Question

Blue Eagle Media just bought a new ropes course. To pay for the ropes course, the company took out a loan that requires Blue Eagle Media to pay the bank a special payment of 6,700 dollars in 2 month(s) and also pay the bank regular payments.   The first regular payment is expected to be 1,440 dollars in 1 month and all subsequent regular payments are expected to increase by 0.2 percent per month forever. The interest rate on the loan is 1.36 percent per month. What was the price of the ropes course?

Raj has an investment worth 171,638 dollars. The investment will make a special payment of X to Raj in 3 quarters in addition to making regular quarterly payments to Raj forever. The first regular quarterly payment to Raj is expected to be 3,600 dollars and will be made in 3 months. All subsequent regular quarterly payments are expected to increase by 0.54 percent per quarter forever. The expected return for the investment is 2.83 percent per quarter. What is X, the amount of the special payment that will be made to Raj in 3 quarters?

Answer 1

Part-1 of the question

Given Facts

1. Upfront Special Payment = $6700
2. Payment after one month = $1440
3. Monthly Interest Rate is 1.36%
4. Growth Rate is 0.2% per month

Problem is to identify price of the ropes course

Solution -

The price of ropes course should be the summation of the present value of all the payments made to bank.

Two types of payments are made/will be made to the bank :

1. Upfront Special Payment = $ 6700

2. Perpetual Monthly Payments

We already have the value of Upfront Special Payment and therefore, we should calculate the present value of perpetual payments to be made to the bank. With the help of following formulae, we can calculate the Present Value of growing perpetuity :

PV of Growing Perpetuity = d1/(r-g) where,

d1 denotes payment at period1,

r denotes interest rate and

g denotes growth rate

Now using the information available in the problem, we shall calculate as follows :

PV of Perpetual Monthly Payments = $1440/(1.36% - .02%)

PV of Perpetual Monthly Payments = $1440/(1.34%)

PV of Perpetual Monthly Payments = $107462.69 (rounded off 2 decimal)

As we know, the price of ropes course = Upfront Special Payment + PV of Perpetual Monthly Payments

Price of ropes course = $ 6700 + $107462.69

Price of ropes course = $ 114162.69

Part-2 of the question

In this part, Mr. Raj has made some investment against which he will receive quarterly payments in perpetuity (forever) as a return.

Facts Given

1. The investment made $ 171638
2. Special Payment in 3 quarters = X
3. First Quarterly Regular Payment = $ 3600 ( in perpetuity)
4. Growth Rate = 0.54% quarterly
5. Expected ROI = 2.83% quarterly

The problem to be resolved is to Calculate the value of X being the value of special payment in 3 quarters.

The current value of Investment equals to present value of all future payments at a given rate of return on investment.

Current Value of Investment = PV of Special Payments in 3 quarters ( + ) PV of quarterly regular payment in perpetuity

$171638 = PV of Special Payments in 3 quarters ( + ) PV of quarterly regular payment in perpetuity

PV of Special Payments in 3 quarters = $171638 ( - ) PV of quarterly regular payment in perpetuity

Now to calculate PV of quarterly regular payment in perpetuity, we shall use the following formulae:

PV of Growing Perpetuity = d1/(r-g) where,

d1 denotes payment at period1,

r denotes interest rate and

g denotes growth rate

Now using the information available in the problem, we shall calculate as follows :

PV of quarterly regular payment in perpetuity = $3600/(2.83% - 0.54%)

PV of quarterly regular payment in perpetuity = $3600/(2.29%)

PV of quarterly regular payment in perpetuity = $157205.24

Therefore, PV of Special Payments in 3 quarters = $171638 ( - ) $157205.24

PV of Special Payments in 3 quarters = $ 14432.76

If the payment at end of each quarter for next 3 quarter is given X, then the PV be as follows :

PV = P[(1 - (1 / (1 + r)ⁿ)) / r] , where

P = Periodic Payment

R = Rate of Interest

N = Number of period

Using information is given in a problem

PV of Special Payments in 3 quarters = X[(1 - (1 / (1 + .0283)^3)) / .0283]

$ 14432.76 = 2.84X

X = $ 14432.76 /2.84

X = $ 5081.95

Hence , value of X being the value of special payment in 3 quarters = $ 5081.95