Question

You are evaluating two alternative financing arrangements. The first arrangement requires twenty annual payments with the first payment of $10,000 made in a year, while the second arrangement requires that each payment be made a year earlier but is otherwise similar to the first arrangement.
(a) If payments subsequent to the first increase at an annual rate of 5%, and the financiers require a 10% return on both arrangements, calculate how much more capital the financiers would be willing to provide under the second arrangement.
(b) If payments are level, and the financiers require a 10% return on both arrangements, calculate how much more capital would the financiers be willing to provide under the second arrangement?

(c) Explain why financiers are willing to provide more capital under the second arrangement in parts (a) and (b).   

Answer 1

a)

We are given : CF = 10000 ; r = 10% ; g = 5% and t = 20

where CF is the periodic cash flow, r is the applicable interest rate, g is the growth rate and t is the time period

i) First Arrangement - It is a case of Ordinary growing/increasing Annuity

PV of increasing annuity = CF * [ 1 - (1+g)t*(1+r)-t ] / (r - g)

PV of increasing annuity = 10000 * [ 1 - (1+5%)20 * (1+10%)-20 ] / (10% - 5%)

PV of increasing annuity = 121120.84

ii) Second Arrangement - It is a case of Growing/Increasing annuity due

PV of increasing annuity due = CF * (1+r)* [ 1 - (1+g)t*(1+r)-t ] / (r - g)

PV of increasing annuity due = 10000 * (1+10%) * [ 1 - (1+5%)20 * (1+10%)-20 ] / (10% - 5%)

PV of increasing annuity due = 133232.92

Hence the financiers should be willing to put additional (133232.92 - 121120.84) = 12112.08

b)

i) First Arrangement - It is a case of Ordinary Annuity

PV of ordinary annuity = CF * [1 - (1+r)-t ]/ r

where CF is the periodic cash flow, r is the applicable interest rate, g is the growth rate and t is the time period

We are given : CF = 10000 ; r = 10% ; g = 5% and t = 20 ; now we can solve and answer:

PV of ordinary annuity = 10000 * [1 - (1+10%)-20 ]/ 10%

PV of ordinary annuity = 85135.64

ii) Second Arrangement - It is a case of Ordinary annuity due

PV of ordinary annuity due = CF * (1+r) * [1 - (1+r)-t ]/ r

PV of Ordinary annuity due = 10000 * (1+10%)* [1 - (1+10%)-20 ]/ 10%

PV of Ordinary annuity due = 93649.20

Hence the financiers should be willing to put additional (93649.20 - 85135.64) = 8513.56

C)

The financiers should be willing to pay more under the second arrangement since they would be getting their principal amount back earlier which improves their over all IRR due to which they would be willing to invest more.

In other words, lower discounting will be applicable and so present value is higher.

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