Money and Banking Time Value of Money Understanding
Money and Banking Time Value of Money (Understanding Interest Rates) Mr. Vaughan Updated: 2/16/09
Learning Objectives Time Value of Money (TVM) After this lecture, you will understand: • How to put cash flows received/paid out at different times in common time denominator. • How to use TVM framework to compute present/ value. • How to price stocks. • How to evaluate profitability of capital projects using net-present-value (NPV). • How to price bonds. • How to compute “yield to maturity” on bonds. 2 - 36
Time Value of Money • Firms/investors often receive/pay out cash flows at different times. • Other things equal, people prefer cash flows received sooner to cash flows received later. This idea is called “time value of money (TVM). ” • Future value/present value calculations put cash flows in “common time denominator. ” 3 - 36
What is “future value? ” • A dollar in hand today is worth more than a dollar promised tomorrow because dollar today will earn interest. • Amount an asset is worth at a specific time in the future is its future value. 4 - 36
Example: Calculating Future Value To calculate future value (FV) one period ahead: • Multiply initial, or present value, of cash flow (PV) by appropriate interest rate (r). • Add product (PV x r) to initial cash flow (PV). 5 - 36
Example: Calculating Future Value • Factoring out initial cash flow (PV) yields: • We can compute value of cash flow after more than one compounding period by recognizing interest will earn interest (i. e. , compound). 6 - 36
Example: Calculating Future Value • To account for compounding, multiply initial cash flow (PV) by one plus interest rate (1 + r) for each period. • Therefore, future value of cash flow after “n” periods is given by: 7 - 36
Example: Calculating Future Value • Suppose $100 is invested at 5%, compounded yearly. • After one year, investment will be worth: 8 - 36
Example: Calculating Future Value • Over second year, $105 grows by another 5%, yielding: • Note: Number could have obtained from general formula, with n=2: 9 - 36
Example: Calculating Future Value Note: Future value ($110. 25) has three components: • Original principal: $100 • Simple interest: 0. 05 x $100 x 2 = $10 • Compound interest: $110. 25 - $100 - $10 = 0. 25 10 - 36
Power of Compound Interest Example • In 1626, Peter Minuit of Dutch West Indies Company bought Manhattan island from Manhattan Indians for $24 worth of trinkets. • Suppose Indians had invested funds and earned 10. 1% annually (average nominal return on U. S. stocks from 1926 to 2006) since … • What would $24 be worth today, assuming annual compounding? 11 - 36
Power of Compound Interest Future value = $24 (1. 101)377 = $24 (1. 431 x 1016) = $3. 434 x 1017 The $24 would now be worth $3. 434 x 1017. Note: 17 Zeros! Enough to buy all real estate on earth, with change. 12 - 36
What is “present value? ” • Present value is inverse of future value. • If we know when cash flow will be received, we can use present-value framework to calculate what cash flow is worth to us now. • Present value allows us to determine “fundamental value” of assets. 13 - 36
Calculating “Present Value” For single future payment (“n” periods in future), present value can be obtained by: • Rearranging future-value equation. • Plugging anticipated future cash flow in for FV and solving for present value. 14 - 36
Example: Calculating Present Value • What is present value of $100 to be received in one year if rate of return available on investments of comparable risk is 10%? • What is intuition behind number? 15 - 36
Example: Calculating Present Value Solution: Intuition: $90. 91 in hand today, invested at 10% (compounded annually), would grow to $100 in one year. 16 - 36
Example: Calculating Present Value What about cash flows received over multiple periods? • Calculate present value of each cash flow (CF) and sum discounted values: 17 - 36
Example: Calculating Present Value What is present value of following stream of cash flows? • $100 received one year from now • $100 received two years from now Assume: • return available on investments with comparable risk is 10%. • annual compounding. 18 - 36
Example: Calculating Present Value Solution: 19 - 36
Applications of Present Value Pricing Stocks Stock = Price Like any financial asset, stocks are priced at present value of future cash flows [expected dividends (D) and expected resale price of stock (R)]. § In well-functioning markets, resale price in year “n” depends on dividends to be received from “n+1” to infinity. § When dividends are expected to grow at a constant rate to infinity, the stockpricing equation reduces to: Stock = Price where: “g” is the constant growth rate of dividends 20 - 22
Economic News and Stock Prices Stock = Price § An increase in return on investments of comparable risk (r) causes present value of cash flows from stock (and, therefore, stock price) to fall. Ø Such an increase could be caused by rise in market interest rates § Bad economic news causes investors to revise expected dividends (D) downward, thereby lowering present value and stock price. Note: Stock prices are more volatile than bond prices because numerator and denominator fluctuate with market conditions (only denominator fluctuates for bonds) 21 - 22
