Note on Present Value

Transcription

1 Note on Present Value Demonstration Workbook John Joseph Crump, 2001 This workbook has been prepared for use in class discussion for Prof. Richard Dole's course in Sales and Leasing given at the University of Houston Law Center, Summer This Workbook contains the following tabs: Title Block RE Opportunity Valuing a hypothetical real estate investment opportunity Compounding Illustrates how compounding works, with several examples Inflation v. Real Two examples comparing actual growth, inflation, and real gain Discrete Factor Tables Tables, graphs, examples and notes for four PV and FV tables Annuity PV Factor Plot Plots annuity PV factor v. r and n PV with Periodic CF Example of an ordinary annuity problem Actual & PV CF Plot Graph from ordinary annuity problem IRR Plot Graph from ordinary annuity problem Martha's Lease Problem 2.1 from Keating CB Charlie's Lease Problem 23.1 from Keating CB Note on PV.xls Title Block 8/9/01 1 / 26

2 PV Example 2 Real Estate Investment Decision Interest Rate (r) 6.0% 7.0% % 17.2% 1+r Discount Factor (1/[1+r]) U.S. Government Securities Given Initial Investment (C 0 ) $350,000 $350,000 $350,000 $350,000 Compute Future Value (FV) After One Year (C 1 ) $371,000 $374,500 $400,000 $410,200 Initial Investment (C 0 ) Required to Receive Target FV $377,358 $373,832 $350,000 $341,297 Target Future Value (FV) After One Year (C 1 ) $400,000 $400,000 $400,000 $400,000 Rebuilt Apartment House Initial Real Estate Investment (C 0RE ) $250,000 $250,000 $250,000 $250,000 Initial Securities Investment (C 0S ) $100,000 $100,000 $100,000 $100,000 Total Initial Securities Investment (C 0 ) $350,000 $350,000 $350,000 $350,000 Real Estate Value After One Year (C 1RE ) $293,000 $293,000 $293,000 $293,000 Return on Real Estate Investment (r RE = C 1RE /C 0RE 1) % % % % Securities Value After One Year (C 1S ) $106,000 $107,000 $114,286 $117,200 Return on Securities Investment (r S = C 1S /C 0S 1) % % % % Total Value After One Year (C 1 ) $399,000 $400,000 $407,286 $410,200 Return on Portfolio of Investments (r = C 1 /C 0 1) % % % % Present Value (PV) of Amount C 1 $376,415 $373,832 $356,375 $350,000 Net Present Value (NPV) $26,415 $23,832 $6,375 $0 New Office Building Initial Real Estate Investment (C 0 ) $350,000 $350,000 $350,000 $350,000 Total Value After One Year (C 1 ) $400,000 $400,000 $400,000 $400,000 Return on Real Estate Investment (r) % % % % Present Value (PV) of Amount C 1 $377,358 $373,832 $350,000 $341,297 Net Present Value (NPV) $27,358 $23,832 $0 ($8,703) Note on PV.xls RE Opportunity 8/9/01 2 / 26

3 Compounding and Comparison of Factors FV Factor F(SP,r A,n) comparison for n = 1 Compare Discrete and Continuous FV Factors Annual vs. Continuous Compounding Compare Discrete and Continuous Return Rates Annual vs. Continuous Compounding Continuous Compounding, exp(r A n) Computed with n = 1 Perfect Match (1:1) Equivalent Continuous Return (r C ) Computed, n irrelevant Perfect Match (1:1) Discrete Compounding, F(SP,r A,n) Nominal Return, Compounded Annually (r A ) Note on PV.xls Compounding 8/9/01 3 / 26

