Present Value

Time Value of Money Financial managers compare the marginal benefits and marginal cost of investment projects. Projects usually have a long-term horizon: timing of benefits and costs matters. Time-value of money concerned with adjusting for different timing of benefits and costs.

Future Value depends on: The value of a lump sum or stream of cash payments at a future point in time: FVn = PV x (1+r)n Future Value depends on: Interest rate Number of periods Compounding interval Formula can be rearranged to compute required return, if price and dividend known: Equity Valuation As will be discussed in chapter 5, the required return on common stock is based on its beta, derived from the CAPM Valuing CS is the most difficult, both practically & theoretically Preferred stock valuation is much easier (the easiest of all) Whenever investors feel the expected return, rˆ, is not equal to the required return, r, prices will react: If exp return declines or reqd return rises, stock price will fall If exp return rises or reqd return declines, stock price will rise Asset prices can change for reasons besides their own risk Changes in asset’s liquidity, tax status can change price Changes in market risk premium can change all asset values Most dramatic change in market risk: Russian default Fall 98 Caused required return on all risky assets to rise, price to fall

Future Value of $200 (4 Years, 7% Interest ) FV4 = $262.16 FV3 = $245.01 FV2 = $228.98 FV1 = $214 PV = $200 0 1 2 3 4 End of Year Compounding: the process of earning interest in each successive year

Compounding Earns 7% interest on initial $200 FV1 = $200+$14 = $214 Year 1: FV1 = $214 Earns 7% interest on initial $200 FV1 = $200+$14 = $214 Year 2: FV2 = $228.98 Earn $14 interest again on $200 principal Earns $0.98 on previous year’s interest of $14: $14 x 7% = $0.98 FV2 = $214+$14+$0.98 = $228.98 Year 3: FV3 = $245.01 Earn $14 interest again on $200 principal Earns $2.03 on previous years’ interest of $28.98: $28.98 x 7% = $2.03 FV3 = $228.98+$14+$2.03 = $245.01 Earn $14 interest again on $200 principal Earns $3.15 on previous years’ interest of $45.01: $45.01 x 7% = $3.15 FV4 = $245.01+$14+$3.15 = $262.16 Year 4: FV4 = $262.16

The Power of Compound Interest 20% Future Value of One Dollar ($) 15% 10% 5% 0% Periods

Present Value Today's value of a lump sum or stream of cash payments received at a future point in time: Formula can be rearranged to compute required return, if price and dividend known: Equity Valuation As will be discussed in chapter 5, the required return on common stock is based on its beta, derived from the CAPM Valuing CS is the most difficult, both practically & theoretically Preferred stock valuation is much easier (the easiest of all) Whenever investors feel the expected return, rˆ, is not equal to the required return, r, prices will react: If exp return declines or reqd return rises, stock price will fall If exp return rises or reqd return declines, stock price will rise Asset prices can change for reasons besides their own risk Changes in asset’s liquidity, tax status can change price Changes in market risk premium can change all asset values Most dramatic change in market risk: Russian default Fall 98 Caused required return on all risky assets to rise, price to fall

Present Value of $200 (4 Years, 7% Interest ) Discounting 0 1 2 3 4 FV4 = $200 PV = $152.58 FV3 = $200 PV = $163.26 PV = $186.92 FV1 = $200 FV2 = $200 PV = $174.69 End of Year Discounting: the process of converting a future cash flow into a present value

The Power of High Discount Rates 1.00 0% 0.75 Present Value of One Dollar ($) 0.5 5% 10% 0.25 15% 20% 0 2 4 6 8 10 12 14 16 18 20 22 24 Periods

Future Value and Present Value of an Ordinary Annuity Compounding Future Value $1,000 $1,000 $1,000 $1,000 $1,000 0 1 2 3 4 5 End of Year Present Value Discounting

Future Value of Ordinary Annuity (End of 5 Years, 5.5% Interest Rate) $1,238.82 $1,174.24 $1,113.02 $1,055.00 $1,000.00 $1,000 $1,000 $1,000 $1,000 $1,000 0 1 2 3 4 5 End of Year How is annuity due different ?

Future Value of Annuity Due (End of 5 Years, 5.5% Interest Rate) $1,306.96 $1,238.82 $1,174.24 $1,113.02 $1,055.00 $1,000 $1,000 $1,000 $1,000 $1,000 0 1 2 3 4 5 End of Year Annuity due: payments occur at the beginning of each period

Present Value of Ordinary Annuity (5 Years, 5.5% Interest Rate) 0 1 2 3 4 5 $1,000 $1,000 $1,000 $1,000 $1,000 End of Year $947.87 $898.45 $851.61 $807.22 $765.13

Present Value of Annuity Due (5 Years, 5.5% Interest Rate) 0 1 2 3 4 5 $1,000 $1,000 $1,000 $1,000 $1,000 End of Year $947.87 $898.45 $851.61 $807.22

FV and PV of Mixed Stream (5 Years, 4% Interest Rate) Compounding - $12,166.5 FV $6,413.8 $3,509.6 $5,624.3 $4,326.4 $3,120.0 -$10,000 $3,000 $5,000 $4,000 $3,000 $2,000.0 0 1 2 3 4 5 End of Year $2,884.6 PV $5,271.7 $4,622.8 $3,556.0 $2,564.4 $1,643.9 Discounting

Present Value of Perpetuity ($1,000 Payment, 7% Interest Rate) Stream of equal annual cash flows that lasts “forever” What if the payments grow at 2% per year?

Present Value of Growing Perpetuity CF1 = $1,000 r = 7% per year g = 2% per year 0 1 2 3 4 $1,000 $1,000(1+0.02)1 $1,000(1+0.02)2 $1,000(1+0.02)3 … $1,000 $1,020 $1,040.4 $1,061.2

Compounding Intervals m compounding periods The more frequent the compounding period, the larger the FV!

Compounding More Frequently Than Annually FV at end of 2 years of $125,000 deposited at 5% interest For semiannual compounding, m equals 2: For quarterly compounding, m equals 4:

Continuous Compounding In extreme case, interest compounded continuously: FVn = PV x (e r x n) FV at end of 2 years of $125,000 at 5 % annual interest, compounded continuously:

The Stated Rate versus the Effective Rate Stated rate: the contractual annual rate charged by lender or promised by borrower Effective rate: the annual rate actually paid or earned

The Stated Rate versus the Effective Rate FV of $100 at end of 1 year, invested at 5% stated annual interest, compounded: Annually: FV = $100 (1.05)1 = $105 Semiannually: FV = $100 (1.025)2 = $105.06 Quarterly: FV = $100 (1.0125)4 = $105.09 Stated rate of 5% does not change. What about the effective rate?

Effective Rates: Always Greater Than or Equal to Stated Rates For annual compounding, effective = stated For semiannual compounding For quarterly compounding

Much Of Finance Involves Finding Future And (Especially) Present Values Central to all financial valuation techniques Techniques used by investors and firms alike