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- 1. Investment Tools – Time Value of Money
- 2. <ul><li>Concepts Covered in This Section </li></ul><ul><ul><li>Future value </li></ul></ul><ul><ul><li>Present value </li></ul></ul><ul><ul><li>Perpetuities </li></ul></ul><ul><ul><li>Annuities </li></ul></ul><ul><ul><li>Uneven Cash Flows </li></ul></ul><ul><ul><li>Rates of return </li></ul></ul>Time Value of Money
- 3. <ul><li>Time lines show timing of cash flows. </li></ul><ul><li>Tick marks at ends of periods. </li></ul><ul><ul><li>Time 0 is today; Time 1 is the end of Period 1; or the beginning of Period 2. </li></ul></ul><ul><li>90% of getting a Time Value problem correct is setting up the timeline correctly!!! </li></ul>Interest Rate Cash Flows CF 0 CF 1 CF 3 CF 2 0 1 2 3 i%
- 4. What’s the FV of an initial $100 after 3 years if i = 10%? <ul><li>Finding FVs (moving to the right on a time line) is called compounding. </li></ul><ul><ul><li>Compounding involves earning interest on interest for investments of more than one period. </li></ul></ul>FV = ? Future Values 100 0 1 2 3 10%
- 5. Single Sum - Future & Present Value <ul><li>Assume that you can invest PV at interest rate i to receive future sum, FV </li></ul><ul><li>Similar reasoning leads to Present Value of a Future sum today. </li></ul>1 2 3 0 FV 1 = (1+i)PV FV 3 = (1+i) 3 PV PV FV 2 = (1+i) 2 PV 1 2 3 0 PV = FV 1 /(1+i) FV 1 PV = FV 2 /(1+i) 2 FV 2 PV = FV 3 /(1+i) 3 FV 3
- 6. PV Calculation for $100 received in 3 years if interest rate is 10% Single Sum – FV & PV Formulas FV n = PV(1 + i ) n for given PV $100 = 0.7513 = $75.13. 1.10 PV = $100 1 3
- 7. Question on PV of a given FV <ul><li>Ex 1. An investor wants to have $1 million when she retires in 20 years. If she can earn a 10 percent annual return, compounded annually, on her investments, the lump-sum amount she would need to invest today to reach her goal is closest to: </li></ul><ul><ul><ul><li>A. $100,000. </li></ul></ul></ul><ul><ul><ul><li>B. $117,459. </li></ul></ul></ul><ul><ul><ul><li>C. $148,644. </li></ul></ul></ul><ul><ul><ul><li>D. $161,506. </li></ul></ul></ul><ul><li>This is a single payment to be turned into a set future value FV=$1,000,000 in N=20 years time invested at r=10% interest rate. </li></ul><ul><li>PV =[ 1/(1+r) ] N FV </li></ul><ul><li>PV = [ 1/(1.10) ] 20 $1,000,000 </li></ul><ul><li>PV 10 = [0.14864]($1,000,000) </li></ul><ul><li>PV 10 = $148,644 </li></ul>
- 8. Perpetuities Perpetuity is a series of constant payments, A, each period forever. Intuition: Present Value of a perpetuity is the amount that must invested today at the interest rate i to yield a payment of A each year without affecting the value of the initial investment. PV perpetuity = [A /(1+i) t ] = A [ 1/(1+i) t ] = A/i 1 2 3 4 5 6 7 A 0 A A A A A A PV 1 = A/(1+r) PV 2 = A/(1+r) 2 PV 3 = A/(1+r) 3 PV 4 = A/(1+r) 4 etc. etc.
