The Time Value of Money

The Time Value of Money Intuition behind present value • Why is a dollar today worth more than a dollar tomorrow? – Inflation erodes the purchasing power of a dollar – Even if there is no inflation, people prefer to consume today than tomorrow – Also, there may be uncertainty (risk) about receiving the future cash flow Implication • Given everything else, the value of future cash flows decreases when – People prefer to consume more today – Expected inflation increases – The uncertainty of receiving the cash flows increases Discounting and Compounding • The discount rate is the rate at which present and future cash flows are traded off • It also shows the opportunity cost of the current investment • In other words, it shows the return that we would have made by investing our money elsewhere The discount rate incorporates … • Preference for current consumption – Greater  Higher discount rate • Expected inflation – Greater  Higher discount rate • Uncertainty of future cash flows – Greater  Higher discount rate • Discounting future cash flows converts them into present value dollars • Compounding present cash flows converts them into future value dollars Present value principle • Cash flows at different points in time cannot be compared and aggregated • All cash flows must be brought to the same point in time before comparisons and aggregations are made Types of cash flows • • • • • Simple cash flows Annuities Growing annuities Perpetuities Growing perpetuities Simple cash flows • Present value of simple cash flow PVt = CFt/(1+r)t • Future value of a simple cash flow FVt = CF0(1+r)t Frequency of compounding • The frequency of compounding affects future and present values of cash flows • The stated interest rate can deviate significantly from the true interest rate • Ex. 10% annual rate, with semiannual compounding turns out to be an effective rate of (1+(stated rate/t))t – 1 = 1.052 –1 = .10125 or 10.25% Impact of various compounding intervals on effective rate Frequency Rate t Formula Effective Annual Rate 10% Annual 10% 1 r Semiannual Monthly Daily Continuous 10% 10% 10% 10% 2 12 365 (1+r/2)2-1 (1+r/12)12-1 (1+r/365)365 -1 expr - 1 10.25% 10.47% 10.5156% 10.5171% Annuities • An annuity is a stream of constant cash flows that occur at regular intervals for a fixed period of time • The present value of an annuity can be found by taking each cash flow and discounting it back to present, and then adding up the present values A shortcut for calculating the present value of an annuity • A = annuity; r = discount rate; n = number of years; annuity received or paid at end of period 1  1   1  r n  PV  A, r, n   A  r       • When PV is known, we can solve from above for A • If annuity is received or paid at the beginning of period, then 1 1    1  r n 1  PV ( A, r, n )  A  A  r       • Again, if we know PV, we can solve for A Future value of an annuity • When annuity is received or paid at the end of each period  1  r n  1 FV  A, r, n   A  r   • When annuity is received or paid at the beginning of each period  1  r n  1 FV  A, r, n   A1  r   r   Example 1: Saving for college • Suppose you want to send your child to college (18 years from now). College tuition is $16,000/year and is expected to rise at 5% per year for the next 18 years. Suppose also that you can obtain an 8% after-tax return on your investments – – Expected tuition cost/year 18 years from now: 16000*(1.05)18 = $38,506 At the beginning of the four-year college period, we need to have an amount equal to the PV of the four-year tuition costs. This is equal to the PV of an annuity of A=$38,506 for n=4 discounted at r=8%: $38,506*[1-(1/(1.08)4)]/.08 = $127,537 • If we need to set aside a lump sum amount now, that amount would be equal to the present value of $127,537 with n=18 and r=8% – Amount need to set aside now is $127,537/(1.08)18 = $31,916 • If we want to set aside an annuity, starting a year from now, the amount would be given by solving for A in the formula for the FV of an annuity, where FV=$127,537, r=8%, and n=18. This would be equal to $3,405 Example 2: Valuing a straight bond • Current price (value) of bond: PV of coupon payments + PV of final payment (face value) • The value of a 15-year bond with a face value $1,000 paying annual coupon of 10.75% and, assuming that the current required rate (current rate on bonds of similar risk, which is used as the discount rate) is 8.5%, is given by – P = PV of annuity (coupon payments) + PV of face value = $107.5 * PV (A, 0.085, 15) + $1000/(1.085)15 = $107.5 [1-(1/(1.085)15]/0.085 + $1000/(1.085)15 = $1186.85 Growing annuities • A growing annuity is a cash flow growing at a constant rate (g) for a specified period of time • If A is the current cash flow, then Period Cash Flow 1 A(1+g) 2 A(1+g)2 3 A(1+g)3 … n A(1+g)n Present value of a growing annuity • The PV is given by  1  g n  1   1  r n  PV  A, r, g , n   A1  g   rg      • When r = g, the above equation cannot be used. Instead, the PV is equal to the sum of the annuities over the period • When r < g, we can still compute the PV of a growing annuity • Applications: mines, Perpetuities • A perpetuity is an annuity that lasts forever • The present value of a perpetuity is given by the same equation as for annuity where n= • That is PV = A/r • Typical examples are console bonds and preferred stock Example 3: Valuing a console bond • A console bond has no maturity and pays a fixed coupon every period • Suppose there is a console bond with face value of $1,000 that pays coupon of 6% annually. If the current required rate is 9%, the value of this bond is P = $60/0.09 = $667 Growing perpetuities • A growing perpetuity is a cash flow that is expected to grow at a constant rate forever • The PV of a growing perpetuity is given by the equation for growing annuity where n= • This is given by PV  A1  g  rg Example 4: Valuing a stock with growing dividends • Company A paid a $2.73 dividend per share in 2003. The company’s earnings and dividends have grown at 6% annually during the past four years and are expected to grow at the same rate in the future. Investors require a return of 12.23% for stock of similar risk to that of company A • Price of stock = (Dividend per share * g)/(r – g) = ($2.73*1.06)/(.1223 - .06) = $46.45