The time value of money
the discount rate (r) is the rate that connects the investment today (PV) with that in the future (FV) at the time (t). the discount rate depends on three main factors,
- Risk-free rate
- Inflation
- correlation with the market as a whole
there are two types of interest rates in our calculations which are simple interest rate and compound interest rate, it means that we are investing with the same interest rate in each period of time. the compound interest rate is the interest over the interest.
FV= PV*(1+r)^t
(1+r)^t: Future value interest factor
Example: what is the future value of 1000 USD investments with an annual rate of 12% per annum if the compounding is monthly? How Much is it after 5 years?
FV= PV*(1+12%/12)¹²=1000*1.1268=1126.80
FV= PV*(1+12%/12)⁶⁰=1000*1.8167=1816.70
Example 2: You currently have 100 available for 21 years. at what interest rate it must be invested to have the value up to 500 till then.
FV=PV*(1+r)^t =>
500=100*(1+r)²¹=>1+r=5^(¹/21)=>r=7.9%
Reminder:
Rule of 72 is a handy rule of thumb:
if you earn r percent per year, your money will double in 72/r of year. for example, if you invest at 6 percent your money will be 12 times in 72 years or it will be doubled in 12 years.
Example 3: Suppose you locate a two-year investment that pays 14 percent per year. If you invest $325, how much will you have at the end of the two years? How much of this is simple interest? How much is compound interest?
PV=325, R=14%, FV (t=2)
FV=325*(1+14%)²= 422.37
How much is Simple interest rate?
simple interest rate means the interest of the first year is not reinvested:
for each year= (325*0.14)=45.5
for two years= 91 USD == Simple interest rate
Total value after 2 years= Principal Value+Simple interest + Compound interest=> 422.37=325+91+Compound=> Compound=6.37
here are some of the useful shortcuts in excel:
Examples from Book:
EX1. Compound InterestFirst Tappan Bank pays 5 percent simple interest on its savings account balances, whereas First Mullineaux Bank pays 5 percent interest compounded annually. If you made a $5,000 deposit in each bank, how much more money would you earn from your First Mullineaux Bank account at the end of 10 years?
FTP Bank: FV=Principal+Interest for 5 years= 5000+5*5%*5000=1.25*5000= 6250
FMB Bank: FV=PV*(1+r)^t= 5000*1.05⁵=6381.4
Hence: difference= 131.4 USD
Ex.Calculating Interest Rates Assume the total cost of a college education will be $200,000 when your child enters college in 18 years. You presently have $27,000 to invest. What annual rate of interest must you earn on your investment to cover the cost of your child’s college education?
FV=200,000 , PV= 27,000 , t=18 Y
200000=27000*(1+r)¹⁸ => 1+r=18th root of 200/27
hence 1+r= 1.117=> r= 11.7%
EX. 3. Calculating the Number of Periods At 6 percent interest, how long does it take to double your money? To quadruple it?
use 72 rule=> 72/6= 12 Years to double the money
to quadruple we have to double the new value, hence: 72/6=12 years more to quadruple => 24 years
EX4. Calculating the Number of Periods You’re trying to save to buy a new $120,000 Ferrari. You have $40,000 today that can be invested at your bank. The bank pays 5.5 percent annual interest on its accounts. How long will it be before you have enough to buy the car?
120k=40k*(1+r)^t => 3=(1.055)^t => t=21 Years
Ex.5. Calculating Present Values You have just received notification that you have won the $1 million first prize in the Centennial Lottery. However, the prize will be awarded on your 100th birthday (assuming you’re around to collect), 80 years from now. What is the present value of your windfall if the appropriate discount rate is 13 percent?
FV= 1M, t= 80 , r= 13
1000000=PV*(1.13)⁸⁰ => PV= 1m/17,630.94=56.7 USD
EX.6. Calculating Present Values Suppose you are still committed to owning a $120,000 Ferrari (see EX4). If you believe your mutual fund can achieve an 11 percent annual rate of return and you want to buy the car in 10 years on the day you turn 30, how much must you invest today?
FV=120,000 and r=11%
120k= PV*(1.11)¹⁰=> PV=42,262
Ex.7. Calculating Future Values You have just made your first $2,000 contribution to your individual retirement account. Assuming you earn a 9 percent rate of return and make no additional contributions, what will your account be worth when you retire in 45 years? What if you wait 10 years before contributing? (Does this suggest an investment strategy?
FV= 2000*(1.09)⁴⁵=96,654 USD
case 2: 2000*(1.09)³⁵=40,842
the increase in the value of the interest over interest (Compound interest rate) is exponentially.
EX.8. Calculating the Number of Periods You expect to receive $10,000 at graduation in two years. You plan on investing it at 12 percent until you have $120,000. How long will you wait from now?
120k=10k*(1.12)^t > t= 22 Years from the time you receive.
you have to wait for 24 years.
Ex.9. Calculating Future Values Find the monthly adjusted prices for Tyco International LTD (TYC). If the stock appreciates 11 percent per year, what stock price do you expect to see in five years? In 10 years? Ignore dividends in your calculations.
Ex 10. You think you will be able to deposit $4,000 at the end of each of the next three years in a bank account paying 8 percent interest. You currently have $7,000 in the account. How much will you have in three years? In four years?
Y0:7000
Y1= 7000*(1.08)+4000= 11,560
Y2= 11560*(1.08)+4000=16,484.8
Y3=16484.8*(1.08)+4000= 21,803.58
Y4=21803.58*1.08=23547
Ex. 11. Suppose you need $1,000 in one year and $2,000 more in two years. If you can earn 9 percent on your money, how much do you have to put up today to exactly cover these amounts in the future?
Y0=?, r=9% , Y1=1000, Y2=2000
Discount Y1 to 0=> 1000/1.09=917.43
Discount Y2 to Y0=> 2000/1.09²=1683.35
Total @ Y0= 2600.78
NOTE: two ways of solving this type of problem
