Exercise 1 - compound interest
What would 150m be worth at the end of 5 years if invested at 7%?
Interest rate 8%
Year 0 1 2 3 4 5
Today's value 150 150 150 150 150 150
Multiplication factor = (1+interest rate)^year 1.00 1.08 1.17 1.26 1.36 1.47
Value at end of period (future value) 150.00 162.00 174.96 188.96 204.07 220.40
Answer: value at end of period = 220.40
Note, Excel formula check: 220.40 Formula for compound interest
Exercise 2 - introducing time value of money & discount factors
Cost of funds 8%
Year 0 1 2 3 4
Future value 200 200 200 200 200
Discount factor = 1/(1+interest rate)^year 1.00 0.93 0.86 0.79 0.74
Today's value (present value) 200.00 185.19 171.47 158.77 147.01
Answer: present value = 147.01
Note, Excel formula check: 147.01 Formula for discounting ( = 1 / compound interest)
Exercise 3 - discounted cash flows
Cost of funds 12%
Year 0 1 2 3 4 5
Future value 0 10 10 10 10 10
Discount factor = 1/(1+interest rate)^year 1.00 0.89 0.80 0.71 0.64 0.57
Present value 0.00 8.93 7.97 7.12 6.36 5.67
Answer: present value ( = sum of discounted
cash flows) =
36.05
Note, Excel formula check: 36.05 Formula for NPV
If I had a contract that paid out 10m per year for 5 years, and my cost of funds (or cost of capital) was 12%, what would that contract be
worth to me today?
If I had to pay back 200m in four years' time, and I anticipated interest rates of 8%, how much money would I have to set aside today?
239272472.xls.ms_office Page 1 of 6
Exercise 4 - discounted cash flows (a)
Cost of funds 9%
Year 0 1 2 3 4
Future value 0 10 20 15 8
Discount factor 1.00 0.92 0.84 0.77 0.71
Present value 0.00 9.17 16.83 11.58 5.67
Answer: present value = 43.26
Note, Excel formula check: 43.26 Formula for NPV
Exercise 5 - net present value
If I paid 30m for the business above, what would the net present value of the investment be?
Cost of funds 9%
Year 0 1 2 3 4
Future value -30 10 20 15 8
Discount factor 1.00 0.92 0.84 0.77 0.71
Present value -30.00 9.17 16.83 11.58 5.67
Answer: net present value = 13.26
Check: present value of cash flows 43.26 Formula for NPV
Price paid -30.00 Price paid
Answer: net present value of investment = 13.26 NPV
Note, how to get it wrong using Excel: 12.16 Why do you think this happens in Excel?
Exercise 6 - net present value (a)
Cost of funds 14%
Year 0 1 2 3
Future value -40 20 26 34
Discount factor 1.00 0.88 0.77 0.67
Present value -40.00 17.54 20.01 22.95
Answer: net present value = 20.50
Check: present value of cash flows 60.50 Formula for NPV
Price paid -40.00 Price paid
Answer: net present value of investment = 20.50 NPV
If I paid 40m for a business that was expected to generate the cash flow stream below, and my cost of capital was 14%, what would I
expect the NPV of the investment to be?
If I had a business that generated the cash flow stream below, and my cost of capital was 9%, what would I be prepared to pay for that
business?
239272472.xls.ms_office Page 2 of 6
Exercise 7 - IRR
What would my IRR on the investment be (= discount rate where NPV = zero)?
Answer: IRR (= discount rate) = 40% Iterate and change cost of funds until NPV = zero
Year 0 1 2 3
Future value -40 20 26 34
Discount factor = 1/(1+interest rate)^year 1.00 0.71 0.51 0.37
Present value -40.00 14.30 13.29 12.42
Net present value = 0.00
Note, Excel formula check: 40% Formula for IRR
Exercise 8 - IRR (a) - semi-annual cash flows
What would my IRR on the investment be if the cash flows above were actually received at six monthly rather than annual intervals?
