The Crisis of Credit Visualized

Just came across an excellent video explaining the credit crunch.


The Crisis of Credit Visualized from Jonathan Jarvis on Vimeo.

EMI Calculation formula -- proof

=== I'm confident that this proof is correct; but it is possible that there is a simpler proof existing for the same ==

P ==> Loan amount (principal)
E ==> EMI per month (to be derived)
n ==> tenure in months
R ==> rate of interest per month (to avoid cluttering of derivations, R is assumed to have been divided by 100. ie., for 12% p.a interest rate, it is 1% per month, so R is 0.01)

Let us derive the formula for outstanding-principal at the end of n EMIs (months):

outstanding-principal after month n = outstanding-principal after month n-1 minus principal component of the n'th EMI.

First month: P-(E-P*R) ==> P-E+PR ==> P(R+1)-E

Second month (n=2):
=> P-(E-PR)-(E-(P-E+PR)*R) => P-E+PR-(E+ER-(P+PR)*R) => -E-E-ER + P + PR + (P+PR)*R
=> -2E-ER + (P+PR)(1+R) => -E(R+2) + P(1+R)(R+1) = P(R+1)² - E(R+2)

Third month (n=3):
=> P(R+1)² - E(R+2) - E + PR(R+1)² - ER(R+2)
=> P(R+1)² + PR(R+1)² -E -E(R+2) -ER(R+2) ==> P(R+1)³ - E(1+(R+2)(1+R))

Fourth month (n=4):
=> P(R+1)³ - E(1+(R+2)(R+1)) - E + PR(R+1)³ - ER(1+(R+2)(R+1))
=> P(R+1)⁴ -E -E(1+(R+2)(R+1)) -ER(1+(R+2)(R+1))
=> P(R+1)⁴ - E(1+(1+(R+2)(R+1))(R+1)) -------- (1)

Clearly the first portion of the outstanding principal has become a function of 'n'. I had to formulate the second portion ie., the co-efficient of E -> (1+(1+(R+2)(R+1))(R+1)) so I can come up with a outstanding-principal formula after n EMIs.

First approach:

1 ==> (R+1)⁰
R+2 ==> (R+1)⁰ + (R+1)¹
1+(R+2)(R+1) == > 1 + R² + R + 2R+2 ==> (R+1)⁰ + (R+1)¹ + (R+1)²
1+(1+(R+2)(R+1))(R+1) ==> 1 + ((R+1)²+R+2)(R+1)) == > 1 + ((R+1)³+(R+2)(R+1))
==> (R+1)⁰ + (R+1)¹ + (R+1)²+ (R+1)³

This translates to something like a⁰+a¹+a²+a³...+aⁿ ; but I have no idea of any means to change this to a simple function of n.

So tried another approach to deduce:

R+2 => (R²+2R+1-1)/R ==> ((R+1)²-1)/R
1 + (R+2)(R+1) ==> 1 + ((R+1)²-1)*(R+1)/R ==> 1 + ((R+1)³-R-1)/R
==> 1 + ((R+1)³-1)/R - 1 == > ((R+1)³-1)/R
Similarly it can also be proved that for n=4 it is ((R+1)⁴-1)/R

Proof my mathematical induction:

Lets call this function P(k):

P(1) = ((R+1)¹-1)/R => R+1-1/R => 1
Assuming P(k) is true, lets prove p(k+1).

based on the above recurrence, p(k+1) = 1+(R+1)P(k)
==> 1+(R+1) * ((R+1)^k-1)/R ==> 1 + ((R+1)^(k+1)-(R+1))/R
==> 1 + ((R+1)^(k+1)-1)/R - 1 ==> ((R+1)^(k+1)-1)/R ==> P(k+1)


Substituting the result in equation (1), the outstanding principal after n EMIs is: => P(R+1)ⁿ - E((R+1)ⁿ-1)/R

For the loan to end correctly, the outstanding principal at the end of n EMIs should be less than or equal to zero (if -ve, last month EMI will be smaller appropriately).

P(R+1)ⁿ - E((R+1)ⁿ - 1)/R <= 0 or E >= P(R+1)ⁿ / (((R+1)ⁿ - 1)/R) or
E >= PR(R+1)ⁿ / ((R+1)ⁿ -1)

EMI has to be as small as possible while completing the loan on time. So, computationally, for E to be a smallest possible round integer, >= can be converted to = by doing a ceil on the result.

