Simple vs Compound Interest

Under simple interest only the original principal earns interest, so a unit grows linearly; under compound interest the interest earned each period is reinvested and itself earns interest, so a unit grows exponentially. At rate $i$ the two accumulation functions are:

$a_{\text{simple}}(t) = 1 + it$

$a_{\text{compound}}(t) = (1+i)^t$

For $i > 0$ the two agree at $t = 0$ and $t = 1$ ; compound interest gives less than simple interest for $0 < t < 1$ and more for $t > 1$ :

$(1+i)^t \begin{cases} < 1 + it & 0 < t < 1 \\ = 1 + it & t = 0 \text{ or } t = 1 \\ > 1 + it & t > 1 \end{cases}$

Simple interest adds a constant amount each period, so its effective rate in period $n$ , $i_n = \dfrac{i}{1 + i(n-1)}$ , falls over time. Compound interest adds a constant proportion, so its effective rate is $i$ in every period.
With compound interest an Equation of Value gives the same answer at any comparison date; with simple interest the answer depends on the comparison date chosen.
The discount counterparts are simple discount, $a(t)^{-1} = 1 - dt$ , and compound discount, $a(t)^{-1} = (1-d)^t$ (see Effective Discount Rate).

Short and Long Horizons $1{,}000$ is invested at 8% a year. Find the accumulated value after 6 months and after 3 years under simple and under compound interest.
Answer After 6 months ( $t = 0.5$ ):
 $\begin{align*} \text{Simple: } 1000(1 + 0.08 \times 0.5) &= 1040.00 \\ \text{Compound: } 1000(1.08)^{0.5} &= 1039.23 \end{align*}$ 
After 3 years ( $t = 3$ ):
 $\begin{align*} \text{Simple: } 1000(1 + 0.08 \times 3) &= 1240.00 \\ \text{Compound: } 1000(1.08)^{3} &= 1259.71 \end{align*}$ 
Simple interest is ahead for the half year and behind over three years, as the comparison above says.

The Comparison Date Matters Under Simple Interest An insurer owes a claimant $500$ now and $500$ in 4 years, and agrees instead to pay a single amount $P$ at time 6. Interest is 6% simple from the date each payment is made. Find $P$ using a comparison date of time 6, then time 10. Compare with 6% compound interest.
Answer Comparison date 6:
 $\begin{align*} P &= 500(1 + 0.06 \times 6) + 500(1 + 0.06 \times 2) \\ &= 680 + 560 \\ &= 1240.00 \end{align*}$ 
Comparison date 10, where $P$ itself earns 4 years of simple interest:
 $\begin{align*} P(1 + 0.06 \times 4) &= 500(1 + 0.06 \times 10) + 500(1 + 0.06 \times 6) \\ 1.24P &= 800 + 680 \\ P &= 1193.55 \end{align*}$ 
Under 6% compound interest, $P = 500(1.06)^6 + 500(1.06)^2 = 709.26 + 561.80 = 1271.06$ at time 6, and the same $1271.06$ from a comparison date of time 10.