Present value, annuity, perpetuity Financial Economics 2012 höst

How to Calculate Present Values LEARNING OBJECTIVES  Time value of money how to calculate present value of future cash flows. To calculate the present value of perpetuities, growing perpetuities, annuities and growing annuities. Compound interest and simple interest Nominal and effective interest rates. To understand value additive property and the concept of arbitrage. the net present value rule and the rate of return rule.

Time value of money The value of 1 € today is not the same as 1 € in a year’s time Suppose the interest rate on savings is 2% per year. At the end of year 1: 1 € × (1+0,02)=1,02 At the end of year 2: 1,02€ × (1+0,02) =1,0404 year 3: 1,0404 × (1+0,02)=1,0612 i.e. FV = 1€× (1 + 0,02)3 year FV 1 2 1,02 3 1,0404 4 1,061208 5 1,082432 6 1,104081 7 1,126162

Topics Covered Future Values and Compound Interest Present Values Multiple Cash Flows Level Cash Flows Perpetuities and Annuities Effective Annual Interest Rates Inflation & Time Value 2

Calculating future value of 1$ Note that the 1 $ doubled in about 37 year’s time, given interest rate 2%.

future value and present value FV = PV× (1 + r)t where FV = Future value PV = Present value r = interest rate t = number of years (Periods) It is readily seen, PV = FV/ (1 + r)t

Manhattan Island Sale The Power of Compounding! Peter Minuit bought Manhattan Island for $24 in 1626. Was this a good deal? To answer, determine $24 is worth in the year 2008, compounded at 8%. FYI - The value of Manhattan Island land is well below this figure. Obs! 8% is way to high as historical return! Test 5% 3% or 2 % instead. The value of the purchase then becomes catastrophically exponentially lower!

Future Values Compound Interest - Interest earned on interest. Simple Interest - Interest earned only on the original investment. Future Value - Amount to which an investment will grow after earning interest. 3

Future Values Interest Earned Per Year = 100 × .06 = $ 6 Example - Simple Interest Interest earned at a rate of 6% for five years on a principal balance of $100. Interest Earned Per Year = 100 × .06 = $ 6 5

Future Values Example - Simple Interest Interest earned at a rate of 6% for five years on a principal balance of $100. Today Future Years 1 2 3 4 5 Interest Earned Value 100 6 106 6 112 6 118 6 124 6 130 Value at the end of Year 5 = $130 12

Future Values Example - Compound Interest Interest earned at a rate of 6% for five years on the previous year’s balance. Today Future Years 1 2 3 4 5 Interest Earned Value 100 6 106 6.36 112.36 6.74 119.10 7.15 126.25 7.57 133.82 Value at the end of Year 5 = $133.82 12

Future Values Future Value of $100 = FV 21

Future Values Example - FV What is the future value of $100 if interest is compounded annually at a rate of 6% for five years? 23

Future Values with Compounding Interest Rates 24

Present Values Present Value Value today of a future cash flow. Discount Factor Equals to Present value of a $1 future payment. Discount Rate Interest rate used to compute present values of future cash flows. r 31

Present Values 32

Present Values Example 5.2 You just bought a new computer for $3,000. The payment due in 2 years. That means you need to pay 3000$ after 2 years. If you can earn 8% on your money, how much money should you set aside today in order to make the payment when due in two years? 34

Present Values Discount Factor = DF Note that discount factor is the same as PV of $1 in t year´s time Discount Factors can be used to compute the present value of any cash flow. PV=DF*FV 36

Time Value of Money (applications) The PV formula has many applications. Given any variables in the equation, you can solve for the remaining variable. 38

PV of Multiple Cash Flows Example Your auto dealer gives you the choice to pay $15,500 cash now, or make three payments: $8,000 now and $4,000 at the end of the following two years. If your cost of money is 8%, which do you prefer? 42

Present Values $8,000 $4,000 $ 4,000 Present Value Year 0 4000/1.08 4000/1.082 Total = $3,703.70 = $3,429.36 = $15,133.06 Year 0 1 2 $8,000

PV of Multiple Cash Flows PVs can be added together to evaluate multiple cash flows. 43

Perpetuities & Annuities Perpetuity A stream of level cash payments that never ends. Annuity Equally spaced level stream of cash flows for a limited period of time. 44

Perpetuities & Annuities PV of Perpetuity Formula C = cash payment r = interest rate 45

Perpetuities & Annuities Example - Perpetuity In order to create an endowment, which pays $100,000 per year, forever, how much money must be set aside today in the rate of interest is 10%? 47

Perpetuities & Annuities Example - continued If the first perpetuity payment will not be received until three years from today, how much money needs to be set aside today? 49

Perpetuities & Annuities PV of Annuity Formula C = cash payment r = interest rate t = Number of years cash payment is received 50

Perpetuities & Annuities PV Annuity Factor (PVAF) - The present value of $1 a year for each of t years. 51

Perpetuities & Annuities Example - Annuity You are purchasing a car. You are scheduled to make 3 annual installments of $4,000 per year. Given a rate of interest of 10%, what is the price you are paying for the car (i.e. what is the PV)? 53

Perpetuities & Annuities Applications Value of payments Implied interest rate for an annuity Calculation of periodic payments Mortgage payment Annual income from an investment payout Future Value of annual payments 54

Perpetuities & Annuities Example - Future Value of annual payments You plan to save $4,000 every year for 20 years and then retire. Given a 10% rate of interest, what will be the FV of your retirement account? The number e is an important mathematical constant, approximately equal to 2.71828, that is the base of the natural logarithm.[1] It is thelimit of (1 + 1/n)n as n becomes large, an expression that arises in the study of compound interest, and can also be calculated as the sum of the infinite series 56

Effective Interest Rates Effective Annual Interest Rate - Interest rate that is annualized using compound interest. Annual Percentage Rate - Interest rate that is annualized using simple interest. 26

Effective Interest Rates example Given a monthly rate of 1%, what is the Effective Annual Rate(EAR)? What is the Annual Percentage Rate (APR)? 27

Effective Interest Rates example Given a monthly rate of 1%, what is the Effective Annual Rate(EAR)? What is the Annual Percentage Rate (APR)? 28

Inflation Inflation - Rate at which prices as a whole are increasing. Nominal Interest Rate - Rate at which money invested grows. Real Interest Rate - Rate at which the purchasing power of an investment increases. 57

Inflation Annual U.S. Inflation Rates from 1900 - 2007 Annual Inflation, %

Inflation approximation formula 59

Inflation Example If the interest rate on one year govt. bonds is 6.0% and the inflation rate is 2.0%, what is the real interest rate? Savings Bond 62

Inflation Remember: Current dollar cash flows must be discounted by the nominal interest rate; real cash flows must be discounted by the real interest rate.