I came across the "Dimensions" video via Reddit. It tries to help you visualise four-dimensional objects via their projections onto ordinary three-dimensional space. If you have even the slightest interest in mathematics, you should check it out. You can either view it on-line or order a reasonably-priced DVD. The video is about two hours long.
Showing posts with label math. Show all posts
Showing posts with label math. Show all posts
2008-12-25
2007-08-20
Calculating Interest Rates
You want to buy that fancy LCD TV that costs Rs 60,000 but you do not have that much money with you. You see an advertisement in a newspaper for the TV from a dealer who offers to sell it to you if you make a down-payment of Rs 10,000 and pay Rs 4,380 every month for one year. You see another advertisement in the newspaper for the same TV from another dealer who offers to sell it to you if you make a down-payment of Rs 20,000 and pay Rs 1,330 every month for three years. How do you calculate the rate of interest each dealer is charging you for what is, in effect, a loan?
In "Calculating EMIs", we derived the formula for calculating the "Equated Monthly Installment" (EMI) on a loan. If "E" represents the EMI, "P" represents the principal amount in the loan, "r" represents the monthly rate of interest (one way of arriving at it is to divide the annual rate of interest, quoted as a percentage, by 1,200) and "n" represents the number of months in the tenure of the loan, then:
E = P × r × (1 + r)n / ((1 + r)n - 1)
In the current example, we know the values for "E", "P" and "n" and wish to calculate "r". Unfortunately it is not that simple to calculate "r" using just the high-school algebra that most of us manage to remember. Fortunately there is a simple algorithm that can help us in this situation.
Let us first rewrite the formula above as an equation:
P × r × (1 + r)n / ((1 + r)n - 1) - E = 0
Our task now is to find the roots of this equation - that is, the values of "r" that will make the left-hand-side (LHS) of this equation evaluate to zero.
To find the roots of a given equation "f(x) = 0", the algorithm in question can be described as follows:
Note that this is just a binary search algorithm. By "close enough to zero", we mean that the absolute value of "f(c)" is less than some value, usually called "epsilon", that can be as small as we please. The algorithm given above can be rewritten as a function in a pseudo-language as follows:
You can implement this in your favourite programming language along with a function that calculates the LHS of the equation given earlier. You can choose a value of "epsilon" according to your preference - the smaller the value of "epsilon", the more accurate is the result and the longer it takes to compute it. The time taken for the computation is also affected by how wide is the range between "a" and "b". Note that Newton's method is a much faster way of computing the roots of such equations, though it involves calculating derivatives.
How do you arrive at the values for "a" and "b"? This differs for each function. For our example, we can start with a low guess of "0.001%" ("0%" gives an undefined result) as the annual rate of interest and a high guess of "100%" and this gives us a negative and a positive value for the LHS respectively. With an "epsilon" of "0.00001", a C programme computes the answer in around 25 iterations.
In our example, the first dealer is offering us an effective loan of Rs 50,000 for 12 months with an EMI of Rs 4,380 and the effective annual rate of interest comes to about 9.32%. The second dealer is offering us an effective loan of Rs 40,000 for 36 months with an EMI of Rs 1,330 and the effective annual rate of interest comes to about 12.08%. In terms of the interest rates being charged by the dealers, you should now be able to tell that the first dealer has a better proposition for you when compared to the second dealer.
In "Calculating EMIs", we derived the formula for calculating the "Equated Monthly Installment" (EMI) on a loan. If "E" represents the EMI, "P" represents the principal amount in the loan, "r" represents the monthly rate of interest (one way of arriving at it is to divide the annual rate of interest, quoted as a percentage, by 1,200) and "n" represents the number of months in the tenure of the loan, then:
E = P × r × (1 + r)n / ((1 + r)n - 1)
In the current example, we know the values for "E", "P" and "n" and wish to calculate "r". Unfortunately it is not that simple to calculate "r" using just the high-school algebra that most of us manage to remember. Fortunately there is a simple algorithm that can help us in this situation.
Let us first rewrite the formula above as an equation:
P × r × (1 + r)n / ((1 + r)n - 1) - E = 0
Our task now is to find the roots of this equation - that is, the values of "r" that will make the left-hand-side (LHS) of this equation evaluate to zero.
To find the roots of a given equation "f(x) = 0", the algorithm in question can be described as follows:
- Find a value "a" for which "f(a)" evaluates to a negative value.
- Find a value "b" for which "f(b)" evaluates to a positive value.
- Let "c" be the average of "a" and "b".
