At an annual effective interest rate of i, i > 0 all the following are equal:
i. the present value of 10,000 at the end of 6 years
ii. the sum of the present values of 6000 at the end of year t and 56,000 at the end of year 2t
iii. 5000 immediately
Calculate $v^t$.
Note the following equations:
$FV= PV(1+i)^t\tag{1}$
or we can express this in terms of present value:
$PV=\cfrac{FV}{(1+i)^t}=FV \cdot v^t\tag{2}$
Where $v^t$ is known as the present value factor. From the three equations, I wrote the following:
$10,000 v^6=6,000v^t + 56,000 v^{2t}=5000\tag{3}$
So we have a systems of equations and I broke it down to solve for $v$:
$10,000 v^6 = 5000 \implies v=\left(\cfrac{5}{10}\right)^{1/6}=0.891\tag{4}$
Now I used the second part of the equation:
$6,000v^t + 56,000 v^{2t}=5000$
$v^t ( 6 + 56 v^2) = 5 \implies v^t= \cfrac{5}{6+56 v^2}=\cfrac{5}{6+56 (0.891)^2} \approx 0.1$
So I got 0.1 but the answer is 0.25. Can someone please tell me what I'm doing wrong? Thank you in advance.
Best Answer
Let $x = v^t$. From $56 v^{2t} + 6v^t = 5$, we have $56x^2 + 6x - 5 = 0$.
If you recall the quadratic formula, you should get one positive root, which is exactly what you need.