Question

1. You will receive $100 per year for 10 years. The discount rate is 10%. What is the present value of this stream?

2. Using the previous information, assume now the 100 will increase at a 5% per year from year 1. What is the new present value?

3. Now assume compute the present value for the same information using a perpetuity without and with growth. Compare the 4 present values. What would you rank those?

(Please explain step by step)

Answer 1

Part 1:
Present value is calculated as the discounted values of future cash flows.
Given that discount rate=10%
Given that per year cash flow for 10 years is $100
Formula to calculate present value =Cash flow in time period 1/(1+ discount rate)^1+Cash flow in time period 2/(1+ discount rate)^2+..+Cash flow in time period n/(1+ discount rate)^n
Here, we will have the following calculations:
Present value =$100/(1+10%)^1+$100/(1+10%)^2+$100/(1+10%)^3+$100/(1+10%)^4+$100/(1+10%)^5+$100/(1+10%)^6+$100/(1+10%)^7+$100/(1+10%)^8+$100/(1+10%)^9+$100/(1+10%)^10
=$100/(1.1)^1+$100/(1.1)^2+$100/(1.1)^3+$100/(1.1)^4+$100/(1.1)^5+$100/(1.1)^6+$100/(1.1)^7+$100/(1.1)^8+$100/(1.1)^9+$100/(1.1)^10
=$100/(1.1)+$100/1.21+$100/1.331+$100/1.4641+$100/1.61051+$100/1.771561+$100/1.9487171+$100/2.14358881+$100/2.357947691+$100/2.59374246
=$90.90909091+$82.6446281+$75.13148009+$68.30134554+$62.09213231+$56.44739301+$51.31581182+$46.65073802+$42.40976184+$38.55432894
=$614.4567106 or $614.46 (rounded upto two decimal places)

Part 2: $100 will increase at a 5% per year from year 1.

Discount rate will be same as 10%
Then the cash flows will become:
Year1:$100
Year2:$100*(1+5%)=$105
Year3:$105*(1+5%)=$110.25
Year4:$110.25*(1+5%)=$115.7625
Year5:$115.7625*(1+5%)=$121.550625
Year6:$121.550625*(1+5%)=$127.6281563
Year7:$127.6281563*(1+5%)=$134.0095641
Year8:$134.0095641*(1+5%)=$140.7100423
Year9:$140.7100423*(1+5%)=$147.7455444
Year10:$147.7455444*(1+5%)=$155.1328216

New present value =$100/(1+10%)^1+$105/(1+10%)^2+$110.25/(1+10%)^3+$115.7625/(1+10%)^4+$121.550625/(1+10%)^5+$127.6281563/(1+10%)^6+$134.0095641/(1+10%)^7+$140.7100423/(1+10%)^8+$147.7455444/(1+10%)^9+$155.1328216/(1+10%)^10
=$100/(1.1)^1+$105/(1.1)^2+$110.25/(1.1)^3+$115.7625/(1.1)^4+$121.550625/(1.1)^5+$127.6281563/(1.1)^6+$134.0095641/(1.1)^7+$140.7100423/(1.1)^8+$147.7455444/(1.1)^9+$155.1328216/(1.1)^10

=$100/(1.1)+$105/1.21+$110.25/1.331+$115.7625/1.4641+$121.550625/1.61051+$127.6281563/1.771561+$134.0095641/1.9487171+$140.7100423/2.14358881+$147.7455444/2.357947691+$155.1328216/2.59374246
=$90.90909091+$86.7768595+$82.8324568+$79.06734513+$75.47337489+$72.04276697+$68.76809574+$65.6422732+$62.65853351+$59.81041834
=$743.981215 or $743.98 (rounded upto two decimal places)

Part 3:
Calculating present value using perpetuity without and with growth.

Present value of perpetuity without growth is given by:
Cash flow/Discount rate
Here, $100 payment is the cash flow and discount rate=10%. Perpetuity means the payments will continue forever.

Present value of perpetuity without growth=$100/10%=$100/.1=$1000

Present value of perpetuity with growth is given by:
Cash flow of period 1/(Discount rate - growth rate)
Here, cash flow of period 1 is calculated as first period cash flow*(1+growth rate)
=$100*(1+5%)=$105
The value of growth rate as 5% is given in question 2.

Present value of perpetuity with growth=$105/(10%-5%)=$105/5%=$2100

Ranking between the present values in the above four cases:

Part 1: $614.46
Part 2: $743.98
Part 3: Present value without growth is $1000 and with growth is $2100

Rank 1: Present value of perpetuity with growth that is $2100
Rank 2: Present value of perpetuity without growth that is $1000
Rank 3: Answer for part 2 that is $743.98
Rank 4: Answer for part 1 that is $614.46