In my previous blog, I highlighted two personal loans: the Citi Personal Loan and the POSB Personal Loan. Both are installment loans. The banks’ websites quote two interest rates: a nominal interest rate and an effective interest rate (EIR).

This blog gives a fuller explanation as to why there are two interest rates. More importantly, it shows you why the EIR and not the nominal interest rate is the true cost of each loan.

**Citi Personal Loan**

Click here for the **website **for this loan.

The loan tenor is from 12 months to 60 months. The nominal (or stated) interest rate for a 12-month loan of $10,000 is 4.94%. If you click “view repayment schedule”, you will see the following page showing the loan repayment schedule. The effective interest rate (EIR) of 9% is below the nominal rate.

This page is a jungle, but once you know how to read it, you will understand why there are two interest rates. Let’s begin.

If you multiply 1.0494 by $10,000 and divide the result by 12, you get $874.50, which is very close to the monthly installment of $874.51 save for a tiny rounding difference.This is how the bank used the nominal rate of 4.94% to derive the monthly loan installment.

Now refer to the loan schedule. It says that each month, the borrow has to repay Citibank $874.51. Columns 4 and 5 splits this amount into a principal portion and an interest portion. For example, for the first month, the principal portion is $799.51. The balance of $874.51 minus $799.51 or $75 is the interest portion. After one month, the principal goes down from $10,000 to $9,200.49 ($10,000 minus $799.51). Repeating the same calculations for the other 11 months completes the repayment schedule.

The loan repayment schedule makes it clear that this loan is a monthly reducing installment loan. The phrase “monthly reducing” is appropriate because after each payment of $874.51, the outstanding principal goes down.

The formula for the present value of a monthly reducing loan of T months, and a monthly interest rate r%/12 is:

PMT x PVIFA (12, 9%/12) = 10,000.

PMT stands for payment (in the present context, it is the monthly installment) and PVIFA stands for “**present value interest factor of an annuity**“.

An annuity is a regular stream of fixed payments. An installment loan with fixed monthly payment is an example of an annuity.

PVFIA(12, 9%/12) gives the present value of 12 payments of $1 each by discounting them at 9%/12 per month. Thus PMT times this interest factor gives the present value of the loan which is also the principal of $10,000.

Using a financial calculator, we have PVIFA(12, 9%/12) = 11.4349126.

Therefore, the monthly installment is:

PMT = 10,000/11.4349126 = $874.51 which reconciles with the number obtained at the beginning. In short, by paying $874.51 each month, you are *effectively* paying an interest rate of 9% p.a. on this loan.

Note: Although I used a financial calculator to work out PMT, you can also use the formula for PVIFA.

Why did Citibank quote a nominal interest rate of 4.94% when the effective rate is almost twice as high? From a marketing viewpoint, I think the answer is obvious.

**POSB Personal Loan **

Click here for the **website** for this loan.

Other than the nominal interest rate, the main difference between this loan and the Citi Personal Loan is that there is a processing fee of 3% which is deducted from the principal. As before, I will use the example of a $10,000 for a tenor of 12 months.

The nominal interest rate is 7.99%. The actual loan amount received by the borrower is $9,700 ($10,000 less 3% of that). Principal and interest is repaid via monthly installments of $900. The monthly installment is derived as follows:

PMT = (10,000 x 1.0799)/12 = $899.92. (POSB rounds this up to $900).

To get the EIR, we use the PVIFA formula with $899.92 as the PMT, and $9700 as the loan present value. We then back out the interest rate r in the formula. r is is the EIR for this installment loan. The calculations are as follows:

899.92 x PVIFA (12, r%/12) = 9,700

Using a financial calculator, we have r = 20.29% = EIR

Note two things:

- The EIR for the POSB loan is much higher than that of the Citibank loan although the difference in monthly installment is just $25.41 (899.92 – 874.51). This is due to two factors: (a) the higher nominal rate of the POSB loan (7.99%) compared to that of the Citibank loan (4.94%), and (b) the additional processing fee charged on the POSB loan. This is why the EIR is so useful – it incorporates all charges into a single number that reflects the true cost of a loan.
- In contrast to the EIR, the nominal interest rate understates the true cost of each loan.