Corporate Finance Lecture 2 Dr. Solt Eszter BME 2017

The time value of money Investments: Companies invest in e.g. tangible assets or intangible assets Individuals also make investments e.g. college education in the hope of receiving more money later Fianacial decisions require comparisons of cash payments at different dates

The time value of money The scope of our examination: How funds invested at a specific interest rate will grow over time How much you would need to invest today to produce a specified future sum of money Shortcuts for working out the value of a series of cash payments How inflation affects the financial calculations

Future values and compound interest FV: Amount to which an investment will grow after earning interest Compound interest: interest earned on interest Simple interest: interest earned only on the original investment; no interest is earned on interest Compound growth: the value increases each period by the factor (1+growth rate). The growth rate = the interest rate when investing at compound interest. After t periods: Future value= the initial value x (1+growth rate)t Interest= interest rate x initial investment

Future values and compound interest Example: You have $100 invested on a bank account. Suppose banks are currently paying an interest rate (r) of 6 percent (.06) per year on deposits. So after a year your account will earn interest of $6: .06 x $100=$6 Value after 1 year: $100+$6= $106 Value after 1 year: initial investment x (1+r)= $100 x 1.06= $106 Initial investment can be referred to as C0

Compound interest Suppose you leave this money ($106) in the bank for a second year Interest in year2 = .06 x $106 = $6.36 So by the end of the second year the value of your account will grow to $106 + $6.36 = $112.36 Value of account after 2 years: $100 x (1.06)2 = $112.36 Value of account after 3 years: $100 x (1.06)3 = $119.10

Present values A dollar today is worth more than a dollar tomorrow Money in hand today has a time value We have seen that $100 invested for 1 year at 6 percent will grow to a future value of $100 x (1 + .06) = $106 Let’s turn this around: How much do we need to invest now in order to produce $106 at the end of the year? This is referred to as the present value (PV) of the $106 payoff.

Present value = future value after t periods/(1 + r)t Present values To calculate present value (PV), we simply reverse the process of calculating the future value and divide the future value by (1 + .06): PV = $106/1.06 = $100 How much do we need to invest now to produce 112.36 after 2 years? PV = $112.36/1.062 = $100 In general, PV for a future value or payment t periods away: Present value = future value after t periods/(1 + r)t

Discount rate discounted value discount factor Discount rate: interest rate used to compute present values of future cash flows (in this context the interest rate (r) is known as the discount rate) The present value is often called the discounted value of the future payment To calculate present value, we discounted the future value at the interest r The expression 1/(1 +r)t is called the discount factor: PV = future payment/(1 +r)t = future payment x 1/(1 +r)t It measures the present value of $1 received in year t

Futute value of multiple cash flows The capitalized value of an asset Most real world investments will involve many cash flows over time referred to as a stream of cashflows You should never compare cashflows occuring at different times without first discounting them to a common date The present value of a stream of future cash flows is the amount you would have to invest today to generate that stream: PV = C1/(1+r)+C2/(1+r)2+C3/(1+r)3 Where C2, C2 and C3 are the cashflows/yields of each year. This formula provides the capitalized value of an asset of an economic life span of 3 years.

Net present value and the net present value rule Net present value: present value or cash flows minus initial investment [or: initial outlay (IO)] The net present value rule states that managers increase shareholders’ wealth by accepting all projects that are worth more than they cost. Therefore they should accept all the projects with a positive net present value. Simply put: NPV = PV – IO In general: NPV= C0+C1/(1+r)+C2/(1+r)2+…Ct/(1+r)t where C0 is the yield of the period zero/the present/ the price of the asset/I O, which is negative !

The market of the assets Assets provide a stream of cashflows The present value of a stream of future cash flows is the amount you would have to invest today to generate that stream, which is the capitalized value of the asset The maximum price the investors will pay for an asset equals its capitalized value (the present value (PV)of a stream of future cash flows) In that case NPV equals zero NPV = PV – IO (initial outlay that is the price of the asset) If he has to pay less, the NPV becomes positive (see net present value rule)

Example for capitalized value Suppose an additional part for cars is available in the market provides expected savings on petrol for 3 years as follows. (at expected run and petrol prices) (HUF 000): year 1 year 2 year év 48 72 86,4 What is the maximum price the car owner will pay for the part if the interest rate is constantly 20%? Solution: The PV of the stream of expected savings („yields”) when r = 0.2 (20%): PV = 48/1,2+72/1,22+86,4/1,23=40+50+50=140 (HUF 000) So this is the maximum price he will pay, if he has to pay less, the NPV turns to positive

Level cash flows Level cash flows: A stream of equal cash flows Perpetuities Example: Some time ago the British government borrowed by issuing perpetuities.Instead of repaying these loans, the British government pays the investors holding these securities a fixed annual payment in perpetuity (forever). The rate of interest on a perpetuity is equal to the promised annual payment „C” (yield) divided by the present value,e.g. if a perpetuity pays $10 per year and you can buy it for $100, you will earn 10 percent interest each year on your investment.

Perpetuities In general: Interest rate on a perpetuity = cash payment/present value r = C/PV By rearranging this relationship to derive the present value of a perpetuity, given the interest rate r and the cash payment C: PV of perpetuity = C/r = cash payment/ interest rate

Example for perpetuity A landowner hires out his land during his entire lifetime, which pays him HUF 50.000 per year adjusted with the inflation rate. Suppose he decides to sell his land, what is the minimum selling price he should get if the expected real rate of interest is constantly 2%? Solution: He should get minimum the capitalized value of the land, which is: PV=C/r, that is: 50.000/0,02= HUF 2.500.000 If he gets less, the NPV will be negative

Inflation and the time value of money Inflation: rate at which prices as a whole are increasing Tracking the general level of prices: use of different price indexes CPI (consumer price index): the most common index, it measures the number of dollars that it takes to buy a specified basket of goods and services that is supposed represent the typical family’s purchases. Thus the percentage increase in the CPI from one year to the next measures the rate of inflation. „nominal” dollars refer to the actual number of dollars while „real” dollars refer to the amount of purchasing power.

Inflation and interest rates Nominal interest rate: rate at which money invested grows Real interest rate: rate at which the purchasing power of an investment grows The real interest rate is calculated by: Real interest rate = (1 + nominal interst rate/1 + inflation rate) – 1 Example: a bank offers 4% nominal rate per annum on deposits and the inflation rate in that economy is 2.5%; what real interest rate (real yield for capital owners) does it mean? (Assume there is no tax on interest) (1.04/1. 025)-1= 1.01463-1=0.01463=1,463%