QS 5.16 in CT1

[Soumya]

Hi All,

I am failing to understand the answer to Qs 5.16 (iii)

True or false:

v(t2 ) = v(t1)v(t2 - t1) is answered as False

According to me if t2>t1, the above statement is True
If t1>t2 then V(t2) should be equal to v(t1)/V(t1-t2) - hence the above statement is false.

Plz provide your thougts.

 

[mugono]

Hi All,

I am failing to understand the answer to Qs 5.16 (iii)

True or false:

v(t2 ) = v(t1)v(t2 - t1) is answered as False

According to me if t2>t1, the above statement is True
If t1>t2 then V(t2) should be equal to v(t1)/V(t1-t2) - hence the above statement is false.

Plz provide your thougts.

Write the question here so pple without the notes can help and explain what you're having trouble with.

In regards to what you have written above, the best thing would be to work it out from first principles. Under your latter situation where t1>t2, you'd simply have v^t2.

 

[Soumya]

Write the question here so pple without the notes can help and explain what you're having trouble with.

In regards to what you have written above, the best thing would be to work it out from first principles. Under your latter situation where t1>t2, you'd simply have v^t2.

Hi,

To clarify the question is: whether the below statement is true or false:
v(t2 ) = v(t1)v(t2 - t1)
The answer is given as False. I feel it can be both, depending upon whether t1>t2 or t1<t2

 

[Mark Mitchell]

The notation v(t) gives the discount factor used to discount a cashflow from time t to time 0.

So, v(t2-t1) is the discount factor from time t2-t1 to time 0.

The relationship v(t2) = v(t1)v(t2-t1) is false in general as:
- the LHS discounts from t2 to 0, and
- the RHS discounts from t1 to 0 and from t2-t1 to 0
which don't match up.

What you need on the RHS to make the relationship true is a factor which discounts from time t2 to time t1, multiplied by a factor which discounts from time t1 to time 0 (v(t1)).

Note that the relationship stated is true if we have constant interest rates.

 

[Soumya]

The notation v(t) gives the discount factor used to discount a cashflow from time t to time 0.

So, v(t2-t1) is the discount factor from time t2-t1 to time 0.

The relationship v(t2) = v(t1)v(t2-t1) is false in general as:
- the LHS discounts from t2 to 0, and
- the RHS discounts from t1 to 0 and from t2-t1 to 0
which don't match up.

What you need on the RHS to make the relationship true is a factor which discounts from time t2 to time t1, multiplied by a factor which discounts from time t1 to time 0 (v(t1)).

Note that the relationship stated is true if we have constant interest rates.

Thanks a lot Mark. So basically what you are saying is the relation is true for constant interest rates.

This is the Qs iii in 5.16 and I will now give a snapshot of whole Qs 5.16 and their sol in the attachment.

Apologies, if I am mistaken but I feel (i) also assumes there is a constant rate and hence answers it true - in that case should not that one be also marked as false?

 

[Mark Mitchell]

So basically what you are saying is the relation is true for constant interest rates.

I am saying that (iii) is not true in general (which is why the answer given is false), but is true in the special case of constant interest rates.

(i) is true in general. It rearranges to:

A(0,t1)A(t1,t2) = A(0,t2)

That is: accumulating money from time 0 to time t1, and then accumulating that value from time t1 to time t2, is the same as accumulating from time 0 to time t2 in one go.

This does not require an assumption of constant interest rates to be true.

Note that v(t2-t1) is not equal to 1/A(t1,t2) in general. v(t2-t1)=1/A(0,t2-t1). Perhaps this is the source of your confusion.

 

[Soumya]

I am saying that (iii) is not true in general (which is why the answer given is false), but is true in the special case of constant interest rates.

(i) is true in general. It rearranges to:

A(0,t1)A(t1,t2) = A(0,t2)

That is: accumulating money from time 0 to time t1, and then accumulating that value from time t1 to time t2, is the same as accumulating from time 0 to time t2 in one go.

This does not require an assumption of constant interest rates to be true.

Note that v(t2-t1) is not equal to 1/A(t1,t2) in general. v(t2-t1)=1/A(0,t2-t1). Perhaps this is the source of your confusion.

Thanks a lot, Mark - I was trapped in assuming constant interest rate and thus was always thinking V(0, t2-t1) should be always equal to V(t1,t2).. perhaps that was the cause of my error.