How did my lecturer get this equation ?

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Skyun

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Hi everyone,

I hope I am not posting in the wrong category, please do let me know and accept my apologies if I did.

I am trying to understand how did my lecturer get the equation he used to answer an exercise involving bonds. I have attached three pictures below to better illustrate what I am referring to:

Picture 1 - The initial formula that was given during our lectures (C = Coupon payment; P_0 = Price of the bond):

Screenshot 2021-01-30 at 16.09.29.png

Picture 2 - The exercise, and how I used the formula to find the answer (we assume that the coupon's Face value is 100):

Screenshot 2021-01-30 at 16.13.51.png

Picture 3 - The solution

Screenshot 2021-01-30 at 16.16.56.png

As you can see, the solution does not use the same formula. Although both methods lead to the same answer and I could just stick with the initial formula, I cannot sleep at night if I do not understand how his method works. I have tried to figure out how did he go from what I have done in Picture 2, to what he has done in Picture 3.
So far, I have only come to the conclusion that his method is basically following this "template":

Screenshot 2021-01-30 at 16.22.56.png

However, I do not understand how do we go from the formula in Picture 1, to the formula above. There must be a relation between the two as they give the same answers. I just can't figure it out, I think I must be missing something or lack knowledge.

Could anyone help me understand please ?

Regards.
 
Last edited:
Subhotosh Khan said:
Look up the equation for summation of Geometric Series - in your textbook/class-notes or Google.

If you are still stuck - come back and show us what you found.

View attachment 24778

=

View attachment 24779
Click to expand...
Thank you for your reply.

I looked it up and it makes much more sense already. However, I think I might have done something wrong as I do not find the same answer when using the equation for summing a geometric series.

Here is what I found:
Screenshot 2021-01-30 at 17.49.29.png
I believe this is because the denominator in the geometric series equation is 1-1/1.08 (which is the 1-r); whereas the equation I found by "reverse engineering" my lecturer's solution had a denominator that was equal to 0.08 only. Perhaps I have made a mistake somewhere, if you could please point it out ?
 
Skyun said:
Thank you for your reply.

I looked it up and it makes much more sense already. However, I think I might have done something wrong as I do not find the same answer when using the equation for summing a geometric series.

Here is what I found:
View attachment 24781
I believe this is because the denominator in the geometric series equation is 1-1/1.08 (which is the 1-r); whereas the equation I found by "reverse engineering" my lecturer's solution had a denominator that was equal to 0.08 only. Perhaps I have made a mistake somewhere, if you could please point it out ?
Click to expand...
I have just noticed that the geometric sequence in my exercise does not include the first term (which is 'a' in the formula for computing geometric series). The sequence starts directly with 'ar'. I believe this is most likely the reason why I do not find the same answer (?). In this case, how would this affect the process ?
 
Let S = 1+r + r^2 + r^3 + ... + r^n
Then rS = r + r^2 + r^3 + ... + r^n + r^(n+1)

Now subtract, bottom eq - top equation

rS-S = (1-r)S = 1- r^(n+1)

dividing by 1-r

S = [1- r^(n+1)]/[1-r]
 
Jomo said:
Let S = 1+r + r^2 + r^3 + ... + r^n
Then rS = r + r^2 + r^3 + ... + r^n + r^(n+1)

Now subtract, bottom eq - top equation

rS-S = (1-r)S = 1- r^(n+1)

dividing by 1-r

S = [1- r^(n+1)]/[1-r]
Click to expand...
Thank you for your help, I understood how we get to this formula.

I believe you made a mistake in line 4. Shouldn't it be S-rS = 1-r^(n+1) ?
Because rS-S = r^(n+1)-1
 
Skyun said:
I have just noticed that the geometric sequence in my exercise does not include the first term (which is 'a' in the formula for computing geometric series). The sequence starts directly with 'ar'. I believe this is most likely the reason why I do not find the same answer (?). In this case, how would this affect the process ?
Click to expand...
In your problem, the first term is not 5, so that's not what a is in your formula. Rather, a = 5/1.08. Right?

Alternatively, you could either factor r out, apply the formula, and multiply by r; or add a to the start, apply the formula, and subtract a; or divide the whole series by its first term, apply Jomo's formula that assumes the first term is 1, and then multiply by the first term.

In any case, when you apply any formula, you have to check the definitions of the variables.
 
Thank you all for your support. I have figured it out.

Here is the process for anyone in the future that might have a similar question:
Screenshot 2021-01-30 at 20.03.36.png
 
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