Investment Tools – Time Value of Money

2 Concepts Covered in This Section –Future value –Present value –Perpetuities –Annuities –Uneven Cash Flows –Rates of return Time Value of Money

3 Time lines show timing of cash flows. CF 0 CF 1 CF 3 CF i% Tick marks at ends of periods. Time 0 is today; Time 1 is the end of Period 1; or the beginning of Period 2. 90% of getting a Time Value problem correct is setting up the timeline correctly!!! Interest Rate Cash Flows

4 Finding FVs (moving to the right on a time line) is called compounding. Compounding involves earning interest on interest for investments of more than one period. What’s the FV of an initial $100 after 3 years if i = 10%? FV = ? % Future Values

5 Single Sum - Future & Present Value Assume that you can invest PV at interest rate i to receive future sum, FV Similar reasoning leads to Present Value of a Future sum today FV 1 = (1+i)PV FV 3 = (1+i) 3 PV PV FV 2 = (1+i) 2 PV 1230 PV = FV 1 /(1+i) FV 1 PV = FV 2 /(1+i) 2 FV 2 PV = FV 3 /(1+i) 3 FV 3

6 FV n = PV(1 + i ) n for given PV $100   = = $ PV= $100 1       3 PV Calculation for $100 received in 3 years if interest rate is 10% Single Sum – FV & PV Formulas

7 Question on PV of a given FV 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: A.$100,000. B.$117,459. C.$148,644. D.$161,506. 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. PV =[ 1/(1+r) ] N FV PV = [ 1/(1.10) ] 20 $1,000,000 PV 10 = [ ]($1,000,000) PV 10 = $148,644

8 Perpetuities Perpetuity is a series of constant payments, A, each period forever A 0 AAAAAA 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 PV 1 = A/(1+r) PV 2 = A/(1+r) 2 PV 3 = A/(1+r) 3 PV 4 = A/(1+r) 4 etc.

9 finiteRegular 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. finite immediatelyAn 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. Annuities

10 Time line for an ordinary annuity of $100 for 3 years. $ i% Ordinary Annuity Timeline

11 PMT 0123 i% PMT Annuity Due PVFV Ordinary Annuity PMT 0123 i% Difference between an ordinary annuity and an annuity due? Ordinary Annuity vs. Annuity Due

12 Annuity Formula and Perpetuities Perpetuity of A per period in Period 0 -- PV 1 = A/i A 0 AAAAAAAAAAAAA Perpetuity of A per period in Period 8 -- PV 8 = [1/(1+i)] 8 x (A/i) 0 AAAAAA PV = PV 1 – PV 8 = (A/i) x { 1 – [1/(1+i)] 8 } 3. Annuity of A for 8 periods -- PV = PV 1 – PV 8 = (A/i) x { 1 – [1/(1+i)] 8 } A 0 AAAAAAA 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

13 Annuities & Perpetuities Again Rather than memorize the annuity formula, it is easier to calculate it as the difference between two perpetuities with the same payment. PV of an N-period annuity of $A per period is: PV N = (A/i) x { 1 – [1/(1+i)] N } Calculating the PV of an annuity has 3 steps: 1.Calculate (A/i) –PV of a Perpetuity with payments of $A per period. 2.Calculate [1/(1+i)] N –Discount factor associated with end of the annuity. 3.Calculate PV N = (A/i) x { 1 - [1/(1 + i)] N } –I think this is easier under pressure than memorizing the formula.

