Introduction to Present Value Text: Chapter 4

How much are you willing to pay for a project with payoff of $40,000 in a year?

Present Value If the project is riskless, then use return of government bond as a guide, say 8%, how much are you willing to pay? $40,000/1.08 = 37,037 [ $40,000 * (1/1.08) = $40,000 * 0.926] Present value of this project is $37,037 Future value of this project is $40,000 Discount rate Discount factors

Present Value PV = C1 * [1 / (1+r)] = C1 * discount factor where r is discount rate, opportunity cost of capital [1/1+r] is discount factor

Future Value Future Value = PV * (1+r)

The First Rule in Finance One dollar today is worth more than one dollar tomorrow

Present Value and Future Value

Present Value What if the project is risky? Suppose the return is as risky as the stock index, which offers 12% in the long run, what is the PV of $40,000 next year? $40,000 / (1.12) = $35714 < $37037 (8% discount rate)

The Second Rule in Finance A safe dollar is worth more than a risky one!

Introduction to Net Present Value If the riskless project is sold at $37,000, is this a good deal? Net Present Value = C0 + C1* (1/1+r) NPV = -37000 + 37037 = 37 Accept any project with positive net present value!

NPV > 0 <==> rate of return > opportunity cost of capital NPV and Cost of Capital In the previous example, the rate of return [40,000 / 37,000] - 1 = 8.1% > 8% (discount rate) NPV > 0 <==> rate of return > opportunity cost of capital

Opportunity Cost of Capital Example You may invest $100,000 today. Depending on the state of the economy, you may get one of three possible payoffs:

Opportunity Cost of Capital Example - continued The stock is expected to have a 15% rate of return. Discounting the expected payoff at the expected return leads to the PV of the project Initial Investment: $100,000

Opportunity Cost of Capital Example - continued Notice that you come to the same conclusion if you compare the expected project return with the cost of capital. But ….Cost of capital: 15%

No money machine exists for a lasting time. A simple question ….. Should DF2 always be less than DF1? [ie, 1/(1+r2)2 > 1/(1+r1) ]? If not, say DF2 = .8, and DF1=.7, then we can 1. Borrow $.8 and return $1 in period 2 2. Lend $.7 and get $1 back at period 1, then hold it to period 2 We end with $.1 without investing anything! But if everybody does this, then r2  (DF2 ) and r1  (DF1 ) until no arbitrage conditions exists any more No money machine exists for a lasting time.

The Calculation of Present Value

Present Values PVs can be added together to evaluate multiple cash flows.

Present Value of Multiple Cash Flow $200 $100 Present Value Year 0 100/1.07 200/1.0772 Total = $93.46 = $172.42 = $265.88 Year 0 1 2

Short Cuts Sometimes there are shortcuts that make it very easy to calculate the present value of an asset that pays off in different periods. These tolls allow us to cut through the calculations quickly.

Perpetuity A Perpetuity is a constant stream of CF without end. PVt = Ct+1 / r 0 1 2 3 …forever... |---------|--------|---------|--------- (r = 10%) $100 $100 $100 ...forever… PV0 = $100 / 0.1 = $1000

Growing Perpetuity a stream of cash flows that grows at a constant rate forever. Simplification: PVt = Ct+1 / (r - g) 0 1 2 3 …forever... |---------|---------|---------|--------- (r = 10%) $100 $102 $104.04 … (g = 2%) PV0 = $100 / (0.10 - 0.02) = $1250

Annuity Annuity - An asset that pays a fixed sum each year for a specified number of years. 0 1 2 3 |---------|---------|---------| (r = 10%) $100 $100 $100

Short Cuts Annuity - An asset that pays a fixed sum each year for a specified number of years. Asset Year of Payment 1 2…..t t + 1 Present Value Perpetuity (first payment in year 1) Perpetuity (first payment in year t + 1) Annuity from year 1 to year t

Short Cuts Annuity - An asset that pays a fixed sum each year for a specified number of years.

Annuity Short Cut Example You agree to lease a car for 4 years at $300 per month. You are not required to pay any money up front or at the end of your agreement. If your opportunity cost of capital is 0.5% per month, what is the cost of the lease?

Annuity Short Cut Example - continued You agree to lease a car for 4 years at $300 per month. You are not required to pay any money up front or at the end of your agreement. If your opportunity cost of capital is 0.5% per month, what is the cost of the lease?

Compound Interest i ii iii iv v Periods Interest Value Annually per per APR after compounded year period (i x ii) one year interest rate 1 6% 6% 1.06 6.000% 2 3 6 1.032 = 1.0609 6.090 4 1.5 6 1.0154 = 1.06136 6.136 12 .5 6 1.00512 = 1.06168 6.168 52 .1154 6 1.00115452 = 1.06180 6.180 365 .0164 6 1.000164365 = 1.06183 6.183

Compound Interest

Effective Annual Interest Rates Compounding periods (m) How often is interest computed Stated (nominal) Annual Percentage Rate (r) What the bank usually quotes Effective Annual Interest Rate (EAR): EAR = (1 + r / m) m - 1 As m approaches infinity, (1 + r / m) m -----> er EAR of continuous compounding = er - 1

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 Example If the interest rate on one year govt. bonds is 5.9% and the inflation rate is 3.3%, what is the real interest rate? Savings Bond