- If "f(c)" is close enough to zero, "c" is the desired root.
- Otherwise, if "f(c)" is a negative value, substitute "c" for "a" and repeat the procedure from step #3 and if "f(c)" is a positive value, substitute "c" for "b" and repeat the procedure from step #3.
Note that this is just a binary search algorithm. By "close enough to zero", we mean that the absolute value of "f(c)" is less than some value, usually called "epsilon", that can be as small as we please. The algorithm given above can be rewritten as a function in a pseudo-language as follows:
guessRoot( f, a, b)
{
c := (a + b) / 2;
if( absoluteValue( f( c)) < EPSILON)
return c;
else if( f(c) < 0)
return guessRoot( f, c, b);
else
return guessRoot( f, a, c);
}
You can implement this in your favourite programming language along with a function that calculates the LHS of the equation given earlier. You can choose a value of "epsilon" according to your preference - the smaller the value of "epsilon", the more accurate is the result and the longer it takes to compute it. The time taken for the computation is also affected by how wide is the range between "a" and "b". Note that Newton's method is a much faster way of computing the roots of such equations, though it involves calculating derivatives.
How do you arrive at the values for "a" and "b"? This differs for each function. For our example, we can start with a low guess of "0.001%" ("0%" gives an undefined result) as the annual rate of interest and a high guess of "100%" and this gives us a negative and a positive value for the LHS respectively. With an "epsilon" of "0.00001", a C programme computes the answer in around 25 iterations.
In our example, the first dealer is offering us an effective loan of Rs 50,000 for 12 months with an EMI of Rs 4,380 and the effective annual rate of interest comes to about 9.32%. The second dealer is offering us an effective loan of Rs 40,000 for 36 months with an EMI of Rs 1,330 and the effective annual rate of interest comes to about 12.08%. In terms of the interest rates being charged by the dealers, you should now be able to tell that the first dealer has a better proposition for you when compared to the second dealer.
2007-03-11
Investing For Retirement
(Note: This post might not be of interest to those not from India.)
Most of us do not even think about planning for retirement until we reach the age of 30. Some of us "live for the moment" and don't care for the future, some of us feel uncomfortable thinking about retirement and pretend like the proverbial cat that closing our eyes to the problem will make it go away and some of us just do not know how to assess our financial requirements three decades into the future.
Unfortunately for us, there is not much of a government-provided social security in India for old folks, we cannot realistically expect our children to take care of all our expenses, inflation constantly lessens the value of our savings and interests on assured-return investments (fixed-deposits, EPFs, etc.) keep falling. We must have some idea of our needs at the time of our retirement and know how much to invest now to be able to afford the same lifestyle that we are currently used to.
The good news is that we can use basic mathematics to calculate these figures. We will make use of two equations. The first equation (call it "E1") tells us the final amount "S" that an initial amount "P" grows to if it grows at a compounded rate of "r" over "n" years:
The second equation (call it "E2") tells us the final amount "S" that a regular annual investment of "P" over "n" years gives if it grows at a compounded rate of "r":
Note that since the rates are usually quoted as percentages, you need to divide them by 100 to get the value of "r" usable in these equations. For example, a quoted rate of 8% translates to "r" equal to 0.08.
Now assume that you are aged 30 years, plan to retire at the age of 60 years, have a montly expenditure of 20,000 rupees and the rate of inflation is about 5% on the average. Using E1, you can see that at the time of your retirement 30 years hence, your monthly expenditure would become about 86,438.85 rupees simply because of inflation! That translates to about 10,37,266 rupees in annual expenditure. With old age come many an ailment for which you would need to spend money - at about 1,00,000 rupees per year at today's rates, you would need about 4,32,194 rupees at the time of your retirement to meet medical expenses. So you would need an annual income of at least 14,69,460 rupees at the time of your retirement just to sustain your current lifestyle and cope with the inevitable medical expenses!
How will you generate an income like this at that time? It is very likely that your appetite for risk would have considerably diminished at that time and you would only be willing to invest for assured-returns and thus lower rates of interest, say, about 5%. This in turn means that you would need a sum of 2,93,89,200 rupees (5% of which is the amount you need per year) at the time of your retirement. You need to have raised about 3 crore rupees by the time you retire just to be able to afford your current lifestyle!