- 9. <ul><li>Regular or ordinary annuity is a finite set of sequential cash flows, all with the same value A , which has a first cash flow that occurs one period from now. </li></ul><ul><li>An annuity due is a finite set of sequential cash flows, all with the same value A, which has a first cash flow that is paid immediately . </li></ul>Annuities
- 10. Time line for an ordinary annuity of $100 for 3 years. $100 $100 $100 i% Ordinary Annuity Timeline 0 1 2 3
- 11. Difference between an ordinary annuity and an annuity due ? Ordinary Annuity vs. Annuity Due PMT PMT 0 1 2 3 i% PMT Annuity Due PV FV Ordinary Annuity PMT PMT PMT 0 1 2 3 i%
- 12. Annuity Formula and Perpetuities Intuition : Formula for a N-period annuity of A is: PV of a Perpetuity of A today minus PV of a Perpetuity of A in period N 2 4 6 8 10 12 14 1. Perpetuity of A per period in Period 0 -- PV 1 = A/i A 0 A A A A A A A A A A A A A 2 4 6 8 10 12 14 2. Perpetuity of A per period in Period 8 -- PV 8 = [1/(1+i)] 8 x (A/i) 0 A A A A A A 2 4 6 8 10 12 14 3. Annuity of A for 8 periods -- PV = PV 1 – PV 8 = (A/i) x { 1 – [1/(1+i)] 8 } A 0 A A A A A A A
- 13. Annuities & Perpetuities Again <ul><li>Rather than memorize the annuity formula, it is easier to calculate it as the difference between two perpetuities with the same payment. </li></ul><ul><li>PV of an N-period annuity of $A per period is: </li></ul><ul><li>PV N = </li></ul><ul><li>(A/i) x { 1 – [1/(1+i)] N } </li></ul><ul><li>Calculating the PV of an annuity has 3 steps: </li></ul><ul><li>Calculate (A/i) </li></ul><ul><ul><li>PV of a Perpetuity with payments of $A per period. </li></ul></ul><ul><li>Calculate [1/(1+i)] N </li></ul><ul><ul><li>Discount factor associated with end of the annuity. </li></ul></ul><ul><li>Calculate PV N = </li></ul><ul><li>(A/i) x { 1 - [1/(1 + i)] N } </li></ul><ul><ul><li>I think this is easier under pressure than memorizing the formula. </li></ul></ul>
- 14. Question on FV of Annuity Due <ul><li>Ex 2.An individual deposits $10,000 at the beginning of each of the next 10 years, starting today, into an account paying 9 percent interest compounded annually. The amount of money in the account at the end of 10 years will be closest to: </li></ul><ul><ul><ul><li>A. $109,000. </li></ul></ul></ul><ul><ul><ul><li>B. $143,200. </li></ul></ul></ul><ul><ul><ul><li>C. $151,900. </li></ul></ul></ul><ul><ul><ul><li>D. $165,600. </li></ul></ul></ul><ul><li>This is an annuity due of A=$10,000 for N=10 years at i=9% interest rate. </li></ul><ul><li>Annuity due must be adjusted by (1+i) to reflect payment is made at beginning rather than end of period. </li></ul><ul><li>Also must adjust PV formula by (1+i) N for FV of annuity. </li></ul><ul><li>PV N = (1+i) N (1+i) [ ( A/i) { 1 – [1/(1+i)] N } ] </li></ul><ul><li>PV 10 = (1.09) 11 ($10K/.09) {1 – [1/1.09] 10 } </li></ul><ul><li>PV 10 = (2.58)($111,111){1 – [0.42]} </li></ul><ul><li>PV 10 = $165,601 </li></ul>
- 15. Time line for uneven CFs: $100 at end of Year 1 (t = 1), $200 at t=2, and$300 at the end of Year 3. $100 $300 $200 Uneven Cash Flows 0 1 2 3 i%
- 16. Question on Uneven Cash Flows <ul><li>Ex 3.An investment promises to pay $100 one year from today, $200 two years from today, and $300 three years from today. If the required rate of return is 14 percent, compounded annually, the value of this investment today is closest to: </li></ul><ul><ul><ul><li>A. $404. </li></ul></ul></ul><ul><ul><ul><li>B. $444. </li></ul></ul></ul><ul><ul><ul><li>C. $462. </li></ul></ul></ul><ul><ul><ul><li>D. $516. </li></ul></ul></ul><ul><li>This is a set of unequal cash flows. You could do it as a sum of annuities but it is easier to calculate it directly in this case. </li></ul><ul><li>Interest rate is i =14%. </li></ul><ul><li>PV = [ 1/(1+i) ] t FV t </li></ul><ul><li>PV = $100/(1.14) + $200/(1.14) 2 + $300/(1.14) 3 </li></ul><ul><li>PV = $87.72 + $153.89 + $202.49 </li></ul><ul><li>PV = $444.10 </li></ul>
- 17. Uneven Cash Flows Intuition : PV of uneven cash flows is equal to the sum of the PV’s of regular cash flows that sum to the uneven cash flows. 2 4 6 8 10 12 14 1. Uneven cash Flows over 10 periods – PV = PV 10 + PV 4 5 0 $100 $100 $100 $100 $100 $500 $500 $500 $100 $500 2. Annuity of $100 per period for 10 periods -- PV 10 = { 1 - [1/(1+i)] 10 } x (A/i) 2 4 6 8 10 12 14 0 $100 $100 $100 $100 $100 $100 $100 $100 $100 $100 3. Annuity of $400 per period for 4 periods from period 5 -- PV 4 5 = [1/(1+i)] 5 x [ (A/i) x { 1 – [1/(1+i)] 4 } ] 2 4 6 8 10 12 14 0 $400 $400 $400 $400
