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## Present value, annuity, perpetuity

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**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**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**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 ratet = number of years (Periods) It is readily seen, PV = FV/ (1 + r)t**Manhattan Island SaleThe 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.**Future Values**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**Future Values**Example - Simple Interest Interest earned at a rate of 6% for five years on a principal balance of $100. Today Future Years 12345 Interest Earned Value 100 6 106 6 112 6 118 6 124 6 130 Value at the end of Year 5 = $130**Future Values**Example - Compound Interest Interest earned at a rate of 6% for five years on the previous year’s balance. Today Future Years 12345 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**Future Values**Future Value of $100 = FV**Future Values**Example - FV What is the future value of $100 if interest is compounded annually at a rate of 6% for five years?**Future Values with Compounding**Interest Rates**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**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?**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**Time Value of Money(applications)**• The PV formula has many applications. Given any variables in the equation, you can solve for the remaining variable.**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?**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.**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.**Perpetuities & Annuities**PV of Perpetuity Formula C = cash payment r = interest rate**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%?**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?**Perpetuities & Annuities**PV of Annuity Formula C = cash payment r = interest rate t = Number of years cash payment is received**Perpetuities & Annuities**PV Annuity Factor (PVAF) - The present value of $1 a year for each of t years.**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)?**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**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?**Effective Interest Rates**Annual Percentage Rate - Interest rate that is annualized using simple interest. Effective Annual Interest Rate - Interest rate that is annualized using compound interest.**Effective Interest Rates**example Given a monthly rate of 1%, what is the Effective Annual Rate(EAR)? What is the Annual Percentage Rate (APR)?**Effective Interest Rates**example Given a monthly rate of 1%, what is the Effective Annual Rate(EAR)? What is the Annual Percentage Rate (APR)?**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.**Inflation**Annual U.S. Inflation Rates from 1900 - 2007 Annual Inflation, %**Inflation**approximation formula**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**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.