BUS-F 371 Chapter 4 Introduction to Valuation: The Time Value of Money

Mutual fund calcuation You invest $10,000 in a mutual fund that returns 12% per year, how much will your investment be worth after 30 years?

FV=PV x 〖(1+r)〗^t=$10,000 x〖(1+0.12)〗^30=$299,599.20

FV Paper Calc If you invest $10,000 in a mutual fund that returns 12% per year, how much will your investment be worth after 30 years? FV=PV x〖(1+r)〗^t

FV=PVx〖(1+r)〗^t =$10,000 x 〖(1+0.12)〗^30=$299,599.20

What annual rate has been earned if $1,000 grows into $4,000 in 20 years?

r=〖(FV/PV)〗^(1/t)−1=($4,000/$1,000)^(1/20)−1=0.0718= 7.18%

How many years will you have to wait for your bank account to grow to $1M if your relatives originally invested $3 and the annual interest rate is 5%?

t=(ln⁡(FV/PV))/(ln⁡(1+r))= (ln⁡(1,000,000/3))/(ln⁡(1.05))= 260.65 years

How many years will you have to wait for your bank account to grow to $1M if your relatives originally invested $3 and the annual interest rate is 5%?

t=(ln⁡(FV/PV))/(ln⁡(1+r))= (ln⁡(1,000,000/3))/(ln⁡(1.05))= 260.65 𝑦𝑒𝑎𝑟𝑠

How many years will you have to wait for your bank account to grow to $1M if your relatives originally invested $3 and the annual interest rate is 5%? How many years to grow to $2M?

t=(ln⁡(FV/PV))/(ln⁡(1+r))= (ln⁡(2,000,000/3))/(ln⁡(1.05))= 274.85 years

How many years will you have to wait for your bank account to grow to $1M if your relatives originally invested $3 and the annual interest rate is 5%? How many years to grow to $2M?

t=(ln⁡(FV/PV))/(ln⁡(1+r))= (ln⁡(2,000,000/3))/(ln⁡(1.05))= 274.85 𝑦𝑒𝑎𝑟𝑠

How long will it take your $10,000 to double in value if it earns 5% annually?

t=(ln⁡(FV/PV))/(ln⁡(1+r))= (ln⁡(20,000/10,000))/(ln⁡(1.05))= 14.2 years

You want to purchase a new car and you are willing to pay $20,000. If you can invest at 10% per year and you currently have $15,000. How long will it take before you can pay cash for this car?

t=(ln⁡(FV/PV))/(ln⁡(1+r))= (ln⁡(20,000/15,000))/(ln⁡(1.1))= 3.02 years

Suppose that you want to buy new furniture. You currently have $500 and the furniture that you want costs $600. If you can earn 6%, how long will you have to wait to purchase this furniture?

t=(ln⁡(FV/PV))/(ln⁡(1+r))= (ln⁡(600/500))/(ln⁡(1.06))= 3.13 years

Present Value: Formula

FV = PV x (1 + r)t PV= FV/〖(1+r)〗^t

Set payments to occur at the end of the period

(Do this only if you see 'BGN' in the upper right of your screen) 2nd BGN 2nd SET 2nd QUIT

CLEAR YOUR CALCULATOR

2ND CLR TVM 2ND CLR WORK

Set # of compounding periods to 1

2nd P/Y 1 ENTER 2nd QUIT

You have $10,000 to invest for 5 years. How much additional interest will you earn if the investment provides a 5% annual return versus a 4.5% annual return?

At I/Y = 5, FV = $12,762.82 At I/Y = 4.5, FV = $12,461.82 Difference = $301.00

Time Value of Money (TVM)

Costs (outflows) and benefits (inflows) usually occur in different time periods

Present Value and Discounting

Discounting = opposite of compounding Finding the present value of a future amount

A firm currently pays a cash dividend of $5 per share. You believe the dividend will be increased by 4% each year indefinitely. How big will the dividend be in year eight?

