Chapter Organization. The future value (FV) is the cash value of. an investment at some time in the future.

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Luke Walters
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1 Chapter 5 The Time Value of Money Chapter Organization 5.2. Present Value and Discounting The future value (FV) is the cash value of an investment at some time in the future Suppose you invest 100 in a saving account that pays 10% interest per year. How much will you have in one year? 1 2 The future value (FV) is the cash value of an investment at some time in the future. Suppose you invest 100 in a saving account that pays 10% interest per year. How much will you have in one year? 110 The future value (FV) is the cash value of an investment at some time in the future. Suppose you invest 100 in a saving account that pays 10% interest per year. How much will you have in one year? x 1,1 = 110 (1+r) r an interest rate 3 4 1

2 If you leave 110 in bank for the next year, how much will you have in two years? If you leave 110 in bank for the next year, how much will you have in two years? 121 = 110 x 1,1 These 121 consists of four parts: 100 the original principal 10 10% interest you earned the first year 10 10% interest you earned the second year 1 interest you earned the second year on the interest paid the first year 5 6 The compound interest means earning interest on interest. The simple interest is the interest which is not reinvested and it is earned each period only on the original principal. Now take a closer look at how we calculated the 121 future value. 121 = 110 x 1,1= 100 x 1,1 x 1,1 = = 100 x 1,1 2 If we invest 121 for one year again we will have 133,10 = 121 x 1,1 = 100 x 1,1 3 Future value = 100 x (1+r) t 7 8 2

3 Future value = 100 x (1+r) t Future value = 100 x (1+r) t (1+r) t future value interest factor What would your 100 be worth after five years? FV = 100x(1+0,1) 5 = 100 x 1,6105 =161,05 1,1 y x 5 = 1, Present Value and Discounting How much do we have to invest today at 10% to get 100 in one year? FV = 100 x (1+r) t 100 = Present value x (1+0,1) Present value = = 90,909 1,

4 5.2. Present Value and Discounting To discount means to calculate the present value of some future amount 1 / (1+r) t discount factor 5.2. Present Value and Discounting Suppose you need to have in two years. If you can earn 7%, how much do you have to invest to make sure that you have the when you need it? = PV x 1,07 2 PV = / 1,1449 = 873,44 1,07 y x 2 1/x x 1000 = 873, PV = FV t / (1+r) t Your company proposes to buy an asset for 335. This investment is very safe. You would sell off the asset in three years for 400. You know you could invest the $335 elsewhere at 10 percent with very little risk. What do you think of the proposed investment? FV = 335 x 1,1 3 = 335 x 1,331 = 445,89 This is not a good investment. How much should you invest in order to make this investment be as good as the other one? PV = FV t / (1+r) t PV= 400 / (1+0,1) 3 = 400 / 1,331 = 300,

5 PV = FV t / (1+r) t Determining the Discount Rate r We are offered an investment that costs us and will double our money in eight years. To compare this to other investments, we would like to know what discount rate is implicit in these numbers? = x (1+r) 8 2 = (1+r) = 1+r 2 1/8 = 1+r 1, = 1+r r = 9,0508% The Rule of 72. For reasonable rates of return, the time it takes to double your money is given approximately by 72 / r%. t 72/r% r 72/8 9% 8 72/r% This rule is fairly accurate for discount rates in the range of 5 percent to 20 percent. Finding the number of periods t Suppose we are interested in purchasing an asset that costs 50,000. We currently have 25,000. If we can earn 12 percent on this 25,000, how long will it take to have 50,000? t 72/r% t 72/12% 6 years

6 = x (1+0,12) t (1+0,12) t = 2 log 1,12 t = log 2 t x log 1,12 = log 2 t = log 2 / log 1,12 t = 0, / 0, = 6,11625 years 21 6

Simple Interest: Interest earned only on the original principal amount invested.

53 Future Value (FV): The amount an investment is worth after one or more periods. Simple Interest: Interest earned only on the original principal amount invested. Compound Interest: Interest earned on