To raise this kind of money, you either need to invest a certain amount annually till the time you retire or do a one-time investment. If you assume an annual return of 8% on your investments, you either need to invest about 2,59,431 rupees annually for the next 30 years (using E2) or about 29,20,620 rupees at a single shot (using E1). If you assume a more aggressive (though riskier) annual return of 15% on your investments, the amounts change to about 67,601 rupees and about 4,43,867 rupees respectively.
If you had started at the age of 25 years, you would have had 35 years to raise the money. At a per-annum return of 8%, you would have either invested about 1,70,533 rupees annually or about 19,87,725 rupees at a single shot. At a per-annum return of 15%, these figures become about 33,352 rupees and about 2,20,680 rupees respectively.
If you postpone investing for your retirement by another five years, you would have 25 years to raise the money. At a per-annum return of 8%, you would need to either invest about 4,02,008 rupees annually or about 42,91,349 rupees at a single shot. At a per-annum return of 15%, these figures become about 1,38,112 rupees and about 8,92,774 rupees respectively.
So the earlier you start investing for your retirement, the better it is for you. The folks at Personalfn.com have a report titled "Retirement Planning and You" that provides a more detailed analysis of this situation as well as the available investment options suitable for retirement planning.
Of course, these are simplified calculations. They do not take into account the fact that you will very likely have to pay income tax on the returns from this investment. They also do not take into account the fact that because of inflation, you would need slightly more and more every year after you retire instead of the fixed amount assumed here. Hopefully the average rate of inflation for India for the next 30 years will be less than the 5% assumed here.
Most of us do not even think about planning for retirement until we reach the age of 30. Some of us "live for the moment" and don't care for the future, some of us feel uncomfortable thinking about retirement and pretend like the proverbial cat that closing our eyes to the problem will make it go away and some of us just do not know how to assess our financial requirements three decades into the future.
Unfortunately for us, there is not much of a government-provided social security in India for old folks, we cannot realistically expect our children to take care of all our expenses, inflation constantly lessens the value of our savings and interests on assured-return investments (fixed-deposits, EPFs, etc.) keep falling. We must have some idea of our needs at the time of our retirement and know how much to invest now to be able to afford the same lifestyle that we are currently used to.
The good news is that we can use basic mathematics to calculate these figures. We will make use of two equations. The first equation (call it "E1") tells us the final amount "S" that an initial amount "P" grows to if it grows at a compounded rate of "r" over "n" years:
S = P × (1 + r)n
The second equation (call it "E2") tells us the final amount "S" that a regular annual investment of "P" over "n" years gives if it grows at a compounded rate of "r":
S = P × ((1 + r)n - 1) / r
Note that since the rates are usually quoted as percentages, you need to divide them by 100 to get the value of "r" usable in these equations. For example, a quoted rate of 8% translates to "r" equal to 0.08.
Now assume that you are aged 30 years, plan to retire at the age of 60 years, have a montly expenditure of 20,000 rupees and the rate of inflation is about 5% on the average. Using E1, you can see that at the time of your retirement 30 years hence, your monthly expenditure would become about 86,438.85 rupees simply because of inflation! That translates to about 10,37,266 rupees in annual expenditure. With old age come many an ailment for which you would need to spend money - at about 1,00,000 rupees per year at today's rates, you would need about 4,32,194 rupees at the time of your retirement to meet medical expenses. So you would need an annual income of at least 14,69,460 rupees at the time of your retirement just to sustain your current lifestyle and cope with the inevitable medical expenses!
How will you generate an income like this at that time? It is very likely that your appetite for risk would have considerably diminished at that time and you would only be willing to invest for assured-returns and thus lower rates of interest, say, about 5%. This in turn means that you would need a sum of 2,93,89,200 rupees (5% of which is the amount you need per year) at the time of your retirement. You need to have raised about 3 crore rupees by the time you retire just to be able to afford your current lifestyle!
To raise this kind of money, you either need to invest a certain amount annually till the time you retire or do a one-time investment. If you assume an annual return of 8% on your investments, you either need to invest about 2,59,431 rupees annually for the next 30 years (using E2) or about 29,20,620 rupees at a single shot (using E1). If you assume a more aggressive (though riskier) annual return of 15% on your investments, the amounts change to about 67,601 rupees and about 4,43,867 rupees respectively.
If you had started at the age of 25 years, you would have had 35 years to raise the money. At a per-annum return of 8%, you would have either invested about 1,70,533 rupees annually or about 19,87,725 rupees at a single shot. At a per-annum return of 15%, these figures become about 33,352 rupees and about 2,20,680 rupees respectively.