4 Comparison of Single Payment Future Value Factors with Different Compounding Plans Discrete Compounding Compounding None Simple Interest Annually Semiannually Quarterly Monthly Daily Continuously Compounded Return Equation for F SP,r,n 1 + r A n (1 + r A ) 1n (1 + r A 2) 2n (1 + r A 4) 4n (1 + r A 12) 12n (1 + r A 365) 365n exp(r A n) X N/A n = 1 X(n) N/A r A 0% F(S$1,r A,n) % Increment 0.000% 0.000% 0.000% 0.000% 0.000% 1% F(S$1,r A,n) % Increment 0.002% 0.004% 0.005% 0.005% 0.005% 2% F(S$1,r A,n) % Increment 0.010% 0.015% 0.018% 0.020% 0.020% 3% F(S$1,r A,n) % Increment 0.022% 0.033% 0.040% 0.044% 0.044% 4% F(S$1,r A,n) % Increment 0.038% 0.058% 0.071% 0.078% 0.078% 5% F(S$1,r A,n) % Increment 0.060% 0.090% 0.111% 0.121% 0.121% 6% F(S$1,r A,n) % Increment 0.085% 0.129% 0.158% 0.173% 0.173% 7% F(S$1,r A,n) % Increment 0.114% 0.174% 0.214% 0.234% 0.234% 8% F(S$1,r A,n) % Increment 0.148% 0.225% 0.278% 0.303% 0.304% 9% F(S$1,r A,n) Note on PV.xls Compounding 8/9/01 4 / 26

5 % Increment 0.186% 0.283% 0.349% 0.382% 0.383% 10% F(S$1,r A,n) % Increment 0.227% 0.347% 0.428% 0.469% 0.470% 11% F(S$1,r A,n) % Increment 0.273% 0.416% 0.515% 0.564% 0.566% 12% F(S$1,r A,n) % Increment 0.321% 0.492% 0.609% 0.667% 0.669% 13% F(S$1,r A,n) % Increment 0.374% 0.573% 0.711% 0.779% 0.781% 14% F(S$1,r A,n) % Increment 0.430% 0.660% 0.819% 0.899% 0.901% 15% F(S$1,r A,n) % Increment 0.489% 0.752% 0.935% 1.026% 1.029% 16% F(S$1,r A,n) % Increment 0.552% 0.850% 1.058% 1.161% 1.165% 17% F(S$1,r A,n) % Increment 0.618% 0.953% 1.187% 1.304% 1.308% 18% F(S$1,r A,n) % Increment 0.686% 1.061% 1.324% 1.455% 1.459% 19% F(S$1,r A,n) % Increment 0.758% 1.174% 1.466% 1.613% 1.618% 20% F(S$1,r A,n) % Increment 0.833% 1.292% 1.616% 1.778% 1.784% 21% F(S$1,r A,n) % Increment 0.911% 1.415% 1.772% 1.951% 1.957% 22% F(S$1,r A,n) % Increment 0.992% 1.543% 1.934% 2.131% 2.137% 23% F(S$1,r A,n) % Increment 1.075% 1.676% 2.103% 2.318% 2.325% 24% F(S$1,r A,n) % Increment 1.161% 1.813% 2.278% 2.512% 2.520% 25% F(S$1,r A,n) % Increment 1.250% 1.954% 2.459% 2.713% 2.722% Note on PV.xls Compounding 8/9/01 5 / 26