14 Question on FV of Annuity Due 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: A.$109,000. B.$143,200. C.$151,900. D.$165,600. This is an annuity due of A=$10,000 for N=10 years at i=9% interest rate. Annuity due must be adjusted by (1+i) to reflect payment is made at beginning rather than end of period. Also must adjust PV formula by (1+i) N for FV of annuity. PV N = (1+i) N (1+i) [ ( A/i) { 1 – [1/(1+i)] N } ] PV 10 = (1.09) 11 ($10K/.09) {1 – [1/1.09] 10 } PV 10 = (2.58)($111,111){1 – [0.42]} PV 10 = $165,601

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 $ i% Uneven Cash Flows

16 Question on Uneven Cash Flows 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: A.$404. B.$444. C.$462. D.$516. 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. Interest rate is i =14%. PV =  [ 1/(1+i) ] t FV t PV = $100/(1.14) + $200/(1.14) 2 + $300/(1.14) 3 PV = $ $ $ PV = $444.10

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 Uneven cash Flows over 10 periods – PV = PV 10 + PV $100 $500 $100$ Annuity of $100 per period for 10 periods -- PV 10 = { 1 - [1/(1+i)] 10 } x (A/i) $ 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 } ] $400

18 Comparison of Compounding Periods Annually: FV 3 = $100(1.10) 3 = $ Semiannually: FV 6 = $100(1.05) 6 = $ % %

19 Questions on Time Value Develop an approach to problems on Time Value. 1. Draw the Time line for the cash flows. 2. Put in the cash flows from the problem. 3. Identify if single payment, annuity, annuity due, or perpetuity. If uneven cash flows can you break it into sums of annuities? 4. Identify what is to be calculated – PV, FV, N or i ? 5. Write out the appropriate formula, put in values for the variables, and calculate. Best Study Tip: Do the problems, and then do some more and then do some more!! Practice using your calculator!!

20 Possible Time Value Questions Present Value Formula 1.Given FV N, i, N – solve for PV N 2.Given PV N, i, N – solve for FV N 3.Given PV N, FV N, N – solve for i 4.Given PV N, FV N, i – solve for N Perpetuity Formula 1.Given A, i – solve for PV per 2.Given PV per, i – solve for A 3.Given PV per, A – solve for i Annuity Formula 1.Given A, i, N – solve for PV 2.Given A, i, N – solve for FV 3.Given PV, i, N – solve for A

21 Bonds and Their Valuation Key features of bonds Bond valuation Measuring yield Assessing risk

22 1.Par value: 1.Par value: Face amount; paid at maturity. Assume $1, Coupon interest rate: 2.Coupon interest rate: Stated interest rate. Multiply by par value to get dollars of interest. Often fixed but can float with market rate. 3.Maturity: 3.Maturity: Years until bond must be repaid. Declines. 4.Issue date: 4.Issue date: Date when bond was issued. 5.Default risk: 5.Default risk: Risk that issuer will not make interest or principal payments. Key Features of a Bond

23 Valuing a 5-Period Bond Time = Bond Price, P B t Discounted Cash Flow Approach Current Bond Price = Present value of all future Cash Flows (Interest & Principal) at required return, k B. Coupon Interest, CP Face Value, FV

24 The Right Discount Factor 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. k i = k* + IP + DRP + MRP + LP k* = Real rate of interest IP = Inflation risk premium DRP = Default risk premium MRP = Maturity premium LP = Liquidity risk premium

25 What’s the value of a 10-year, 10% coupon bond if k d = 10%? $100 $100 + $1,000 V B = ? %... = $ $ $ = $1,000. V kk B dd  $100$1, (1 000 ( $100 (1 + k d )) ) Bond Valuation Example

26 Stocks and Their Valuation Features of common stock Determining common stock values

27 Represents ownership. Ownership implies control. Stockholders elect directors. Directors hire management. Management’s goal: Maximize stock price. Features of Common Stock

28 Valuing Common Stock Time = Stock Price, P S t Uncertain Dividends, D t+i Dividend Discount Model Current Stock Price = Present value of all future Expected Cash Flows (Dividends) at required return, k S.

29 Constant Growth stock One whose dividends are expected to grow forever at a constant rate, g. Can link this to earnings by assuming that firm pays out a fixed percentage of earnings as dividends i.e. D t = k x E t where k equals payout ratio Stock Value = PV of Dividends