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!

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So first we need to work out the amount of each installment, $R$ , thus $1000000=Ra_{25:0.08} \implies R = 93678.779$

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, $n$ be the number of periods and $i$ 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 $iL = i(Ra_{n:i}) = iR\left(\frac{1-v^n}{i}\right) = R(1-v^n)$ where $v = (1+i)^{-1}$ . Then clearly the capital component paid at the end of the first period is $Rv^n$ since $R(1-v^n) +Rv^n = R$ , we can show in general that for payment period k, the interest component paid at the end of the period is $R(1-v^{n-k+1})$ and the capital component paid at the end of the period is $Rv^{n-k+1}$

Now back to the question: let $u = (1+0.09)^{-1}$ and $v = (1+0.08)^{-1}$

The capital payments of each period starting from 1986 until the end of 2005 when discounted back to 1986 form the cash flow stream:

$u(Rv^{19}) + u^2(Rv^{18}) + \cdots + u^{19}(Rv)$

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)

$u(0.55R(1-v^{19})) + u^2(0.55R(1-v^{18})) + \cdots + u^{19}(0.55R(1-v))$

Now we need to combine both of these cash flow streams and form one single expression to evaluate it.

Let us match together the $v^{k}$ components:

$u(Rv^{19}) + u(0.55R(1-v^{19})) = 0.45uRv^{19} + 0.55uR$

$u^2(Rv^{18}) + u^2(0.55R(1-v^{18})) = 0.45u^2Rv^{18} +0.55u^2R$

And so on... so let us now group together the expressions with 0.55 as the coefficient:

$0.55uR + 0.55u^2R + \cdots + 0.55u^{19}R = 0.55R(u+u^2+ \cdots + u^{19}) \cdots [1]$

Now group together the expressions with 0.45 as the coefficient:

$0.45uRv^{19} + 0.45u^2Rv^{18} + \cdots + 0.45u^{19}Rv = 0.45R(uv^{19} + u^2v^{18} + \cdots + u^{19}v) \cdots [2]$

$[1] = 0.55Ra_{19:0.09} = 0.55(93678.779)\left(\frac{1-(1.09)^{-19}}{0.09}\right) = 461139.703695$

$[2] = 0.45R\left(uv^{19}\left(\frac{1-(uv^{-1})^{19}}{1-uv^{-1}}\right)\right) = 156912.68$

Thus $[1]+[2] = 618052.3837$

Now let the lump sum paid on April 1995 be L, L must satisfy the following equation of value: (Let $q = 1.1^{-1}$ )

$\left[0.55R(q+q^2+\cdots+q^9)\right] + \left[0.45R(qv^{19}+q^2v^{18} + \cdots + q^9v^{11}\right] + Lq^{9.25} = 618052.3837$

$0.55R\left(\frac{1-1.1^{-9}}{0.1}\right) + 0.45R\left[qv^{19}\left(\frac{1-(qv^{-1})^9}{1-qv^{-1}}\right)\right] + Lq^{9.25} = 618052.3837$

Solving the above yields $L = 596411.93$

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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 $(1+0.1) = (1-d)^{-1}$ where d is the effective annual rate of discount.

$d = \frac{1}{11}$

$10000=R\ddot{a}_{20:0.1} \implies 10000=R\left(\frac{1-1.1^{-20}}{d}\right) \implies R = 1067.81477$

Now we need to find on 1 Jan 1990 the total outstanding capital left to be repaid.

This is given by $R\ddot{a}_{15:0.1} = R\left(\frac{1-1.1^{-15}}{d}\right) = 8934.07$

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:

(Note $(1+0.12) = (1-d)^{-1} \implies d = \frac{3}{28}$ )

$8934.07 = R\ddot{a}_{15:0.12} \implies 8934.07 = R\left(\frac{1-1.12^{-15}}{\frac{3}{28}}\right) \implies R = 1171.19499$

Now let A's final payment be L, L must satisfy the following equation of value:

$R\ddot{a}_{5:0.12} + L(1.12)^{-5}= 8934.07$

$\implies R\left(\frac{1-1.12^{-5}}{\frac{3}{28}}\right) + L(1.12)^{-5} = 8934.07$

$\implies L = 7411.61448$

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Let $C_n$ be the capital component paid in arrear for the nth period and $I_n$ be the interest component paid in arrear for the nth period. Let $L_0$ be the loan outstanding.

From the question $C_1+(1-0.3)I_1 = 5000$

But $I_1 = 0.1L_0 \implies C_1 +0.07L_0 = 5000$

Likewise, $C_2+0.7I_2=5000$

But $I_2 =0.1L_1 = 0.1(L_0 - C_1)$ [This is obvious as $L_1$ 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]

So $C_2+0.07(L_0-C_1) = 5000$

Furthering this pattern, $C_3+0.7I_3 = 5000$

But $I_3 = 0.1L_2 = 0.1(L_1-C_2) = 0.1(L_0-C_1-C_2)$

Hence $C_3+0.07(L_0-C_1-C_2)=5000$

So we have:

$C_1 +0.07L_0 = 5000 \cdots [1]$

$C_2+0.07(L_0-C_1) = 5000 \cdots [2]$

$C_3+0.07(L_0-C_1-C_2)=5000 \cdots [3]$

From [1] we have $C_1 = 5000 - 0.07L_0$

So [2] becomes $C_2+0.07(L_0 - 5000 +0.07L_0) = 5000 \implies C_2 + 0.07L_0 - 0.07(5000) + 0.07^2L_0 = 5000 \implies C_2 = 5000+0.07(5000) - (0.07+0.07^2)L_0 \implies C_2 = 1.07(5000-0.07L_0) = 1.07C_1$

In [3] we have:

$C_3 = 5000- 0.07(L_0-C_1-C_2) \implies C_3 = 0.07C_1+0.07(1.07)C_1+C_1 \implies C_3 = 1.07C_1 + 0.07(1.07)C_1 = 1.07^2C_1$

This forms a recurrence relation and we can show that $C_n = 1.07^{n-1}C_1$

Now we need to find what $L_0$ is. Note that if we assume the person who bought this decreasing annuity does not pay tax, then we should have:

$\sum_{i=1}^{10} \left(C_i+I_i\right)v^i = L_0$ where $v = 1.1^{-1}$

$\implies \sum_{i=1}^{10} \left(C_i + \frac{1}{0.7}(5000-C_i)\right)v^i=L_0 \implies \sum_{i=1}^{10} \left(1.07^{i-1}C_1 + \frac{1}{0.7}(5000-1.07^{i-1}C_1)\right)v^i=L_0 \implies \sum_{i=1}^{10} \left(1.07^{i-1}(5000-0.07L_0) + \frac{1}{0.7}(5000-1.07^{i-1}(5000-0.07L_0))\right)v^i=L_0$

Now we can solve for $L_0, L_0 = 35117.0977$

The price that someone would pay for this decreasing annuity if they do not pay tax is given by:

$\sum_{i=1}^{10} \left(1.07^{i-1}(5000-0.07L_0) + \frac{1}{0.7}(5000-1.07^{i-1}(5000-0.07L_0))\right)v^i$ where $v^i = 1.08^{-i}$

The price is 38252.909

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