6 Comparison of Effective Rates of Return with Different Compounding Frequencies Notes: The Nominal Rate of Return with annual compounding is equivalent to the annual Effective Rate of Return. Where r A is a Nominal Rate of Return compounded annually, r X = X(1+r A ) 1/X X is the equivalent Effective Return Rate for period X, where X is the actual number of compounding periods per year. Since the continuous compounding factor, exp(r C ), for an effective continuous return rate must equal F(SP,r A,1) where the equivalent effective annual interest rate is r A, we get exp(r C ) = 1 + r A, and r C = ln(1+r A ). Discrete Compounding Compounding Simple Interest Annually (Effective = Nominal) Semiannually Quarterly Monthly Daily Continuously Compounded Return Equation for F SP,r,n r A (1 + r A ) 1 1 2(1 + r A ) 1/2 2 4(1 + r A ) 1/4 4 12(1 + r A ) 1/ (1+r A ) 1/ ln(1+r A ) X N/A r A 0% Effective Return 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% % Change 1% Effective Return 1.0% 1.0% % % % % % % Increment % % % % % 2% Effective Return 2.0% 2.0% % % % % % % Increment % % % % % 3% Effective Return 3.0% 3.0% % % % % % % Increment % % % % % 4% Effective Return 4.0% 4.0% % % % % % % Increment % % % % % 5% Effective Return 5.0% 5.0% % % % % % % Increment % % % % % 6% Effective Return 6.0% 6.0% % % % % % % Increment % % % % % 7% Effective Return 7.0% 7.0% % % % % % % Increment % % % % % Note on PV.xls Compounding 8/9/01 6 / 26

7 8% Effective Return 8.0% 8.0% % % % % % % Increment % % % % % 9% Effective Return 9.0% 9.0% % % % % % % Increment % % % % % 10% Effective Return 10.0% 10.0% % % % % % % Increment % % % % % 11% Effective Return 11.0% 11.0% % % % % % % Increment % % % % % 12% Effective Return 12.0% 12.0% % % % % % % Increment % % % % % 13% Effective Return 13.0% 13.0% % % % % % % Increment % % % % % 14% Effective Return 14.0% 14.0% % % % % % % Increment % % % % % 15% Effective Return 15.0% 15.0% % % % % % % Increment % % % % % 16% Effective Return 16.0% 16.0% % % % % % % Increment % % % % % 17% Effective Return 17.0% 17.0% % % % % % % Increment % % % % % 18% Effective Return 18.0% 18.0% % % % % % % Increment % % % % % 19% Effective Return 19.0% 19.0% % % % % % % Increment % % % % % 20% Effective Return 20.0% 20.0% % % % % % % Increment % % % % % 21% Effective Return 21.0% 21.0% % % % % % % Increment % % % % % 22% Effective Return 22.0% 22.0% % % % % % % Increment % % % % % 23% Effective Return 23.0% 23.0% % % % % % % Increment % % % % % 24% Effective Return 24.0% 24.0% % % % % % % Increment % % % % % 25% Effective Return 25.0% 25.0% % % % % % % Increment % % % % % Note on PV.xls Compounding 8/9/01 7 / 26

8 Inflation and Real Interest Two Examples 1. If inflation is expected to average d percent per year, long term, what rate or return (r percent per year) is required to realize a real gain of 5 percent after acounting for inflation? Here, d = 5.00% We know that (1 + r) = (1 + r real ) (1 + d) (1 + r) = Target r real = 5.00% Therefore, r = 10.25% 2. Our factory will require addition of a new crane at time T (< 10). The cost today of such a crane is C. If equipment costs are expected to grow at an average of d percent per year between now and time T and the discount rate is r percent per year, what is the PV cost of the crane i.e., how much should we invest now to assure that we can afford the crane at time T? T = 5 C = $20,000 C infl = Inflated crane cost in T years = C (1 + d) T = $26, d = 6.00% r = 11.00% PV cost of crane (sinking fund investment today) = = C infl (1 + r) T = $15, r real = 4.72% Time (year) T Inflation d 6.00% 6.00% 6.00% 6.00% 6.00% 6.00% 6.00% 6.00% 6.00% 6.00% 6.00% Deflator (1 + d) T Discount Rate r 11.00% 11.00% 11.00% 11.00% 11.00% 11.00% 11.00% 11.00% 11.00% 11.00% 11.00% FV Factor (1 + r) T Discount Factor 1/(1 + r) T Real Growth Factor Difference Real Growth Rate r real 4.72% 4.72% 4.72% 4.72% 4.72% 4.72% 4.72% 4.72% 4.72% 4.72% 4.72% Real Growth Factor (1 + r real ) T Crane cost today C $20,000 Inflated crane cost C infl $20,000 $21, $22, $23, $25, $26, $28, $30, $31, $33, $35, Investment today $20,000 $19, $18, $17, $16, $15, $15, $14, $13, $13, $12, Note on PV.xls Inflation v. Real 8/9/01 8 / 26

