Loan calculation; what went wrong?

Question

I am attempting a question given as follows

A loan is payable over 20 years by level installments of $\$1000$ per annum made annually in arrear. Interest is charges at $5\%$ per annum effective for the first 10 years and increasing to $7\%$ for the remaining term. Show that the amount of the original loan is about $\$12,033.56$.

My solution

I took this as an immediate annuity $a_{\bar{n}|}$ of 10 years (the term with $5\%$ interest) and deferred annuity of $_m|a_{\bar{n}|}$ of 10 years for 10 years(at $7\%$ interest). Namely, the first 10 years is simply an immediate annuity, then since the rate changes, we use the new rate for another 10 years as an immediate annuity but "deferred" for 10 years(which used $5\%$ instead).

So basically, $a_{\bar{10}|}=\frac{u(u^{10}-1)}{u-1}$ where $u=\frac{1}{1.05}$ and $_{10}|a'_{\bar{10}|}=u'^{10}a'_{\bar{10}|}=u'^{10}\frac{u'(u'^{10}-1)}{u'-1}$ where $u'=\frac{1}{1.07}$. And so my answer is,

$$1000a_{\bar{10}|}+1000_{10}|a'_{\bar{10}|}=1000 \bigg\{\frac{u(u^{10}-1)}{u-1}+u'^{10}\frac{u'(u'^{10}-1)}{u'-1} \bigg\}$$

(I've used the formula for immediate annuity and deferred annuity here for the results).

However, direct calculation by inserting values to the above expression yields $11292.16763$ which is far off from the value I must have obtained. What have I done wrong? Is there a problem with my theory and reasoning up there?

I admit I am still not completely comfortable with the topic so please kindly explain and point out if there are any faults.

Thank you so much in advance

Answer 1

Your solution doesn't work because during the deferral period, the interest rate is $5\%$, not $7\%$. Therefore, the correct present value of the loan is $$PV = 1000\left( a_{\overline{10}\rceil .05} + v^{10} a_{\overline{10}\rceil .07} \right),$$ where $v = (1.05)^{-1}$ is the present value discount factor for the initial 10-year repayment period.

Answer 2

I´m not sure what you have calculated because I´m not very familiar with the actuarial notation. The original loan ($\color{red}x$) has to be compounded 20 times. 10 times with $q=1.05$ for the first ten years and ten times with $q=1.07$ for the second ten years :

$\color{red}x\cdot 1.05^{10}\cdot 1.07^{10}$

This is the future value of the loan. This has to be equal to the future value of the installments.

The future value of the first 10 payments is $$1000\cdot \frac{1-1.05^{10}}{1-1.05}\cdot 1.07^{10}$$ You do not multiply it by $1.07$ because the installments are made in arrear.

The future value of the second 10 payments is

$$1000\cdot \frac{1-1.07^{10}}{1-1.07}$$

Therefore the equation is

$$\color{red}x\cdot1.05^{10}\cdot 1.07^{10}=1000\cdot \frac{1-1.05^{10}}{1-1.05}\cdot 1.07^{10}+ 1000\cdot \frac{1-1.07^{10}}{1-1.07}$$