Annuities certain (arrear and due)
Hey guys, I have a few questions that I've been stuck on for a few days, these questions have no solutions provided so I was wondering if anyone can check them for me. The first 3 questions I have completed with full working however I'm not entirely sure if they are correct, the last 4 I have no clue how to solve :3 Any help would be appreciated!
So first we need to work out the amount of each installment, , thus
Now X purchases the annuity on 1 Jan 1986, so we need to work out the price that X bought the annuity for.
Since he purchased it on 1 Jan 1986 and the annuity expires at the end of 2005, the annuity has a maturity left of 19 years.
But since X pays a tax rate of 0.45 on the interest portion of each payment of R, then we need to split up R each period to find the interest payment and capital payment so we can account for the tax.
To derive an expression for the interest payment and capital payment of R each period we let L be the lump sum borrowed at the start of an arbitrary period, be the number of periods and be the effective interest rate per period. Then for the first payment period, the loan outstanding at the start of the period is L, the interest payment for the first period is where . Then clearly the capital component paid at the end of the first period is since , we can show in general that for payment period k, the interest component paid at the end of the period is and the capital component paid at the end of the period is
Now back to the question: let and
The capital payments of each period starting from 1986 until the end of 2005 when discounted back to 1986 form the cash flow stream:
The interest payments of each period starting from 1986 until the end of 2005 when discounted back to 1986 form the cash flow stream: (Note we take into consideration the tax rate of 45% which is (1-0.45) = 0.55 of the interest payments)
Now we need to combine both of these cash flow streams and form one single expression to evaluate it.
Let us match together the components:
And so on... so let us now group together the expressions with 0.55 as the coefficient:
Now group together the expressions with 0.45 as the coefficient:
Now let the lump sum paid on April 1995 be L, L must satisfy the following equation of value: (Let )
Solving the above yields
First we need to work out the amount of each installment paid in advance from 1 Jan 1985 to 2005 (20 years).
Let each installment be R.
First note where d is the effective annual rate of discount.
Now we need to find on 1 Jan 1990 the total outstanding capital left to be repaid.
This is given by
Now from 1 Jan 1990, if B were to increase the yield to 0.12, then we need to find the new installments per period, these new R is given by:
Now let A's final payment be L, L must satisfy the following equation of value:
Let be the capital component paid in arrear for the nth period and be the interest component paid in arrear for the nth period. Let be the loan outstanding.
From the question
But [This is obvious as represent the loan outstanding at the end of the first period, which is equal to the loan outstanding at the beginning of the period minus the capital paid in the first period]
Furthering this pattern,
So we have:
From  we have
So  becomes
In  we have:
This forms a recurrence relation and we can show that
Now we need to find what is. Note that if we assume the person who bought this decreasing annuity does not pay tax, then we should have:
Now we can solve for
The price that someone would pay for this decreasing annuity if they do not pay tax is given by:
The price is 38252.909
Re: Annuities certain (arrear and due)
Gave 1st one a quick read; quite confused:
Mr.X steps in Jan.1/86 and buys annuity (19 annual payments of $93,678.78)
at 9%, thus for $838,435.82.
So whatever happened before is now history; really, the whole problem could
start with the above; no need to bring in anything prior to Jan.1/86.
Then you've calculated that if Mr.X receives $596,411.93 on Apr.1/95 (so after
receiving 9 of the annual payments), then Mr.X has realised 10% on:
"THE ENTIRE TRANSACTION" : what d'heck does that mean?
Also confusing is your v = 1.08^(-1); how is it possible that the 8% (which was
the rate BEFORE Mr.X stepped in) be used in ANY calculation afterwards?
Can you clarify? If I'm dumb for having missed something: tell me (Wink)