9 Total Growth v. Real Growth and the Effect of Inflation Actual Growth Factor, (1 + r)^n Inflation Factor, (1 + d)^n Real Growth Factor, (1 + r-real)^n Actual Growth Factor Inflation Factor Growth Factors Years Note on PV.xls Inflation v. Real 8/9/01 9 / 26

10 Sample Tables for Four Present and Future Value Factors with Discrete Compounding F XY,r,n Notation: P = Present value, the value now, today. S n = Future value, at the end of the nth time period. R = R e = Uniform (i.e., repeated each period) end-of-year amount. Can be positive (revenue) or negative (cost). R b = Uniform beginning-of-year amount. r = Return, discount, interest, or hurdle rate, or the opportunity cost of capital, in % per time period. Note that these tables assume that r = r 1 = r 2 = = r n = constant. In practical capital budgeting, a single discount rate is usually applied, but it is not required. See R.A. BREALEY AND S.C. MEYERS, PRINCIPLES OF CORPORATE FINANCE at (6th ed. 2000). n = Number of periods for discrete (periodic) factors. Most commonly, n is in units of years, but it can be in any time units or it can be continuous. F XY,r,n = Present Value Factor for using X to compute Y, given r and n. Note on PV.xls Discrete Factor Tables 8/9/01 10 / 26

11 Single Payment Future Value Factors with Discrete Compounding F PS,r,n Here, S n = PF PS,r,n = P(1 + r) n, and F PS,r,n is the Future Value of $1.00 today, given r and n. Use the value now, P, to compute the future value, S, if that present value is invested for n time periods, earning interest or a return of r% per period. S n (n 1) n P As shown, this table assumes that payment in (P) and payment out (S n ) are made at the ends of periods 0 and n, respectively. Example $100 invested at 7% interest per year for 4 years is worth $100(1+0.07) 4 = $ = $ at the end of the fourth year. Note: The Rule of 17s or 72s Roughly speaking, F PS,r,n 2 in the range of 6% < r < 11% and 6 < n < 11, where r + n 17. Alternatively, dividing 72 by the interest rate (r) gives the approximate number of periods (n) needed for F PS,r,n to be about 2, and vice versa. Periodr the opportunity cost of capital, in % per time period. Period n 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 14% 15% 16% 17% 18% 19% 20% 21% 22% 23% 24% 25% 30% 35% 40% n E E E E E E E E E E E E E E E E % 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 14% 15% 16% 17% 18% 19% 20% 21% 22% 23% 24% 25% 30% 35% 40% Note on PV.xls Discrete Factor Tables 8/9/01 11 / 26

12 Single Payment Present Value Factors with Discrete Compounding F SP,r,n Here, P = S n F SP,r,n = S n (1 + r) n, and F SP,r,n is the Present Value of $1.00 at the end of period n, given r and n. Use the future (forecast) value, S, to compute the value now, P, that must be invested for n time periods, earning interest or a return of r% per period in order to grow to S. S n (n 1) n P As shown, this table assumes that payment in (P) and payment out (S n ) are made at the ends of periods 0 and n, respectively. Example At 7% interest per year, $100 that you expect to receive at the end of 4 years is worth $100/(1+0.07) 4 = $ = $ = $76.29 right now. Note: F SP,r,n is the inverse of F PS,r,n. F SP,r,n = 1 / F PS,r,n. Periodr the opportunity cost of capital, in % per time period. Period n 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 14% 15% 16% 17% 18% 19% 20% 21% 22% 23% 24% 25% 30% 35% 40% n E E E E E E E E E E E E E E E E E E E E E E E E E E E % 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 14% 15% 16% 17% 18% 19% 20% 21% 22% 23% 24% 25% 30% 35% 40% Note on PV.xls Discrete Factor Tables 8/9/01 12 / 26

