Independent solution

How to solve this Equivalent Interest Measures question

Setup

Setup

Let δ denote the constant annual force. The nominal 4% account earns 2% every half-year, so its one-year accumulation factor is the square of 1.02.

i(2)=0.04i^{(2)}=0.04

Model

Model

Equality at the stated comparison date requires equal accumulation factors. Because both accounts start with the same principal and run for the same time, it is enough to set exp(δ) equal to the nominal-rate account’s annual factor.

eδ=(1+0.04/2)2e^\delta=(1+0.04/2)^2

Compute

Compute

Taking natural logarithms gives δ = 2 ln(1.02) = 0.039605. Retaining the extra digits before rounding avoids confusing the force with either the 4% nominal quote or the 4.04% annual effective rate.

δ=2ln⁡(1.02)=0.039605\delta=2\ln(1.02)=0.039605

Answer

Answer

The calculation gives 0.0396 for equivalent interest measures, matching published choice C.

δ=0.0396(C)\boxed{\delta=0.0396\quad\text{(C)}}
One unit grows to 1.02 at half a year and 1.0404 at one year, either through semiannual compounding or the equivalent constant force.
The two routes use the same dates and accumulation factors. Compare other rate conventions in the free calculator.