Time Value of Money

Time Value of Money Corporate Accounting LIABILITIES: Current Liabilities „ „ „ Obligations that must be discharged in a short period of time (generally less than one year) Reported on balance sheet at nominal value Examples: ƒ ƒ ƒ ƒ ƒ ƒ Accounts payable Short-term borrowings Current portion of long-term debt Deposits Warranties Deferred Revenues / Income LIABILITIES: Long-term Liabilities „ „ „ Obligations spanning a longer period of time (generally more than one year) Generally reported on the balance sheet at present value based on interest rate when initiated Examples: Bonds Long-term loans Mortgages Capital Leases „ How do we compute present values? And interest expense? Time Value Of Money Interest = 10% Time 0 $1.00 Time 1 $1.00 Time 1 $1.10 Future value of $ 1.00 today = $1.00 (1+10%) = $1.10 at the end of one year. What is the present value of $1.10 to be received one year from now? Present value of $1.10 one year from now = $1.10/(1+10%) = $1.00 What is the present value of $1.00 to be received one year from now? Present value of $1.00 one year from now = $1.00/(1.10) = $0.91 Time Value Of Money $0.83 T=0 $1.00 $0.91 T=1 $1.10 $1.00 T=2 $1.21 Future value of $1.00 two years from now = $1.00*(1+10%)*(1+10%) = $1.00*(1.10)2 = $1.21 Present value of $1.00 to be received two years from now = $1.00/[(1.10)2] = $0.83 RECALL: PV of $1.00 to be received a year from now = $0.91 Calculating present values: An example You have just won a lottery. The lottery board offers you three different options for collecting your winnings: (1) Payments of $500,000 at the end of each year for 20 years. (2) Lump-sum payment of $4,500,000 today. (3) Lump-sum payment of $1 million today, followed by $2,100,000 at the end of years 5, 6, and 7. Assume all earnings can be invested at a 10 percent annual rate. Ignoring any tax effects, which option should you choose and why? Future Value of Option 1: $500,000 at the end of each year for 20 years. 0 1 2 3 4 5 6 7 19 years 18 years 17 years 19 20 $0.5m×(1.1)19= $3.06m $0.5m×(1.1)18= $2.78m $0.5m×(1.1)17= $2.53m $0.5m×(1.1)1 = $0.55m $0.5m×(1.1)0 = $0.50m $28.64m Future Value of Option 2: Lump-sum payment of $4,500,000 today 0 1 2 3 4 5 6 7 20 years 19 20 $4.5m ×(1.1)20= $30.27m Future Value of Option 3: $1m today, and $2.1m at the end of years 5, 6, and 7. 0 1 2 3 4 5 6 7 19 20 20 years $1.0m×(1.1)20= $3.36m 15 years 14 years 13 years $2.1m×(1.1)15= $8.77m $2.1m×(1.1)14= $7.97m $2.1m×(1.1)13= $7.25m $30.71m Future Values z If you invest all lottery receipts at 10% per year, how much will you have in 20 years? 1. $500K × (1.10)19 + $500K × (1.10)18 + ... ... + $500K × (1.10)1 + $500K = $28.64m 2. $4,500,000 × (1.10)20 = $30.27m 3. $1m × (1.10)20 + $2.1m × (1.10)15 + $2.1m × (1.10)14 + $2.1m × (1.10)13 = $30.71m Î FV(Option 1) < FV(Option 2) < FV(Option 3) Present Value of Option 1: $500,000 at the end of each year for 20 years. 0 1 2 3 4 5 6 7 $0.5m×(1.1)-1 = $0.45m 1 year $0.5m×(1.1)-2 = $0.41m 2 years $0.5m×(1.1)-3 = $0.38m 3 years $0.5m×(1.1)-19= $0.08m $0.5m×(1.1)-20= $0.07m $4.26m 19 20 19 years 20 years Present Value of Option 2: Lump-sum payment of $4,500,000 today 0 1 2 3 4 5 6 7 $4.5m 19 20 Present Value of Option 3: $1m today, and $2.1m at the end of years 5, 6, and 7. 0 1 2 3 4 5 6 7 $1.0m×(1.1)0 = $1.00m $2.1m×(1.1)-5 = $1.30m $2.1m×(1.1)-6 = $1.19m $2.1m×(1.1)-7 = $1.08m $4.57m 5 years 6 years 7 years 19 20 Present Values z If all lottery receipts can be invested at 10% per year, what is the present value of each option? 1. $500K × (1.10)-20 + $500K × (1.10)-19 + … ... + $500K× (1.10)-2 + $500K × (1.10)-1 = $4.26m 2. $4,500,000 × (1.10)0 = $4.5m 3. $1m × (1.10)0 + $2.1m × (1.10)-5 + $2.1m × (1.10)-6 + $2.1m × (1.10)-7 = $4.57m Î PV(Option 1) < PV(Option 2) < PV(Option 3) Converting Present and Future Values Present Values Option 1 $4.26m 0 1 2 3 4 5 6 7 FV = 4.26 × 1.120 = 28.64 19 20 Future Values Option 1 $28.64m PV = 28.64×1.1-20 = 4.26 Option 2 $4.50m FV = 4.50 × 1.120 = 30.27 Option 2 $30.27m PV = 30.27×1.1-20 = 4.50 Option 3 $4.57m FV = 4.57 × 1.120 = 30.71 PV = 30.71×1.1-20 = 4.57 Option 3 $30.71m Using PV and FV Tables (Appendix) z Table 1: Future Value of $1 z z z z A one-time payment to be received now and held (reinvested) for N periods Compounded at interest rate r% Multiply the dollar amount received by the factor in Row N, Column r% Table 2: Present Value of $1 z z z A one-time payment to be received N periods from now Discounted at interest rate r Multiply the dollar amount to be received by the factor in Row N, Column r Time Value of Money Terminology z Annuity: a stream of fixed-dollar payments made at regular intervals of time z z z Ordinary Annuity (annuity in arrears): payments occur at the end of the period Annuity due (annuity in advance): payments occur at the beginning of the period Formulas: z FV(a) = { [((1+r)^N) - 1] / r} * Fixed Period Cash Flow z PV(a) = { [(1 - (1+r)^(-N))] / r] } * Fixed Period Cash Flow Using PV and FV Tables (Appendix) z Table 3: Future Value of $1 ordinary annuity (annuity in arrears) z z z z Regular payments to be received at end of year for N years and held (reinvested) until time N Compounded at interest rate r% Multiply the dollar amount received by the factor in Row N, Column r% FV of $1 annuity due (annuity in advance) = (FV of an ordinary annuity for N+1 years) - $1 Using PV and FV Tables (Appendix) z Table 4: Present Value of $1 ordinary annuity (annuity in arrears) z z z z Regular payments to be received at end of year for N years Discounted at interest rate r% Multiply the dollar amount to be received by the factor in Row N, Column r PV of $1 annuity due (annuity in advance) = (PV of an ordinary annuity for N-1 years) + $1