years would look like this
payments = []
for x in range(0,5):
payments.append((100,x))
timelinePlot(5,payments)
#You immediately get money in period 0, but get none on period 5 since you already got your 5 payments
The way that this will change TVM is that now each payment is discounted by one less year. For example, the first payment was discounted by 1 year but now it is discounted by 0 years. The second payment was discounted by 2 years but now it will be only discounted by 1 year. Compare how we can calcualte a present value below for an annuity and the same one if it were changed to an annuity due.
#Computing present value for the annuity
PV1 = 0
for t in range(1,6):
PV1 += 100/(1.05)**t
print(PV1)
#Computing present value for the annuity due
PV2 = 0
for t in range(1,6):
#Notice the t-1
PV2 += 100/(1.05)**(t-1)
print(PV2)
One way to think about the difference between an annuity due and an annuity is that every payment is being shifted. Because of this there is a relationship that will hold between the two:
$$a_D = a \cdot (1+r)$$
where
$a_D = \text{Annuity Due}$
$a = \text{Annuity}$
$r = \text{Discount Rate}$
Confirm it with the values below.
#It is the same value just with 1.05 multiplied
print(PV1 * 1.05)