Question

Calculate the future value in 5 years of $2100 today with annual compounding and a 10% annual interest rate. Suppose someone saves $1000 today and will have $1052 one year from today. If compounding is daily (assume 365 days in a year), what must be the interest rate on this account? Jane offers Kathy the following deal. Jane will give Kathy $900 today if Kathy gives Jane $1100 in 2 years-time. Suppose there is quarterly compounding and the quarterly interest rate is 3%. Should Kathy accept this deal? Provide a calculation. Suppose Jane had been thinking about offering Kathy a trade of $900 for $1300 in 2 years-time. With quarterly compounding, what would the quarterly interest rate have needed to be to make that a fair trade? Provide a calculation. Suppose there is a current cash flow of $5 that is expected to grow by 14% each year. If the annual interest rate is 15%, what is the present value of this stream of cash flows? Using the information from part e), what would be the future value of that same stream of cash 2 years from now? Suppose the present value of a growing perpetuity is $2100. If the annual interest rate is 10%, and the current cash flow is $100, what must be the growth rate of this perpetuity?

Answer 1

Solution 4 (a) Future value Number of periods Future value = Present value x (1 + Periodic interest rate) = $2,100 x (1 + 10%) = $2,100 1.610510 = $3,382.07 Therefore, the future value is $3,382.07 Solution 4 (b) Interest rate Number of compounding periods Future value = Present value x 1+; Annual Percentage Rate Number of compounding periods $1,052 = $1,000x[1+ Annual Percentage Rate) ** 365 $1,0526 Annual Percentage Rate $1, 000 3 65 (1.052)36 = 1+ Annual Percentage Rate 365 1.00013889488970 = {1+ Annual Percentage Rate 365 Annual Percentage Rate = (1.00013889488970-1)× 365 = 0.00013889488970x365 = 5.07% Therefore, the interest rate on this account is 5.07%

Solution 4 (c) Future value Number of quarters Future value = Present value x (1 + Quarterly interest rate) in two years) = $900 x (1 + 3%) = $900 x 1.2667700813876 = $ 1,1 40.09 Therefore, the future value is $1,140.09 with quarterly interest rate of 3% for 8 quarter (2 years). Therefore, Kathy should accept the deal as she will have to give only $1,100 instead of $1,140.09. The future value of $900 at quarterly compounding is higher and she will have to return less amount, so, she should Accept the offer. Solution 4 (d) Quarterly rate Number of Future value = Present value x (1 + Quarterly interest rate) in two years $1,300 = $900 x (1 + Quarterly interest rate) (1 + Ouarterly interest rates_ $1,300 $900 1 + Quarterly interest rate= (1.444444444444440) Quarterly interest rate = 1.047038389615530 - 1 = 4.70% Therefore, the quarterly interest rate on this trade must be 4.70% deal. to make it a fair

Solution 4 (e) Present value The dividend discount model can be used to calculate the present value of stream of constantly growing cash flows as follows: Current cash flow (CF, )*(1+ Constant Growth rate) Present value =- (Interest rate - Constant Growth rate) $5(1+14%) (15% - 14%) $5.70 1% $570.00 Therefore, the present value of stream of cash flows is $370.00|| Solution 4 (1) Value two years from now The dividend discount model can be used to calculate the present value of stream of constantly growing cash flows (future value two years from now) as follows: Current cash flow (CF,)(1+ Constant Growth rate) are Present value to year:= (Interest rate - Constant Growth rate) _$5*(1+ 14%) (15% - 14%) $7.4077 1% = $740.77 Therefore, the future value of stream of cash flows two years from now is $740.77. Note: Years taken as 3 because, to calculate the value at the end of second year, year 3 cash flow will be used in dividend discount model.

Alternatively: Future value = Present value x (1 + Grow th rate ) te Number of years = $570.00 x (1 + 14% ) $ 740.777 Therefore, the future value of stream of cash flows two years from now is $740.77 Solution 4 (g) Growth rate The dividend discount model can be used to calculate the growth rate of a growing perpetuity as follows: Current cash flow (CF) (1+ Constant Growth rate) Present value = (Interest rate - Constant Growth rate) $100x(1+ Constant Growth rate) $2,100 = (10% - Constant Growth rate) $2,100 (10% - Constant Growth rate)= $100(1+ Constant Growth rate) $210 - $2,100x Constant Growth rate = $100+ $100x Constant Growth rate $2.100 x Constant Growth rate - $ 210-$100 +$100x Constant Growth rate ($2,200x Constant Growth rate) = $110 $110 Constant Growth rate = - $2,200 = 5.00% Therefore, the growth rate on this perpetuity is