### Effective Interest rates vs Nominal rates vs Force of Interest

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What exactly is Effective interest rate? The
Effective interest rate over a given time period is the amount of interest a
single initial investment will earn at the end of time period.

We will clear now it in detail.

1.

*Effective annual interest had interest paid once at the end of each year.***2.**

*Effective annual Discount had interest paid once at the start of each year.*
So, what will happen if interest paid is not once in a
measurement period, then there comes a

*Nominal Rate.*

**Nominal**is used where is interest is paid more (or less) frequently than once per measurement period.

__Application of Nominal rate in real life__: Bank accounts normally use nominal rates. They quote the annual interest rate but interest is actually added at the end of each month. Thus, here interest is paid more frequently than once per unit time year.

*8 points for Interest Rates: Nominal and Effective*

1) Here is the notation; i

**Nominal rate of interest convertible pthly or compounded pthly (we meant to say that interest is payable p times per period)**^{(p)}=
2) It also means that rate of interest of i

^{(p)}/p is applicable for each p^{th}of a period.
3) For example: Nominal rate of interest is 6%p.a.
convertible quarterly. It means i

**6%. So i**^{(4)}=^{(4)}/4 = 1.5%. So, this is the rate (i.e.1.5) which is applicable for each p^{th}of a period where p=4
4) If you see carefully 3

^{rd}point then we are just annualising a pthly effective interest rate.
5) Let’s see one more example: Suppose we have given monthly effective interest rate of 2% then what will
be the nominal annual interest rate convertible monthly? We have given i

**12 = 2%, and we have to find i**^{(12)}/**2%*12 = 24%**^{(12)}=
6) Effective interest rate will always be greater than
nominal interest rate as we have taken into account the effect of compounding
too.

7) Note the important formula:

**1+i = (1+i**With the help of this formula we can calculate effective pthly rate into effective annual rate.^{p}/p)^{p}.
8) Let’s see example to justify above point: Suppose nominal rate = 8%p.a. convertible half yearly. So
what will be the value of Rs.500 after 3 years? we have given i

^{(2) }= 8%. So i^{(2)}/2 = 4%. Now we can use the above formula: 1+i = (1+.04)^{2}. By solving this we get i= 8.16%. So accumulated value will be 500(1.0816)^{3}= 632.66**9) Point is simple that if half year effective rate is 3% then it does not mean that effective annual rate will be 6%. Rather it is 1.03**

^{2}-1 = 6.09% due to effect of compounding.

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*8 points for Discounting and Force of Interest Rates*
1. Important formula:

**1-d = (1-d**^{(p)}/p)^{p}
2. Discounting for n years denoted by v(n)= (1-d)

**and Accumulating for n years denoted by A(n) = (1+i)**^{n}**(1-d)**^{n }= 1/v(n) =^{-n}
3. We have seen where interest is paid once per
measurement period (effective) and more or less than once per measurement
period (Nominal). What will happen if the interest is paid continuously, then
there comes

**() denoted by delta***Force of Interest.*
4. Force of interest (delta) it is like i

^{(p)}**where p leads to infinity (∞).**
5. Euler’s rule:
limit (n approaches to ∞) (1+

^{x/n})^{n}= e^{x}and we have seen that**1+i = (1+i**Thus^{p}/p)^{p}.**1+i = exp(delta****)**

**6. d<d**

^{(2)}<d^{(4)}<……. delta**< ………**

**<i**Here what we are saying is that if we have suppose i=10% then i

^{(4)}<i^{(2)}<i.^{(2)}< 10%, you can calculate too using formula given in point no.7 of interest. Similarly when you calculate delta, it is also less than i

^{(4)}. You can find both of them using i=10%. Suppose i

^{(2)}comes out to be 9% then delta will be less than 9%.

__Case study question:__

*If you go to bank for a loan, and your banker said either you will get loan at d*^{(4)}= 10% or d^{(12)}= 10%, at which rate will you want the loan?

__Case study answer__**:**Simple thing is that you can convert both the rates into effective annual interest rate i.e. “i” whichever i is lower you will prefer than loan. But without calculating the i we can tell i.e. using point no. 6 result above. d

^{(4)}< d

^{(12)}<i .

So here we suppose when i=10% then we found out that d

^{(12)}=9% and d^{(4)}=8% . (these are not correct figures however you can find them using formulas given above). So, there is a common sense that when d^{(12)}= 10% then i will be more than 10% (let’s say 11% , this is not correct figure) and when d^{(4)}=10% then i will be more than 11% (this 11% means is that it will be more than d^{(12)}=10%) . So, as a rational borrower, I want the loan at less rate of interest so I would prefer d^{(12)}=10%.
If you still didn’t get the above answer that here is
the trick, use the result given in point no.6 that the interest rate which will
be in the right , you will prefer that condition is that both rates should be
same. I am saying is that if d

^{(2)}=10% given and d^{(12)}=10% is given than you prefer which one is in the right in that equation i.e. d^{(12)}. Similarly for i^{(2)}=10% and i^{(12)}=10% , I would prefer i^{(2)}, because it is in the right.
7. Application of force of interest: Although it is a
theoretical measure but can be used as an approximation to interest paid very
frequently i.e. daily or weekly.

8. If force of interest is a function of time then we
can find accumulation factor: A(t

_{1}^{,}t_{2}) = exp(∫^{t2}_{t1 }f(t) dt
Conclusion:

Effective interest: When paid once per
measurement period

Nominal interest: When paid more(or
less) than once per measurement period

Force of interest: When paid very
frequently in a measurement period.

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