Answer: IRR (= discount rate) = 96% Iterate and change cost of funds until NPV = zero
Period 0 0.5 1 1.5
Future value -40 20 26 34
Discount factor = 1/(1+interest rate)^year 1.00 0.71 0.51 0.37
Present value -40.00 14.30 13.29 12.42
Net present value = 0.00
Note, Excel formula check: ??? One reason to avoid the IRR formula!
Exercise 9 - IRR (b)
If a private equity firm put 20m into an investment and generated sales proceeds of 60m after three years, what was its IRR?
Answer: IRR (= discount rate) = 44% Iterate and change cost of funds until NPV = zero
Year 0 1 2 3
Future value -20 0 0 60
Discount factor 1.00 0.69 0.48 0.33
Present value -20.00 0.00 0.00 20.00
Net present value of investment = 0.00
Note, Excel formula check: 44% IRR formula
239272472.xls.ms_office Page 3 of 6
Answer 1 - compound interest
Correct answer: what would 10m be worth at the end of 3 years if invested at 8%?
Interest rate 8%
Year 0 1 2 3
Today's value 10 10 10 10
Multiplication factor = (1+interest rate)^year 1.00 1.08 1.17 1.26
Value at end of period (future value) 10.00 10.80 11.66 12.60
Answer: value at end of period = 12.60
Note, Excel formula check: 12.60 Formula for compound interest
Wrong answer 1 10.80
Wrong answer 2 11.66
Wrong answer 3 12.40
Wrong answer 4 13.60
Answer 2 - compound growth rate
Correct answer 24.57%
Wrong answer 1 11.11%
Wrong answer 2 22.22%
Wrong answer 3 33.33%
Wrong answer 4 66.67%
Answer 5 - NPV
1 2 3 4 5
Cost of funds 8% 10.0 10.0 10.0 10.0 10.0
DCF 39.9 9.3 8.6 7.9 7.4 6.8
Cost (35.0)
NPV 4.9
Check 4.9
Wrong answer 1 4.6
Wrong answer 2 8.1
Answer 6 - IRR
Wrong answer Column B Column C Column D Column E Column F Column G
Y1 Y2 Y3 Y4 Y5
Row 3 (20) 0 0 0 60
31.61%
Right answer Y0 Y1 Y2 Y3 Y4 Y5
Row 7 (20) 0 0 0 0 60
24.57%
Wrong answer Y1 Y2 Y3 Y4 Y5
Row 11 (20) 60
200.00%
Wrong answer Y0 Y5
Row 15 (20) 60
200.00%
239272472.xls.ms_office Page 4 of 6
Answer 7 - IRR with semi-annual cash flows
0.5 1.0 1.5 2.0 2.5 3.0
Cost of funds 42% 50.0 200.0
DCF 100.0 0.0 0.0 29.7 0.0 0.0 70.3
Cost (100.0)
NPV 0.0
Wrong answer 1 19% (100) 0 0 50 0 0 200
Wrong answer 2 24% (100) 0 50 0 0 200 0
Wrong answer 3 34% (100) 50 0 0 200
Wrong answer 4 69% (100) 50 200
Answer 8 - XIRR
27-Jun-08 15-Mar-11
IRR = 81% (1.0) 5.0
Years = 2.7
239272472.xls.ms_office Page 5 of 6
Method (i): NPV = 0
What would my IRR on the investment be (= discount rate where NPV = zero)?
Answer: IRR (= discount rate) = 24.57% Iterate and change cost of funds until NPV = zero
Year 0 1 2 3 4 5
Future value -20 0 0 0 0 60
Discount factor = 1/(1+interest rate)^year 1.00 0.80 0.64 0.52 0.42 0.33
Present value -20.00 0.00 0.00 0.00 0.00 20.00
Net present value = 0.00
Method (ii): Algebra
Algebra: 24.57%
Method (iii): IRR formula
IRR formula: 24.57% Formula for IRR
239272472.xls.ms_office Page 6 of 6