Applications of Present Value Evaluating Capital Projects • A financial manager’s goal is: Ø Maximizing shareholder wealth. • To do this, he should undertake all capital projects that add to total value of firm. • “Adding to value of the firm” means undertaking all capital projects with positive net present value. Present value of cash inflows from capital project (marginal benefit) -Present value of cash outflows on capital project (marginal cost) Net Present Value (NPV) 22 - 36
Example: Net Present Value (NPV) Suppose you are considering buying a home computer system, so you can telecommute. You want to know whether this capital project makes sense from financial standpoint. You should: 1. Calculate NPV. 2. Purchase computer system if NPV > 0 23 - 36
Example: Net Present Value (NPV) Relevant information: • Cost of computer system: $3, 000 • Life of computer: 4 years • Salvage value of the computer: $0 • Yearly cash flows from computer: $900 • Appropriate discount rate: 5% 24 - 36
Example: Net Present Value (NPV) Framework: NPV = Present value of inflows - Present value of outflows where: CFn = Cash inflow in period “n” Co = Cash outflow at outset, year “ 0” r = Appropriate discount rate (opportunity cost of funds) 25 - 36
Example: Net Present Value (NPV) Framework: NPV = Present value of cash inflows - Present value of cash outflows Decision: You should buy computer because it is positive NPV project! 26 - 36
Economic News and Investment • Higher interest rates or bad economic news reduce planned investment in structures and equipment. Ø Increase in current market interest rates reduces present value of cash inflows from project (marginal benefit) and hence, NPV. Ø Bad economic news reduces expected cash inflows from project (marginal benefit) and hence, NPV. • A firm considering investment projects knows cash outlays and current discount rate but must estimate cash inflows. Ø Revisions in CFs due to economic news can be large (Keynes: “Animal Spirits”). Ø Business spending on structures and equipment can be quite volatile. 27 - 22
Application of Present Value Pricing Bonds In well-functioning bond market, price of bond equals present value of its cash flows: where: Cn = Coupon payment each period = coupon rate x face value (cash flows) Pn = Principal returned in “nth” year = face value (cash flow) r = Yield available on bonds with comparable risk 28 - 36
Applications of Present Value Pricing Bonds • You hold a bond with 7% annual coupon and three years remaining until maturity. • Face value of bond is $1, 000, and yield available on comparable, newly issued bonds, is 10%. • In efficient bond market, what should price of bond be? 29 - 36
Applications of Present Value Pricing Bonds Solution: In efficient bond market, price of bond equals present value of all cash flows from bond. Intuition: $925. 39 invested at 10% today would yield cash flows offered by this bond over next three years. 30 - 36
Extension of Present Value Yield to Maturity • Suppose you are considering buying a bond. • You want to determine what the return would be if you bought bond at current market price and held to maturity. This is yield to maturity! 31 - 36
Example: Calculating “Yield to Maturity” Use present-value framework and solve for “r”. where: BP 0 = Actual market price of bond in year “ 0”(given) Cn = Expected coupon payment in year “n” (given) Pn = Expected principal payment in the “nth” year (given) r = Yield to maturity (unknown) 32 - 36
Example: Calculating “Yield to Maturity” Suppose you are considering purchase of bond with following characteristics: • • Annual coupon: 7% Years remaining until maturity: 3 Face value: $1, 000 Current market price: $925. 39 What is yield to maturity on bond? 33 - 36
Example: Calculating “Yield to Maturity” Solution: 34 - 36
Economic News and Bond Prices Bond = Price • Firms selling bonds in primary markets set coupon payments (C) so yield-to -maturity (r) equals going rate on bonds of comparable risk. • After initial sale, (C) and principal (P) remain fixed, but going rate on new bonds of comparable risk [r = opportunity cost of funds] fluctuates with market conditions. • A rise in going rate on new bonds of comparable risk (due to rise in market interest rates or default/liquidity risk) cases present value to fall. • Investors price at present value, so rise in opportunity cost of funds causes bond price to fall. Ø Bond prices and interest rates move inversely. 35 - 36
Bond Facts Bond Prices and Interest Rates Fact: When coupon rate on bond “A” is less than yield available on new bonds of comparable risk, bond “A” will sell at discount. Intuition: • Unattractive coupon rate reduces demand for bond “A. ” • Falling demand depresses price of bond “A. ” • Coupon rate on bond “A” was fixed at issue, so falling price implies rising yield on bond “A. ” • Price of bond “A” falls until yield on bond “A” equals market yield on bonds of comparable risk. 36 - 36
Questions over Time Value of Money (Understanding Interest Rates) Money and Banking Mr. Vaughan
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