13 Equal Payment Series Future Value or Annuity Future Value Factors with Discrete Compounding F RS,r,n Here, S n = RF RS,r,n = R[(1 + r) n 1] r, and F PS,r,n is the Future Value of n end of period payments of $1.00 per period, given r. Use the periodic payment amount, R, to compute the future value, S, if that payment is received at the end of each of n time periods, earning interest or a return of r% per period. S n Annuity Due or Annuity in Advance Payments in at the beginning of each period (n 1) n R = R e R e R e R e R e R e R e R e R e Ordinary Annuity or Annuity in Arrears Payments in at the end of each period. As shown, this table assumes that all payment in (R) and payment out (S n ) are made at the ends of periods, starting with the end of period 1. Example At 7% interest per year, $100 that you pay in at the end of each year will be worth $100 [(1+0.07) 4 1] 0.07 = $ = $ at the end of the 4 th year. Note: The Sinking Fund Factor, F SR,r,n, is the inverse of F RS,r,n. F SR,r,n = 1 / F RS,r,n. Also, F RS,r,n = F RP,r,n (1 + r) n = F RP,r,n F PS,r,n. Finally, if payments are made at the beginning of each period, instead of the end (R b, not R e ), R b (1+r) = R. Use that value in computations. Here, with beginning-of-period payments, the example becomes $100 (1+0.07) = $ Periodr the opportunity cost of capital, in % per time period. Period n 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 14% 15% 16% 17% 18% 19% 20% 21% 22% 23% 24% 25% 30% 35% 40% n E E E E E E E E E E E E E E E E E E E E E % 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 14% 15% 16% 17% 18% 19% 20% 21% 22% 23% 24% 25% 30% 35% 40% Note on PV.xls Discrete Factor Tables 8/9/01 13 / 26

14 Equal Payment Series Present Value or Annuity Present Value Factors with Discrete Compounding F RP,r,n Note: For a Perpetuity, n = and P = R/r. Here, P = RF RP,r,n = R[(1 + r) n 1] [r(1 + r) n ], and F PS,r,n is the Present Value at time t 0 of n end of period payments of $1.00 per period, given r. Compute the present value (P) that must be invested, if the return is r% per period, and the desired periodic payment (the annuity) is R, to be paid out at the end of each of n periods. R = R e R e R e R e R e R e R e R e R e Annuity Due or Annuity in Advance Payments in at the beginning of each period (n 1) n P Ordinary Annuity or Annuity in Arrears Payments in at the end of each period. As shown, this table assumes that payment in (P) is made at the end of period 0, and all payments out (the n R) are made at the ends of n periods thereafter. Example To fund 4 years of $100 year-end payments with 7% interest per year, invest $100 [(1+0.07) 4 1] [0.07(1+0.07) 4 ] = $ = $ now. Note: The Capital Recovery Factor, F PR,r,n, is the inverse of F RP,r,n F PR,r,n = 1 F RP,r,n. Also, F RS,r,n = F RP,r,n (1 + r) n = F RP,r,n F PS,r,n. Finally, if payments are made at the beginning of each period, instead of the end (R b, not R e ), R b (1+r) = R. Use that value in computations. Here, with beginning-of-period payments, the example becomes $100 (1+0.07) = $ Periodr the opportunity cost of capital, in % per time period. Period n 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 14% 15% 16% 17% 18% 19% 20% 21% 22% 23% 24% 25% 30% 35% 40% n , Perp , Perp 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 14% 15% 16% 17% 18% 19% 20% 21% 22% 23% 24% 25% 30% 35% 40% Note on PV.xls Discrete Factor Tables 8/9/01 14 / 26