I am new in finance math, and I am working on this problem. I was wondering if someone could help me solve it.
A loan of $10,000$ is to be repaid with annual payments, at the end of each year, for the next $21$ years. For the first $7$ years the payments are $X$ per year, the second $7$ years they are $2X$ per year, and for the third $7$ years they are $3X$ per year.
a) Find $X$ if $i^{(2)} =10% $
b) Find $X$ if $i^{(2)} = 8%$ for the first 10 years and $i^{(2)} = 12%$ afterwards.
My attempt was
a) My timeline looked like $7$ Xs starting from $t=1$ to $7$, then $2X$ from $8$ to $14$ and then $3X$ from $15$ to $21$.
$$PV = 10000 $$
$$PV = X a_{\overline{7} \rceil.05} + (1.05)^{-7} *2X a_{\overline{14} \rceil.05} + (1.05)^{-14} * 3X a_{\overline{21} \rceil.05} $$
$$10000 = \frac{X[1-(1.05)^{-7}]}{\frac{0.05}{1.05}} + (1.05)^{-7}\frac{2X[1-(1.05)^{-14}]}{\frac{0.05}{1.05}} + (1.05)^{-14}\frac{3X[1-(1.05)^{-21}]}{\frac{0.05}{1.05}} $$
When I calculate it I got $243.70$
I wasn't sure if $(1.05)^{-7}$ for example was needed since I am starting at year 8 and bring it to time 0 , where the present value of 2X is 7 payments. Same for $(1.05)^{-14}$
I appreciate your help
The clearest way to proceed is to first calculate the effective periodic rate--in this case, the rate is annual. Since you are given a nominal rate of interest $i^{(2)}$, we must convert this to the effective annual rate $i$ via the formula $$\left(1 + \frac{i^{(m)}}{m}\right)^m = 1 + i.$$ In the first question, the interest rate is constant throughout the repayment term, which is simpler; we have $$i = \left(1 + \frac{0.10}{2}\right)^2 - 1 = 0.1025.$$ Then our equation of value is $$\require{enclose} 10000 = X a_{\enclose{actuarial}{7}i} + 2X v^7 a_{\enclose{actuarial}{7}i} + 3X v^{14} a_{\enclose{actuarial}{7}i}. \tag{1}$$ This can also be written $$10000 = X a_{\enclose{actuarial}{21}i} + X v^7 a_{\enclose{actuarial}{14}i} + X v^{14} a_{\enclose{actuarial}{7}i}. \tag{2}$$ Your equation is not correct because it is computing the present value of cash flow in which the first $7$ payments are $X$, the next $7$ payments are $2X$, the next $7$ payments are $5X$, and then another $14$ payments are $3X$. This is because the term of the second deferred annuity is $14$ years, so you are paying at times $t = 8, 9, \ldots 21$ the amount $2X$, and then starting from $t = 15$, you are paying another $3X$ from the third annuity.
To avoid this confusion, it is helpful to write out the cash flow: $$PV = (Xv + Xv^2 + \cdots + Xv^7) + (2Xv^8 + 2Xv^9 + \cdots + 2Xv^{14}) + (3Xv^{15} + 3Xv^{16} + \cdots 3Xv^{21}).$$ And now if you factor things out, we get $$PV = X(v + v^2 + \cdots + v^7) + 2Xv^7(v + v^2 + \cdots + v^7) + 3Xv^{14}(v + v^2 + \cdots + v^7).$$ And we can factor this even further to get $$PV = X(1 + 2v^7 + 3v^{14})a_{\enclose{actuarial}{7}i}. \tag{3}$$ We can also factor Equation $(1)$ in the same way.
So where does Equation $(2)$ come from? Well, we could have written the cash flow instead as $$PV = X(v + v^2 + \cdots + v^{21}) + X(v^8 + v^9 + \cdots + v^{21}) + X(v^{15} + v^{16} + \cdots + v^{21}).$$ You can see that the first annuity of $X$ has a $21$-year term, and the second has a $14$-year term deferred by $7$ years, so that these contributions of $X$ will contribute the proper amount to the final $7$ years of the overall repayment.
How do we do the second part? In this case, we must calculate two effective rates of interest. One approach is to calculate the effective annual rate for the first $10$-year period, call that $i = (1 + 0.08/2)^2 - 1 = 0.0816$, and then the effective annual rate for the remainder of the term, call that $j = (1 + 0.12/2)^2 - 1 = 0.1236$. Then the equation of value becomes $$10000 = Xa_{\enclose{actuarial}{7}i} + 2Xv_i^7 (a_{\enclose{actuarial}{3}i} + v_i^3 a_{\enclose{actuarial}{4}j}) + 3Xv_i^{10} v_j^4 a_{\enclose{actuarial}{7}j}, \tag{4}$$ where $v_i = 1/(1+i)$ and $v_j = 1/(1+j)$ are the effective annual present value discount factors for each interest period. Note we must split up the second $7$-year payment period into two deferred annuities due to the change in interest rate after time $t = 10$. Also note that the total present value discount factor of the last $7$ payments is neither $v_i^{14}$ nor $v_j^{14}$ but rather $v_i^{10} v_j^4$ because for the first $10$ years of deferral, we are under rate $i$, and for the next $4$ years of deferral, we are under rate $j$.
This leads us to ask whether there is an analogue of Equation $(2)$ for this question. The answer is yes, and as usual, to understand how to get it, we write out the cash flow: $$\begin{align} PV &= X(v_i + v_i^2 + \cdots v_i^7) + 2X(v_i^8 + v_i^9 + v_i^{10} + v_i^{10}v_j + v_i^{10}v_j^2 + v_i^{10}v_j^3 + v_i^{10}v_j^4) \\ &\quad + 3X(v_i^{10}v_j^5 + \cdots v_i^{10}v_j^{11}) \\ &= X(v_i + v_i^2 + \cdots v_i^{10} + v_i^{10}v_j + \cdots + v_i^{10}v_j^{11}) \\ &\quad + X(v_i^8 + v_i^9 + v_i^{10} + v_i^{10}v_j + \cdots v_i^{10}v_j^{11}) \\ &\quad + X(v_i^{10}v_j^5 + \cdots v_i^{10}v_j^{11}). \end{align} \tag{5}$$ This gives the equation of value $$\begin{align}10000 &= X(a_{\enclose{actuarial}{10}i} + v_i^{10} a_{\enclose{actuarial}{11}j}) + X(v_i^7 a_{\enclose{actuarial}{3}i} + v_i^{10} a_{\enclose{actuarial}{11}j}) + X v_i^{10} v_j^4 a_{\enclose{actuarial}{7}j}.\end{align}\tag{6} $$