FINAN303 Principles of Finance Spring Time Value of Money Part B

Transcription

1 Time Value of Money Part B 1. Examples of multiple cash flows - PV Mult = a. Present value of a perpetuity b. Present value of an annuity c. Uneven cash flows T CF t t=0 (1+i) t 2. Annuity vs. Perpetuity a. Annuity: Equal cash flows between equal time periods over a finite period of time. b. Perpetuity: Equal cash flows between equal time periods over an infinite period of time c. Examples: i. Annuity or Perpetuity? Deriving the present value of a perpetuity a. A multiple cash flow formula will crush all the multiple cash flows into one equivalent value at a specific time period. b. In example below, for a given i there exists a dollar value at time 0 in which you are theoretically indifferent between all the values that go on forever and one value ($X) at t=0. c. Suppose you were given today. You deposit it in a bank and the effective annual interest rate is 10% (i=0.1). i. How much cash would you have in a year? Office: MBEB garrettmcbrayer@boisestate.edu

2 d. At time 1, you spend $10 and leave in the bank. e. How much money will you have in the bank at time 2? f. Again, you decide to spend $10 and leave the rest in the bank. g. At this rate, could you support this spending indefinitely? i. Therefore, with i=0.1, at time 0 provides a cash flow stream of $10 beginning at time 1 forever. 1. Notice that: = $ h. More generally: i. Using symbols: 1. R = the repeated cash flow 2. i = the effective periodic rate 3. PV Perp = the present value of the perpetuity PV Perp = R i i. Two traps with this formula: i. The i and the n must match!!! ii. The solution is one time period prior to the first R. Office: MBEB garrettmcbrayer@boisestate.edu

3 j. Once you have crushed the multiple cash flows into a single number (i.e., PV Perp ) then you can move it along the timeline. k. Perpetuity Example: i. You receive a perpetuity beginning 6 months from today of every quarter. A bank pays 6% compounded quarterly. 1. What is the value of the perpetuity today? 2. What is the value of the perpetuity a year from today? 3. What if the payments are made every month? 4. Deriving the present value of an annuity a. Based on knowing the present value of a perpetuity formula. b. Two types of annuities: i. Ordinary Annuity Payments occur at the end of the period ii. Annuity Due Payments occur at the beginning of the period iii. Which has a higher present value? Future value? c. A multiple cash flow formula will crush all the multiple cash flows into one equivalent value at a specific time period. d. In example below, for a given i there exists a dollar value at time 0 in which you are theoretically indifferent between all the annuity and one value ($X) at t=0. Office: MBEB garrettmcbrayer@boisestate.edu

4 e. Per the perpetuity formula ( R ), the below perpetuity is worth 0 at t=0. i f. Suppose we want to derive the value of the annuity of at t=1 and t=2. i. Let s split the perpetuity (A) into two parts (B) and (C) ii. What is the value of (C)? iii. Which one, (A), (B), or (C) is the annuity? iv. What is the relation between (A), (B), and (C)? 1. A = B + C a. Hence: B = A C 2. PV an = 0 t=0 0 t=2 a. Any Issues? g. Hence the PV Annuity is the present value of a perpetuity today less the present value of a perpetuity in the future discounted to today. PV An = 0 0 ( ) 2 = $ Office: MBEB garrettmcbrayer@boisestate.edu

5 h. How was the 0 calculated? R i = 0.1 = 0 thus PV An = ( ) 2 or PV An = 0.1 (1 1 ( ) 2) i. More generally: PV An = $R i (1 1 ( ) 2) i. Where: n = number of payments to be discounted j. Two traps and one caveat: i. The i and the n must match!!! ii. The solution is one time period prior to the first R. iii. Caveat: no closed for solution for i. k. Once crushed into a single number (i.e., PV An ) you can move it along the timeline. 5. Examples Galore!!! a. You receive an ordinary annuity beginning in one year of every year for 12 years. The effective annual rate is 12%. b. You receive an annuity due beginning today of every year for 12 years. The effective annual rate is 12%. c. You receive an ordinary annuity beginning 6 months from today of $500 every quarter for 10 quarters. The effective quarterly rate is 3%. d. We need to accumulate $10,000 and have it available 5 years from now. Suppose we can earn 6%, compounded semi-annually, on our savings. i. How much do we need to save annually starting at the end of year one? e. Same problem, now assume that we can only afford $1,500 per year. i. What interest rate do we need to get to $10,000 at the end of year 5? f. Same problem, but now we think that we will not get the 14.43% required to get to $10,000 in 5 year. We know that banks are paying 6% annually. i. At 6% with an annual deposit of $1,500, how long until we have $10,000? 6. Uneven cash flows $300 $200 $400 a. How can we find the PV of the above cash flow stream? i. Discount each cash flow and sum Office: MBEB garrettmcbrayer@boisestate.edu