- 18. Comparison of Compounding Periods Annually: FV 3 = $100(1.10) 3 = $133.10. Semiannually: FV 6 = $100(1.05) 6 = $134.01. 0 1 2 3 10% 100 133.10 0 1 2 3 5% 4 5 6 134.01 1 2 3 0 100
- 19. Questions on Time Value <ul><li>Develop an approach to problems on Time Value. </li></ul><ul><ul><li>Draw the Time line for the cash flows. </li></ul></ul><ul><ul><li>Put in the cash flows from the problem. </li></ul></ul><ul><ul><li>Identify if single payment, annuity, annuity due, or perpetuity. </li></ul></ul><ul><ul><ul><li>If uneven cash flows can you break it into sums of annuities? </li></ul></ul></ul><ul><ul><li>Identify what is to be calculated – PV, FV, N or i ? </li></ul></ul><ul><ul><li>Write out the appropriate formula, put in values for the variables, and calculate. </li></ul></ul><ul><li>Best Study Tip: Do the problems, and then do some more and then do some more!! Practice using your calculator!! </li></ul>
- 20. Possible Time Value Questions <ul><li>Present Value Formula </li></ul><ul><ul><li>Given FV N , i, N – solve for PV N </li></ul></ul><ul><ul><li>Given PV N , i, N – solve for FV N </li></ul></ul><ul><ul><li>Given PV N , FV N , N – solve for i </li></ul></ul><ul><ul><li>Given PV N , FV N , i – solve for N </li></ul></ul><ul><li>Perpetuity Formula </li></ul><ul><ul><li>Given A, i – solve for PV per </li></ul></ul><ul><ul><li>Given PV per , i – solve for A </li></ul></ul><ul><ul><li>Given PV per , A – solve for i </li></ul></ul><ul><li>Annuity Formula </li></ul><ul><ul><li>Given A, i, N – solve for PV </li></ul></ul><ul><ul><li>Given A, i, N – solve for FV </li></ul></ul><ul><ul><li>Given PV, i, N – solve for A </li></ul></ul>
- 21. Bonds and Their Valuation <ul><li>Key features of bonds </li></ul><ul><li>Bond valuation </li></ul><ul><li>Measuring yield </li></ul><ul><li>Assessing risk </li></ul>
- 22. Key Features of a Bond <ul><li>Par value : Face amount; paid at maturity. Assume $1,000. </li></ul><ul><li>Coupon interest rate : Stated interest rate. Multiply by par value to get dollars of interest. Often fixed but can float with market rate. </li></ul><ul><li>Maturity : Years until bond must be repaid. Declines. </li></ul><ul><li>Issue date : Date when bond was issued. </li></ul><ul><li>Default risk : Risk that issuer will not make interest or principal payments. </li></ul>
- 23. Valuing a 5-Period Bond Time = 0 1 2 3 4 5 6 7 <ul><li>Discounted Cash Flow Approach </li></ul><ul><li>Current Bond Price = Present value of all future Cash Flows (Interest & Principal) at required return, k B . </li></ul>Bond Price, P B t Coupon Interest, CP Face Value, FV
- 24. The Right Discount Factor <ul><li>The discount rate (k i ) is the opportunity cost of capital , i.e., the rate that could be earned on alternative investments of equal risk. </li></ul><ul><li>k i = k* + IP + DRP + MRP + LP </li></ul><ul><li>k* = Real rate of interest </li></ul><ul><li>IP = Inflation risk premium </li></ul><ul><li>DRP = Default risk premium </li></ul><ul><li>MRP = Maturity premium </li></ul><ul><li>LP = Liquidity risk premium </li></ul>
- 25. What’s the value of a 10-year, 10% coupon bond if k d = 10%? V B = ? Bond Valuation Example $100 $100 $100 + $1,000 0 1 2 10 10% ... = $90.91 + . . . + $38.55 + $385.54 = $1,000. V k k B d d $100 $1 , (1 000 (1 1 10 10 . . . + $100 (1 + k d + + + + ) ) )
- 26. Stocks and Their Valuation <ul><li>Features of common stock </li></ul><ul><li>Determining common stock values </li></ul>
- 27. <ul><li>Represents ownership. </li></ul><ul><li>Ownership implies control. </li></ul><ul><li>Stockholders elect directors. </li></ul><ul><li>Directors hire management. </li></ul><ul><li>Management’s goal: Maximize stock price. </li></ul>Features of Common Stock
- 28. Valuing Common Stock Time = 0 1 2 3 4 5 6 7 Uncertain Dividends, D t+i <ul><li>Dividend Discount Model </li></ul><ul><li>Current Stock Price = Present value of all future Expected Cash Flows (Dividends) at required return, k S . </li></ul>Stock Price, P S t
- 29. <ul><li>Constant Growth stock </li></ul><ul><ul><li>One whose dividends are expected to grow forever at a constant rate, g. </li></ul></ul><ul><ul><li>Can link this to earnings by assuming that firm pays out a fixed percentage of earnings as dividends </li></ul></ul><ul><ul><ul><li>i.e. D t = k x E t where k equals payout ratio </li></ul></ul></ul>Stock Value = PV of Dividends

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