Financial Mathematics

Question 1

How is the discounted present value expressed?

Answer 1

Cv(t) where v(t)=exp(-integral from 0 to t of delta(s) ds)=(1+i)^-t

For time t1 of C due at time t2 the integral spans from t1 to t2 and it is equal to C/A(t1,t2)=Cv(t2)/v(t1)

Question 2

What is the present value of a discrete and continuous cash flow?

Answer 2

Discrete: sum of Cv(t)
Continuous: integral from 0 to T of v(t)p(t) dt where p(t) is the rate of payment

Question 3

What are the formulas for the rate of
Interest
Discount
Force of interest

Answer 3

Interest (i): C0(1+i)^n=C1
Discount (d):
C0=C1(1-d)^n
Force of interest (delta): C0e^(n*delta)=C1

Question 4

The value of a series of payments one unit of time Before the first payment is made (in arrear) is a_n, which can be written as?
How can a:_n be written?

Answer 4

a_n=v+v^2+…+v^n
=(1-v^n)/i
And a:_n is a_n/v=1+v+v^2+…+v^(n-1)

Question 5

How are series of payments written?

Answer 5

S_n=
(1+i)^(n-1)+(1+i)^(n-2)+...+1
=((1+i)^n-1)/i 
s:_n=(1+i)^n+(1+i)^(n-1)+...+(1+i) 
=s_n/v

Question 6

What is the theorem for the Force of Interest?

Answer 6

If delta(t) and A(t0,t) are continuous functions of t for t>=t0 and the principle of consistency holds then for t0

Question 7

Write the formulas for interest payable pthly.

Answer 7

Look in folder.

Question 8

Write the formulas for

a_n (p) and for the 3 others.

Answer 8

Check in folder

Question 9

Write the steps to construct and carry out a payment loan schedule

Answer 9

1. If the annuity is convertible pthly, change i to i^(p)/p
2. Find the amount of pthly repayment, this is: C/a_n^(p)=X and then get this in terms of p (if months divide by 12 etc.)
3. Column headings: n//loan outstanding at start of nth p// interest due at end of nth p// capital repaid at end of nth p
4. Calculating columns: n// initial investment// i (p)/p * previous column // pthly repayment - previous column.

Question 10

What is the proof for the principle of consistency?

Answer 10

Check!!

Question 11

Method for finding the force of interest for a year in between

Answer 11

Check in book

Question 12

Finding the present value for a piece wise force of interest with rate

Answer 12

Check example in folder

Question 13

The expressions for the accumulation A(t,t+h) and ih(t)

Answer 13

Check in folder

Question 14

What is the definition for the net present value?

Answer 14

Check in folder

Question 15

What makes a project profitable?

Answer 15

I’d i1

Question 16

What does purchase price suggest?

Answer 16

When you ensure that the interest that is convertible pthly is divided by this number.

Question 17

How do calculate the discount value.

Answer 17

Sum of cv(t) + integral of v(t) p(t) where p(t) is the rate

Question 18

Value at t1 of C due at t2?

And what is Stoodleys formula?

Answer 18

Cv(t2)/v(t1)
Check in book
Discounted present value is Cv(t) where v(t) is the exponential of the negative interval from 0 to t of the force of interest.

Question 19

Method to find force of interest? (In between year)

Answer 19

Check in book log method