FV= PV x〖(1+r)〗^t=$5 x〖(1+0.04)〗^8=$6.84

You want to purchase a house. You currently have $15,000 and figure that you will need a 10% down payment plus an additional 5% of the loan amount for closing costs. Assume the house costs $150,000 and you can earn 7.5% per year. How long will it take to have enough money for the down payment and closing costs? What is the future value and how long will it take?

FV=$150,000 (0.15)=$22,500 t=(ln⁡(FV/PV))/(ln⁡(1+r))= (ln⁡(22,500/15,000))/(ln⁡(1.075))= 5.61 years

Compound interest

Interest earned on interest reinvested

Simple interest

Interest earned on original investment

Future Value

PV(1+r)^t

How much would your relatives have needed to invest 200 years ago at 3% per year in order for us to receive $1M today?

PV= FV/〖(1+r)〗^t =$1,000,000/〖(1+0.03)〗^200 =$2,707.42

How much would your relatives have needed to invest 200 years ago at 5.5% per year in order for us to receive $1M today?

PV= FV/〖(1+r)〗^t =$1,000,000/〖(1+0.055)〗^200 =$22.36

A 10-year-old boy's mother wants to pay for his college education. She estimates that she will need $100,000 in 8 years in order to do this. b) How much does she need to invest today, if her expected return is 12% per year?

PV= FV/〖(1+r)〗^t =$100,000/〖(1+0.12)〗^8 = -$40,390

A 10-year-old boy's mother wants to pay for his college education. She estimates that she will need $100,000 in 8 years in order to do this. a) How much does she need to invest today, if her expected return is 15% per year?

PV= FV/〖(1+r)〗^t =$100,000/〖(1+0.15)〗^8 = -$32,690

Present Value: What would you be willing to pay TODAY for a lump sum of $121 that you will receive 2 years from now?

PV=$121/〖(1.10)〗^2 =$100

Present Value: Suppose that you need $10,000 in three years when you graduate for the down payment on a new car. If you can earn 7% annually, how much do you need to invest today?

PV=FV/〖(1+r)〗^t =$10,000/〖(1+0.07)〗^3 =$8,162.98

Present Value and Discounting

Present Value: Current value of future cash flows discounted at the appropriate discount rate What would you be willing to pay for a specific cash flow (or series of cash flows), GIVEN your required return? How much do I have to invest today to have some amount in the future?

future value (FV)

The amount an investment is worth after one or more periods. the value of a cash flow at some specific future time

Finding the Number of Periods

t=(ln⁡(FV/PV))/(ln⁡(1+r))

compounding interest

earning interest on interest

Finding the Discount Rate

r=〖(FV/PV)〗^(1/t)−1

What interest rate must you earn for your account to be worth $1M today if your relatives deposited $5 in the bank 200 years ago? = 6.29% What interest rate do you need if they only saved $1?

r=〖(FV/PV)〗^(1/t)−1=($1,000,000/$1)^(1/200)−1=0.0715=7.15%

What interest rate must you earn for your account to be worth $1M today if your relatives deposited $5 in the bank 200 years ago?

r=〖(FV/PV)〗^(1/t)−1=($1,000,000/$5)^(1/200)−1=0.0629= 6.29%

R Example An investment will pay $1,200 in 5 years if you invest $1,000 today What is the implied rate of interest/required rate of return?

r=〖(FV/PV)〗^(1/t)−1=($1,200/$1,000)^(1/5)−1=0.03714= 3.714%

Suppose you are offered an investment that will allow you to double your money in 6 years You have $10,000 to invest What is the implied rate of interest?

r=〖(FV/PV)〗^(1/t)−1=($20,000/$10,000)^(1/6)−1=0.1225= 12.25%

You are offered the following investments: Invest $500 today and receive $600 in 5 years Invest $500 in a bank account paying 4% Which investment should you choose?

r=〖(FV/PV)〗^(1/t)−1=($600/$500)^(1/5)−1=0.03714= 3.71%

Suppose you have a 1-year old son and you want to provide $75,000 in 17 years towards his college education. You currently have $5,000 to invest What interest rate must you earn to accomplish this goal?

r=〖(FV/PV)〗^(1/t)−1=($75,000/$5,000)^(1/17)−1=0.1727= 17.27%