If you postpone investing for your retirement by another five years, you would have 25 years to raise the money. At a per-annum return of 8%, you would need to either invest about 4,02,008 rupees annually or about 42,91,349 rupees at a single shot. At a per-annum return of 15%, these figures become about 1,38,112 rupees and about 8,92,774 rupees respectively.
So the earlier you start investing for your retirement, the better it is for you. The folks at Personalfn.com have a report titled "Retirement Planning and You" that provides a more detailed analysis of this situation as well as the available investment options suitable for retirement planning.
Of course, these are simplified calculations. They do not take into account the fact that you will very likely have to pay income tax on the returns from this investment. They also do not take into account the fact that because of inflation, you would need slightly more and more every year after you retire instead of the fixed amount assumed here. Hopefully the average rate of inflation for India for the next 30 years will be less than the 5% assumed here.
2006-06-23
Mathematical Digressions
In my previous post, I said that the sum of the geometric series:
is (xn - 1)/(x - 1). One of my beliefs (as also shown by that post) is that you should never take a formula at its face value. Here I quickly attempt to show why this particular formula should be correct using a very commonly used proof. Let us assume that S denotes the desired sum. That is:
1 + x + x2 + ... + xn-1
is (xn - 1)/(x - 1). One of my beliefs (as also shown by that post) is that you should never take a formula at its face value. Here I quickly attempt to show why this particular formula should be correct using a very commonly used proof. Let us assume that S denotes the desired sum. That is:
S = 1 + x + x2 + ... + xn-1
If we multiply both sides of this equation by x, we get:
If we subtract each side of the first equation from the corresponding side of the second equation we get:
This simplifies to:
which gives:
as we had claimed.
By the way, while attending a conference call last night where it was particularly hard to concentrate on what was being discussed, I began to doodle and stumbled upon a simple result which might have been interesting about a millennium ago and is probably just an idle curiosity now. I am also sure it is contained somewhere in our high school text books for mathematics, but I cannot recall seeing it before. (Despite what might be indicated by these blog posts, I do not have much of an interest in mathematics itself and have never pursued it for its own sake.)
Anyways, looking at the factorisation of (x2 - y2) into (x - y)×(x + y) and (x3 - y3) into (x - y)×(x2 + x×y + y2), I wondered if (x - y) is always a factor of (xn - yn). By playing around a little bit with the terms, I could write:
confirming my hypothesis. If you set y=1, this reduces to:
which can be rewritten as:
which is the same formula that we derived just a while back for calculating the sum of a geometric series!
If we multiply both sides of this equation by x, we get:
x×S = x + x2 + x3 + ... + xn
If we subtract each side of the first equation from the corresponding side of the second equation we get:
(x - 1)×S = x + x2 + x3 + ... + xn - 1 - x - x2 - ... - xn-1
This simplifies to:
(x - 1)×S = xn - 1
which gives:
S = (xn - 1)/(x - 1)
as we had claimed.
By the way, while attending a conference call last night where it was particularly hard to concentrate on what was being discussed, I began to doodle and stumbled upon a simple result which might have been interesting about a millennium ago and is probably just an idle curiosity now. I am also sure it is contained somewhere in our high school text books for mathematics, but I cannot recall seeing it before. (Despite what might be indicated by these blog posts, I do not have much of an interest in mathematics itself and have never pursued it for its own sake.)
Anyways, looking at the factorisation of (x2 - y2) into (x - y)×(x + y) and (x3 - y3) into (x - y)×(x2 + x×y + y2), I wondered if (x - y) is always a factor of (xn - yn). By playing around a little bit with the terms, I could write:
xn - yn = (x - y)×(xn-1 + xn-2×y + ... + x×yn-2 + yn-1)
confirming my hypothesis. If you set y=1, this reduces to:
xn - 1 = (x - 1)×(xn-1 + xn-2 + ... + x + 1)
which can be rewritten as:
(xn - 1)/(x - 1) = xn-1 + xn-2 + ... + x + 1
which is the same formula that we derived just a while back for calculating the sum of a geometric series!
Calculating EMIs
Most of the software engineers I know here in Bangalore have either already bought a house or are planning to buy one. The biggest incentives are perhaps the easy availability of home loans at interest rates far lower than that available to the previous generation and the tax-breaks one gets here in India on the principal and the interest paid on home loans. Our generation is also not averse to taking on a debt unlike the previous generation. In addition, many of us feel that it is better to take the plunge now and enjoy the comforts of your own house than to diligently save all the required money for years and then buy a house, only to find out that the dream was realised a bit too late in your life.
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