I am working on a problem as follows.
A discounted value $X$ that is due when $t=0.5$ has a present value \$4992. Calculate the value of $X$ when a), the effective annual compound interest rate is 8% b), the effective annual simple interest rate is 8% c), the effective annual compound discount rate is 8% and d), the effective annual simple discount rate is 8%
I am able to calculate a) and b), but am having trouble with the discount rates.
I understand that
$$i= \frac{d}{1-d}$$
so using this I am able to calculate the future value of 4992 in a year. Which is
$$5426.09 = 4992(1+(\frac{.08}{.92}))$$
However, since the problem wants us to find the value of $X$, we have to figure out the value of 4992 half a year later.
So, here is my first question.
Is it conventional to think that when an annual compound interest rate is given by $i$, we are supposed to use $i^{(2)}$ for $t=0.5$? Or do we use $e^{\frac{i}{2}}$?
I do not think the question was not clear about this, but what would be a safe route?
Either way, I tried to calculate the value of $X$ by
$$X = 4992(1+\frac{i^{(2)}}{2})$$
which gives me 5209, but the answer tells me that the value is 5204.
The 5 dollar difference does not seem insignificant and I have a feeling that I am not doing this right.
My other question is for d).
If a simple discount is applied, the present value is calculated as
$$4992 = FV(1-d)$$
where $d=0.08$ so the value of $FV$ must be $5426.09$ which is the exact same value as that of c), because when $t=1$ compound and simple interest (discount) gives the same output.
So I am thinking, the equivalent interest rate must be
$$\frac{5426.09}{4992}-1 = 0.8695$$
which is the same as just doing $\frac{d}{1-d}$. So, did it matter whether it was simple interest or compound interest (discount)?
With this in mind I still tried to calculate the answer by simple interest
$$4992(1+0.5(0.8695))=5426.09$$
which gives me the same result as c), but the answer tells me that it should be $5200$
Can someone help me out?
Best Answer
Your question doesn't really make sense. Since $t=0.5$, annual compounding is equivalent to simple interest since no compounding takes place. Also, the question is saying that $X$ is worth $4,992$ today, so you can answer (a) and (b) as
$$FV=X(1+0.5r)=4992(1+0.5*0.08)=5191.68.$$
(c) and (d) don't really make sense. When you're given a discount rate, you're usually computing present value, but this is already given to you.
It may help you to review the following formulas below the line.
Suppose we are investing an amount worth $X$ today (present value). Then we have with $t$ expressed in years and $r$ the interest rate quoted in the corresponding compounding frequency (otherwise known as the annual percentage rate or APR): $$FV_{\text{simple}}=X(1+rt),$$ $$FV_{\text{m-compounded}}=X\left(1+\frac{r}{m}\right)^{mt},$$ $$FV_{\text{continuously-compounded}}=Xe^{rt}.$$
(Note for the $m$-compounding formula that $mt$ must be an integer, i.e. the time horizon $t$ must satisfy $mt\in\mathbb{N}$. In your question, you cannot substitute $t=\frac{1}{2}$ into this formula since $m=1$ and $1\cdot\frac{1}{2}=\frac{1}{2}\notin\mathbb{N}$. The simple interest formula must be used in this case.)
The effective yield is the APR adjusted for the effects of compounding. It is simply defined as the relative percentage gain (or loss) on your initial investment (i.e. the ratio of the future value to the present value minus $1$) over a period of $t=1$: $$EY=\frac{FV(1)-X}{X}=\frac{FV(1)}{X}-1.$$ Depending on how the APR is quoted, this reduces to (respective to the formulas above): $$EY_{\text{simple}}=r,$$ $$EY_{\text{m-compounded}}=\left(1+\frac{r}{m}\right)^{m}-1,$$ $$EY_{\text{continuously-compounded}}=e^{r}-1.$$