Investment, logs and Geometric sequences Flashcards
What are 2 types of interests?
Simple and Compounded interest
What is simple interest?
tba
What is compounded interest?
TBA
What is the formula for annual compound interest ( the balance after n years?
Given an inital amount invested, what is the quarterely compound interest?
What is the formula for monthly compound interest?
What is the formula for weekly compound interest?
Lets say you invest £500 into a bank, that pays 12% interest per annum compounded quarterly, to find balance after a year what would it be?
The more frequently interest is compounded the what?
The higher the amount you get in a given year.
What is the general formula for compound interest over a given interval?
Say i invest 500 in a bank and they give interest per annum 12%, and it is compounded daily, what is the return i will get in a year and what is the return i will get in 4 years?
500(1+0.12/365)^365 = 563.7373078
500(1+0.12/365)^365 x 4 = 807.9734601
When m the amount of compound periods in a year gets larger and larger what happens?
the value of (1+r/m)^m gets closer to the expotential constant, e which is aproximately 2.7182818
When you are continously compounding, what is happening?
its changing every second, minute etc, like stock markets.
If the bank was to compound continously ( interest was compounded every instant, the balance in the bank at the end of one, 2 3 years would be what?
What is the general formula for compounding interest when you invest a certain amount in the bank?
If i invest 500 at 12% interest per annum with continous compounding what will be the balance after 2 years, 10 years?
500e^2x0.12 = 635.6245752
500e^10x 0.12 = 1660.058461
What are 4 problems involving interest rates?
1) How much do I invest
2) What is the interest rate
3) How long I need to invest
4) How often is the interest compounded m = compounding.
How much i need to invest?
How long do i need to invest to get a return of 1000 if i invest 500 at 12% interest per annum compounded annually?
As you can see using trial and error to solve investments can be time consuming, so what do we use?
We use logs
What is a key rule about logs?
What is the mulitplication rule for logs ( providing they have the same base) ?
What is the log division rule given the bases are the same ?
What is the log power rule?
What is a common log??
We usually use logs in base 10, where they are just denoted as log, called a common log
Logs to the base e are denoted as what and called what?
they are denoted as In and we call this a natural log.
Lets say we have Log base 3 343, what can we do the get the same answer 5?
We can change base of the logs?
What is the changing of the base formula for logs?
the base of the logs on the right hand side can be anything, you will still get the right answer.
Show a change in base for a natural log?
Change the base to get the answer?, change base to 10
Change base of log base 5 (3)? changing base to 3
= log base 3 (3) / log base 3 (5) = 1/log base 5 (3)
Now solve this using logs? ( better way than using trial and error)?
When you buy an asset ( car) what often happens?
It depreciates
If i buy a car for 10000 and the car depreciates at a rate per annum of 5%, whats its value after 1 year and whats its value after 2 years?
Lets say there is compound depreciation, with an inital value and it depreciates per annum, what is the general formula for this and what is the general formula for continously depreciating?
Regular saving plans lets say for example you dont invest a lump sum cash into the bank, but instead you pay every year £600, at a rate 12% per annum, what is the balance at the fourth year?
With regular saving patterns, the act of the working out every year, what will be in the account at the beginning of the year andn the end of the year is quite long, so what can we use?
Geometric sequences and series.
What is the general formula for a geometric sequence?
What is the common ratio here?
What is a Geometric series?
this is what we get when we add up a finite number of terms from a geometric sequence
What is the general formula for a finite geometric sequence?
Lets say we wanted to know the balance after 26 years or n years what would it be?
What does an infinite geometric sequence look like?
What is the general rule for infinite geometric sequences?
What are 2 ways to compare investments to see what will give the best return?
APR
Present and Future value
What is APR?
) is the annual rate charged for borrowing or earned through an investment. ( It gives you standardised numbers to compare e.g. apples and apples, not oranges and apples?
Lets say i invest £1 in an acccount and one bank gives an offer of 10% with daily compounding and the other gives an other of 10.1% with quartertly compounding, which is the best offer to take?, what must you not do?
You must not look at the percentage and think 10.1% is higher, you must, calculate APR
TBA
What is the formula for future and present values of investments ?
What is an annunity?
an account that pays you a fixed income at the end of each year for a certain number of years.
Hint, we can display this very similarily to something we have learnt.
If after 2 years, the amount in the account is £1764, how much did you initally invest?
Move the 3 at the front then times by 2 = 6 -
log base 10 (100 = 2 X 2 = 4
6-4 = 2
use the log bit
1) As it is is depreciation it is Ve^-nr
but -r = 0.2 not 0.8
20000e^(0.2)x3
£10976.23
2)
20000e^(-0.8)n = 100000
e^-0.2 = 10000/200000 = 0.5
log[e^(-0.2)n] = log0.5
nloge^(-0.2) = log0.5
n = log0.5/loge(-0.2)
3.47 years
as it is compounding continousl you have to put down the straight value, you cannot round.
What is the APR for all of these calculations?
APR is annual so
And formula is (1+i/n)^n
Annual APR = (1+0.09) = 9%
Quarterly APR = (1+0.09/4)^4 = 9.308%
Monthly APR = (1+0.09/12)^12 = 9.381%
Continously APR = e^0.09 = 9.417%
tba
