Marco Bigelli Prof. Marco Bigelli

Transcription

1 Present value formulas for financial valuations Marco Bigelli

2 Agenda 1 Principles of valuations 2 Compounding and future values 3 Discounting and present value 4 Annual Perc. Rate and Effective Rate 5 Perpetuities and annuities

3 Valuation A great part of Corporate Finance has to do with valuations: Many times we have to estimate the value of: Bonds Stocks Investments in real assets Firms Financing

4 Valuation basic principle For whichever financial asset (a bond, a stock, an investment in real asset, a firm), the value is given by the present value of the cash flows generated by the asset

5 Present value principles Basic principles of the present value of a sum of cash available in the future are: 1) A dollar today is worth more than a dollar tomorrow 2) A safe dollar is worth more than a risky dollar It follows that present value formulas are the basics for financial valuations

6 Simple interests Let start with the easiest example. We have 100 USD and we invest it at 10% for one year. After one year we will have a capital (Future Value) equal to 110 USD ( ) FV = % = where: FV 1 = Future Value in year 1

7 Compound interests Let now assume that we reinvest the 110 USD (that we will get in one year time) for one more year again at the 10% interest rate. The Future Value after 2 years will be equal to 121 USD ( ) FV = % = where: FV 2 = Future Value in year 2

8 Compound interests By substituting the previous formula to FV 1, we can express FV 2 in this way: ( )( ) FV = % 1+ 10% = ( ) FV = % = This result holds also for other periods and can be generalized as follows: 2 ( ) FV = % = FV 3 t ( ) = % t 3

9 Future Values formula The Future Value formula can be generalized as follows: FV =C ( 1+ i ) t t 0 where: FV t = Future Value in year t C 0 = Capital in year 0 r = Annual interest rate

10 FV and compounding Value % 8% 6% 4 0 4% Years

11 Present value When we have to find the present value of a future cash flow, we just have to reverse the process. Using the previous example: PV0 = = = (1+ 10%) 1.21 Where: PV 0 = Present Value in year 0

12 Present value formula The present value formula of a future cash flow is nothing but the inverse of the Future Value formula: PV 0 = CF t (1+ i ) t Where: CF t = cash flow at year t; r = discount rate.

13 Present value of multiple cash flows And if we have more than one future cash flow? Nothing easier! Each of them can be discounted with the proper present value factor: PV 0 = CF 1 1+ r + CF 2 (1+ r ) 2 + CF 3 (1+ r ) 3

14 Present value factors The present value factors are the proper factors that multiplied for the each single future cash flows return the present values of the future cash flows. For ex., for future cash flows in year 1, 2 and 3 we have the following: PVFs : 1 1+ r ; 1 (1+ r ) ; 1 2 (1+ r ) ;... 3 At the end of the book there are matrix-tables of the PVFs for several years and different discount interest rates.

15 Additivity of PVs Additivity of PVs If we have n cash flows and we need to know the present value of all of them today, we can simple take the sum of the present values of all single cash flows. PV 0 = CF 1 1+ r + CF 2 (1+ r ) CF n (1+ r ) n

16 PV and discounting PV of $ Interest Rates 0% 5% 10% 15% Number of Years

17 Annual Percentage Rate Rates of return are generally stated in annual percentage rate (APR) terms, often referred to as the annual interest rate by definition the APR is the product of the periodic rate of return, r, earned over the period times the number of periods per year, m APR = r x m the periodic rate of return is the rate used to calculate interests in each interest calculation period» when interest is calculated once per year then the APR is the periodic interest rate when interest is calculated more often then we need to know the periodic interest rate to determine the APR or vice versa if the per period rate of return, r, is 1.5% and it is compounded 12 times a year, then the APR = r x m = 1.5% x 12 = 18% if the APR is 8% and it is compounded quarterly then the per-period interest rate is r = APR/m = 8% / 4 = 2%

18 Effective Annual Rate How do we compare rates with different compounding intervals? use Effective Annual Rates (EFF) this is the equivalent annual interest rate if interest were calculated only once a year rather than m times a year EFF = (1 + APR/m) m - 1 example: APR = 8% annual compounding (m=1) => EFF = (1+.08/1) 1 1 = 8% quarterly compounding (m = 4) => EFF = (1+.08/4) 4 1 = 8.24% monthly compounding (m =12) => EFF = (1+.08/12) 12 1 = 8.30% daily compounding (m = 365) => EFF = (1+.08/365) = 8.327% as m, EFF exp APR - 1 => EFF = exp.08 1 = 8.33%» exp is the number 2.718; when m the EFF becomes the continuously compounded annual rate

19 Short Cuts for perpetuities Sometimes there are shortcuts that make it very easy to calculate the present value of an asset that pays off in very long periods and are proxied to perpetuities.

20 Perpetuity After the 5Y budget, when evaluating a firm or an investment project, perpetuity assumptions are often used: CFs 5 Years

21 Perpetuity as a proxy of 100yCFs Since the PV of 100 CFs is almost equal to the PV of infinite cash flows (as far CFs have increasing lower present values), long periods of cash flows in firm s or investment valuations can be proxied by perpetuities

22 Perpetuity A perpetuity it s a cash flow or a payment paid forever (for example once a year) How can we sum up the PVs of infinite cash flows? PV 0 = C 1+ r + C (1+ r ) C (1+ r ) n +

23 Perpetuity The present value of such sequence of equal cash flows received for ever is equal to: PV 0 = C 1 i Where: C 1 = constant cash flow r = discount rate

24 Perpetuity: example If a bond pays once a year and forever a coupon equal to 5% of its par value (100), we would have the following sequence of cash flows (CF): Today Which is the value of the perpetual bond if the proper discount rate is equal to 10%? Easy: 50 (5/10%)

25 Constant growing perpetuity A constant growing perpetuity is a series of cash flows which grows at a constant growth rate equal to g. The value of such a stream of cash flows is also given by an easy shortcut : PV 0 = CF 1 r g

26 Cosnstant growing perpetuity: an example Whcih would be the value of a financial asset that pays the following cash flows: 10 10,5 11,025 today If the proper market discount rate is equal to 10% and all the CFs grow at the 5% growth rate (= g), The value would be equal to 200 (try it!)

27 Present value of an annuity Annuity - An asset that pays a fixed sum each year from year 1 to year t. PV of annuity 1 1 = C t r r 1 r ( + )

28 Short Cuts (no growth)

29 Summary of main formulas Present value (PV) PV 0 = C t / (1+r) t Future value (FV) FV N = C 0 x (1+r) t Annuities fixed payments for from year 1 to t» PV 0 = C[1/r 1/r (1+r) -t ] perpetuities payments that go on forever» PV 0 = C / r growing payments that go on forever» PV 0 = C 1 / (r g)