A perpetuity with the first annual cash flow paid at the beginning of year 4 is equivalent to receiving $103,000 in 15 years’ time. Assume that the perpetuity and the lump sum are of equivalent risk and that j12 = 14.32% pa is the appropriate interest rate. How much is the annual cash flow associated with the perpetuity?
A perpetuity with the first annual cash flow paid at the beginning of year 4 is equivalent to receiving $103,000 in 15 years’ time. Assume that the perpetuity and the lump sum are of equivalent risk and that j12 = 14.32% pa is the appropriate interest rate. How much is the annual cash flow associated with the perpetuity?
To determine the annual cash flow associated with the perpetuity, we need to find the present value of the $103,000 lump sum at the appropriate interest rate.
Using the formula for present value of a lump sum:
PV = FV / (1 + r)^n
Where:
PV = Present Value
FV = Future Value
r = interest rate
n = number of periods
We can rearrange the formula to solve for the present value:
PV = FV / (1 + r)^n
PV = $103,000 / (1 + 0.1432)^15
PV ≈ $103,000 / (1.1432)^15
PV ≈ $103,000 / 4.7386
PV ≈ $21,723.53
Now that we have the present value of the lump sum, we can calculate the annual cash flow associated with the perpetuity.
We'll use the formula for the present value of a perpetuity:
PV = CF / r
Where:
PV = Present Value
CF = Cash Flow
r = interest rate
Rearranging the formula to solve for the cash flow:
CF = PV * r
CF = $21,723.53 * 0.1432
CF ≈ $3,112.08
Therefore, the annual cash flow associated with the perpetuity is approximately $3,112.08.
To calculate the annual cash flow associated with the perpetuity, we can use the concept of present value.
Step 1: Determine the present value of the lump sum payment of $103,000 to be received in 15 years.
PV = FV / (1 + r)^n
Where:
PV = Present Value
FV = Future Value
r = Interest rate
n = Number of years
In this case:
FV = $103,000
r = 14.32% or 0.1432 (expressed as a decimal)
n = 15 years
PV = 103,000 / (1 + 0.1432)^15
PV = 103,000 / (1.1432)^15
PV ≈ 24,346.08
Step 2: Determine the present value of the perpetuity.
Since the perpetuity starts paying from the beginning of year 4, we need to discount the first three cash flows.
PV_perpetuity = CF / r + CF / (1 + r)^2 + CF / (1 + r)^3 + ...
Where:
PV_perpetuity = Present Value of the perpetuity
CF = Cash Flow per year
r = Interest rate
In this case, we need to discount the first three cash flows, so we have:
PV_perpetuity = CF / r + CF / (1 + r)^2 + CF / (1 + r)^3 + ...
To simplify the equation, multiply both sides by r:
r * PV_perpetuity = CF + CF / (1 + r) + CF / (1 + r)^2 + CF / (1 + r)^3 + ...
Now, substitute the present value of the perpetuity calculated in Step 1:
r * 24,346.08 = CF + CF / (1 + r) + CF / (1 + r)^2 + CF / (1 + r)^3 + ...
Step 3: Solve the equation to find the annual cash flow associated with the perpetuity.
Using the given interest rate of j12 = 14.32%, we substitute it as the value for r.
0.1432 * 24,346.08 = CF + CF / (1 + 0.1432) + CF / (1 + 0.1432)^2 + CF / (1 + 0.1432)^3 + ...
Simplifying the equation, we have:
3,487.35 = CF + CF / 1.1432 + CF / (1.1432)^2 + CF / (1.1432)^3 + ...
We can continue this process by dividing the cash flow by successive powers of (1 + r):
3,487.35 = CF * (1 + 1 / 1.1432 + 1 / 1.1432^2 + 1 / 1.1432^3 + ...)
This is a geometric series with a common ratio of 1 / 1.1432. We can use the formula for the sum of an infinite geometric series to solve for CF:
Sum = a / (1 - r)
Where:
Sum = 3,487.35
a = CF
r = 1 / 1.1432
3,487.35 = CF / (1 - 1 / 1.1432)
Simplifying further:
3,487.35 = CF / (1 - 0.8754)
3,487.35 = CF / 0.1246
Solving for CF:
CF = 3,487.35 * 0.1246
CF ≈ 434.60
Therefore, the annual cash flow associated with the perpetuity is approximately $434.60.
i = (1+j12/12)12 -1
= 11.3509
Compare the two different options at the same point in time e.g. both at t = 2 or both at t = 0.
Here we are using t = 2
The Perpetuity
PV2 = CF3/i = CF/.113509
Lump Sum
PV2 = 108000/(1.113509)16 = $19,334
These are same so CF/0.113509= 19,334
Reordering we get CF = 19,334*0.113509 = $ 2195