So, E = ceil(PR(R+1)ⁿ / ((R+1)ⁿ -1))

where ceil(x) ==> smallest integer that is >= x e.g, ceil(1.5) = 2; ceil(2.1) = 3; ceil(2) = 2;

Hence proved :)

Loan EMI calculator

Many times people are in need of this. Be it home loan or personal loan or vehicle loan, the calculation of EMI is the same.

This calculator works for any of these loans with a monthly paid EMI and month reducing balance. Just key in the details and enjoy.

Loan amount

Rate of interest (% p.a)

Tenure (in months)

The calculated EMI is

One step further, if you need to calculate the EMI in your own application/webpage you can use this formula,

EMI = ceil(PR(R+1)ⁿ / ((R+1)ⁿ -1))

where,

P ==> Loan amount (principal)
n ==> tenure in months
R ==> rate of interest per month (to avoid cluttering of formula, it is assumed to have been divided by 100. ie., for 12% p.a interest rate, it is 1% per month, so I is 0.01)
ceil(x) ==> smallest integer that is >= x e.g, ceil(1.5) = 2; ceil(2.1) = 3; ceil(2) = 2;

Note: I'm sure there are other websites that help you in calculating this, but this EMI calculator uses my "own" formula to calculate the EMI (don't get scared, it works :D). I will soon post a proof of my derivation of the formula.

MoneySaver, HomeSaver, MaxGain -- what's special?

=== specific to India ===

Disclaimer: I am not providing investment advice or anything of that sort. I'm just sharing my experience and knowledge on this topic. Use your own conscience and decide on your investments. You can consider my inputs but the decision is solely yours. I'm not in any way responsible for any profit/loss that you might make based on the information available in my blog here.

What are these? These are the names of a special variant in home loans. Inspite of being many years old and having tremendous power/flexibility, this home loan variant usually goes unnoticed.

Even today, not all banks have this option. Even HDFC does not have one. ICICI calls it MoneySaver, Standard Chartered Bank calls it HomeSaver, HSBC calls it SmartHome and SBI recently launched its own version as MaxGain -- this may not be the complete list. Banks are slowly starting to catch up!

So whats so special about it? All the usual terms of a standard mortgage loan applies to this loan too. However, along with the loan you would also get a current account associated with it. The exact linkage between the current account and the home loan is being handled by different banks differently -- but whatever it is the essence and the effect of this loan type is the same.

You would be paying EMIs normally just like how you would on anyother home loan. In addition, you have the option to deposit more money into that current account. Any amount deposited into the current account gets debitted from your home loan's outstanding principal -- interesting? So you would not be paying interest on this portion anymore. In its effect, it is as good as you have prepayed a portion of your home loan without any prepayment penalities. In an investor's perspective, the current account earns your money interest at your home loan interest rate. What is even more interesting is that you have all flexibility to withdraw that money out of the current account anytime you want (maybe you are in a short credit crunch) and deposit it back whenever you want.

For any loan, the EMIs are devised in such a way that it is slightly more than the interest that you are supposed to be paying. So the rate at which the principal component is repayed is very small in the initial stages. Over time, the gradual reduction in the principal decreases the interest liability, which naturally increases the principal component that you repay in every EMI. These types of loans give you the flexibility to increase your principal repayment as you like it anytime you want -- now, this is a boon.

In the kind of downturn that we see and with the increased interest rates on loans, these kind of loans would generate stable risk-free return with total liquidity. Any money that you own and that you think cannot generate a return at the rate of interest of your home loan, would atleast have a reasonable place to reside. Even the high interest rate fixed deposits are not close to the home loan interest rates.

Then why would anyone not take this loan? Yes, there is a catch. This loan is availed at a premium on the interest rate. The interest rate margin might differ from bank to bank but usually it is around 0.5%-0.75% more than the normal interest rate (be it floating or semi-fixed or fixed). I think this loan in benefical for anyone who has a consistent increase in their annual income (ineffect a consistent increase in their repayment capability).

An example: A 20L loan at 11% p.a for a tenure of 240 months (20 years) will attract an EMI of around 20K per month. If you manage to save 5K more every month and deposit in the current account, the same loan would be closed in just around 140 months (~12 years).

Atleast consider this option before making your next choice!