Annuities payable pthly (chapter 6)

Guys need you help; can someone please help me get a visual understanding of this concept?

The core material in chapter 6 page 12 states:

If p and n are positive integers, the notation a p/n is used to denote the value at time 0 of a level annuity payable pthly in arrear at the rate of 1 per unit time over the time interval [0.n]. For this annuity the payments are made at times 1/p, 2/p, 3/p...,n and the amount of each payment is 1/p


It then goes onto say:

By definition, a series of p payments, each of amount i^((p))⁄p in arrear at pthly subintervals over any unit time interval, has the same value as a single payment of amount i at the end of the interval. By proportion, p payments, each of amount 1/p in arrear at pthly subintervals over any unit time interval, have the same value as a single payment of amount i⁄i^((p)) at the end of the interval.

Can some help explain this using some example numbers and interest rates and showing this on a time line.

I thank you in advance for those who can help me.

Kind Regards,

Dublin R

 

Hi dublin,

First off all You should be clear with the meaning of i^(p), Interest rate paid pthly which actually means the effective rate of interest for 1/p period.

For example
If we know that effective interest rate is 10% p.a then effective rate per month should be calculated as follows

Since there are 12 months i 1 year and if we assume that 'x' is effective rate per month then...
1.1 = (1+x)^12
so, 1+x = (1.1)^(1/12)
and hence x = 1.1^(1/12)-1 = 0.8%

To make actuarial work easier, we generally denote '12*x' as i^(12)

Now i think You are clear with the concept behind pthly inerest rates..

Now lets have a look over payments

Assume that You have to pay 1200Rs in one year time. there are several options
1. You pay total amount either at beginning or end of the year
or
2. You pay 1200/12=100 (which was written as 1/p) per month. (Note p=12 here)
or
3. You pay 1200/4 = 300 per quarter. (Note p=4 here)
or
many more....

Lets explore option 2......Note i^(12)/12 = 0.8% as calculated before

Since u paid 100 at the start of each month, the total accumulated value after one year would be
100*1.008^12 + 100*1.008^11 +...........100 = 1200*1.1 = 1320

which is equivalent to accumulated value from option 1

For clarity Note
100 = 1200/12 (equivalent to 1/p)
10% = i
p=12
0.8% = i^(p)/p

Now read the core reading again...u can understand that properly

 

is it clear now

 

Hi jatin! Thank you very much for your reply. I think I’ve got a good idea of the concept now. I think my problem seems to the confusion between nominal interest rates and effective interest rates.

Ok, can you help me with this similar question!

Calculate a_6^ at 1.5 % effective. ( I can do this question by entering the numbers into an annuity formula, but can you show me a time line what the payments look like.)

Eg i=1.5%
Value £1 £1 £1 £1 £1 £1
PV 0
Time n=1 n=2 n=3 n=4 n=5 n=6

From my understanding this question is asking for the PRESENT VALUE of 6 payments of 1 unit at 1.5% effective rate per unit time period. Am I right in thinking this?

Calculate a_6^4 at 1.5 % effective ( How will the time value diagram look like for this payment stream)

 

ok sorry guys the diagram did'nt come out clearly! jatin i have attached the document on pdf.

Let me know what you think!
thanks

 

hi dulin,

Yup..u are right here.

a_6^(4) means payments are done 4 times in a year(means quarterly) and total cash flow of 1 unit in each year(means 0.25 at the end of each quarter) now you can deal in two ways...

1. By using direct formula
or
2.
by first calculating effective quarterly rate as (1+i)^(1/4)-1 and then using following expression...... 0.25 * a_24

Note here, i used 0.25 caz cashflow is 0.25 per quarter and 24 because there are total 24 quarters in 6 years

Regards
Jatin

 

if p payments of i^(p) / p payed in arrear at pthly intervals is equivalent to a payment of i at the end of the interval n then by ratio, p payments of ammount 1/p payed in arrear at pthly subintervals is equivalent to

(i^(p)/p)/(1/p) = i/x at the end of n.

re arranging we find x = i/i^(p) as required.

Can someone tell me if im correct in thinking about it in this way.

 

yup....dear now you are thinking in right direction..