Why Compare the Two Models?
A fixed-period model and a continuous exponential model can describe the same overall growth or decay, but they do not automatically make the same predictions. The comparison depends on how their rates are related and on whether the elapsed time ends at a period boundary.
For example, a quantity might increase by a fixed percentage each month, or it might change continuously at a constant relative rate. To compare the models fairly, use the same starting amount, the same time units, and the same elapsed duration. Then check whether the continuous rate constant corresponds to the discrete multiplier.
Set Up the Models
Let \(P_0\) be the starting amount, let \(T>0\) be the length of one discrete period, and let \(g>0\) be the multiplier applied during each period. After \(n\) whole periods, the discrete model predicts \(P_0g^n\). Since \(n\) periods last \(t=nT\) time units, the equivalent continuous rate constant is \(k=\frac{\ln g}{T}\).
The corresponding continuous exponential model is \(P(t)=P_0e^{kt}\). Here, \(k\) has units of inverse time: if \(t\) is measured in years, \(k\) is measured per year. The condition \(g>0\) is needed for \(\ln g\) and hence for this finite real rate constant. A multiplier above \(1\) represents growth; a multiplier between \(0\) and \(1\) represents decay.
The conversion connects the multiplier for one complete period to the rate constant per unit of time. It does not mean that the numerical percentage change per period is itself the continuous rate constant.
Why the Predictions Agree at Period Boundaries
Suppose the continuous model uses the rate constant \(k=\frac{\ln g}{T}\), and consider a time \(t=nT\), where \(n\) is a whole number of periods. Substituting this time into the continuous formula gives:
The last equality follows from the logarithm and exponential identity \(e^{\ln g}=g\), valid for \(g>0\). The final expression is exactly the discrete prediction after \(n\) periods. Thus, when the models have the same initial amount and the continuous rate is converted from the discrete multiplier, their predictions agree at every whole-period boundary.
Worked Example: Matching Predictions After Whole Years
Worked Example: Matching Predictions After Whole Years
A quantity starts at \(200\) units and is multiplied by \(1.25\) every two years. Find the discrete and equivalent continuous predictions after six years.
First identify the period and number of periods:
The discrete prediction is:
Convert the multiplier to a continuous rate constant:
Use the continuous model at \(t=6\) years:
Check: Using the exact converted rate gives \(e^{6\ln(1.25)/2}=e^{3\ln(1.25)}=(1.25)^3\). The continuous and discrete predictions therefore agree at six years.
Same Numerical Rate Does Not Mean Same Model
A common comparison mistake is to use the percentage change per period as the continuous rate constant without converting it. If a quantity increases by \(4\%\) per month, its discrete multiplier is \(1.04\). The equivalent continuous rate is \(\ln(1.04)\) per month, not \(0.04\) per month. Using \(k=0.04\) describes a different continuous model.
Worked Example: Comparing Unmatched Rates
A balance starts at \(\$1{,}000\). Compare a \(4\%\) monthly multiplier with a continuous rate constant of \(0.04\) per month after 12 months. Round amounts to the nearest cent.
For the fixed-period model, there are 12 monthly periods:
For the continuous model with \(k=0.04\) per month:
The difference is approximately \(\$1{,}616.07-\$1{,}601.03=\$15.04\). These outputs differ because the models were assigned different rates, not because the equivalent-rate conversion failed.
Check: The continuous rate equivalent to the monthly multiplier \(1.04\) is \(k=\ln(1.04)\) per month. After 12 months, that model gives \(1000e^{12\ln(1.04)}=1000(1.04)^{12}\), which is the discrete prediction before rounding.
Predictions Between Period Boundaries
A fixed-period multiplier specifies how the amount changes from one period boundary to the next. By itself, it does not state the amount at intermediate times. For instance, knowing a quarterly multiplier tells you the amount at the end of a quarter, but not exactly how the quantity changed during each month within it.
The equivalent continuous model does make a prediction at every time \(t\), including times between period boundaries. That prediction is one possible continuous interpolation of the discrete data. If someone instead assumes a different within-period pattern, such as linear change, the intermediate prediction may differ. The extra assumption must be made explicit.
If one chooses to extend the discrete formula to fractional periods by defining \(P(t)=P_0g^{t/T}\), this extension agrees exactly with the continuous model whose rate is \(k=\frac{\ln g}{T}\). But the fractional exponent is an added modeling choice; it is not information supplied by a rule that specifies only whole-period multipliers.
Worked Example: A Prediction Within a Quarter
A quantity starts at \(800\) units and is multiplied by \(0.8\) every three months. Find the equivalent continuous prediction after one month, and state what the fixed-period model alone can predict at that time.
First convert the quarterly multiplier to a continuous rate constant:
After one month, the continuous model predicts:
The fixed-period model alone does not specify a one-month prediction: its multiplier applies every three months. At the end of the full quarter, however, it predicts:
Check: The continuous model gives \(P(3)=800e^{3\ln(0.8)/3}=800(0.8)=640\) units at the quarter boundary. Its one-month prediction is consistent with that continuous model, but it is not determined by the quarterly multiplier alone.
A Common Mistake and Its Correction
Suppose a fixed-period model increases by \(5\%\) per year. A tempting but incorrect comparison is to set the continuous rate constant to \(0.05\) per year and claim the models match. The discrete multiplier is \(1.05\), so the equivalent continuous rate is \(\ln(1.05)\) per year, approximately \(0.04879016\) per year.
For a starting amount of \(1{,}000\) units after three years, the incorrect comparison gives:
The predictions do not match. The correction is to convert the multiplier:
The error was treating a discrete percentage increase as though it were already a continuous rate constant. Always identify which kind of rate the number represents.
A Consistent Comparison
Before comparing predictions, check the inputs and assumptions. In particular, confirm that the starting amount and time units agree, that the discrete multiplier is positive, and that the continuous rate constant has been converted from that multiplier if the purpose is to match the models.
Record the length \(T\) and the multiplier \(g\) for one period.
For an elapsed time \(t\) at a period boundary, compute \(n=t/T\).
Use \(k=\frac{\ln g}{T}\) to make the continuous model equivalent at period boundaries.
Evaluate \(P_0g^n\) and \(P_0e^{kt}\), and distinguish a true rate mismatch from an intermediate-time assumption.
This comparison is useful whenever data or assumptions are expressed per fixed interval but a continuous model is used to forecast. It makes clear whether two predictions differ because the rates are genuinely different, because the times do not align with period boundaries, or because an assumption has been added about what happens between measurements.
Check Your Understanding
Use the model definitions and rate conversion to answer these questions.
- A multiplier of \(1.08\) applies every two years. What is the equivalent continuous rate constant, including its units?
- Why do the discrete and equivalent continuous models agree after four whole periods?
- A \(3\%\) monthly increase is compared with a continuous rate of \(0.03\) per month. Are these rates equivalent? Explain.
- A discrete model specifies a multiplier every six months. Does that alone determine the amount after two months? Why or why not?
- If a discrete multiplier is \(0.75\) per year, is the equivalent continuous rate positive or negative? Give a reason.
