Finding the annual yield of an investment with a changing interest rate

Question

Let's say that there was a variable annual effective interest rate on a capital invested for two years that followed these changes:

i(t) =
2%, t ∈ [0, 0.5),
3%, t ∈ [0.5, 1.5),
0.5%, t ∈ [1.5, 2]

How would I approach trying to find the annual yield of this investment given the structure of this interest rate? I know that taking an average of the interest rates won't work but it will come close. Thanks.

Answer 1

I would apply the $\color{grey}{\texttt{geometric mean}}$ to calculate the annual yield. In my view there is no need to use the fractional part of an interest rate since there is only one payment.

The result is very close to $2.12\%$

$$\sqrt{1.02^{0.5}\cdot 1.03^1\cdot 1.005^{0.5}}-1=1.0211985-1=0.0211985\approx 0.0212=2.12\%$$

Answer 2

Assume the initial investment is $1$.

Let $r$ be the effective annual yield based on the given specs.

Then $r$ must be such that $(1+r)^2$ equals the total return, hence $$ (1+r)^2=\left(1+\frac{.02}{2}\right)\left(1+\frac{.03}{1}\right)\left(1+\frac{.005}{2}\right) $$ which gives $r=.02122512210$.