✏️ How to solve for a tax bracket

There are two classic types of taxable yield questions:

• #1: given a taxable (corporate) yield (r_c), what is the equivalent tax free yield (r_m)?
  • → for these questions, you multiply r_C by (1-t) to get the equivalent r_m
• #2: given a tax-free yield (r_m), what is the equivalent taxable yield (r_c).
  • → for these questions, you divide r_m by (1-t) to get the equivalent r_C.

For both of those types of problems, you are given the marginal tax rate, t. What happens if you aren’t given the tax rate and need to find it yourself? You can use algebra to solve these.

✏ Consider two bonds: a corporate bond paying a yield of 6% and a municipal bond paying a yield of 5%. For what marginal tax rate would you be indifferent between these two bonds?

✔Often, if you are stuck, you can try the Plug and Chug approach;

Plug and chug: (help)

1. Equation:
   

   $r_c = \frac {r_m}{1-t}$
   $r_c = 6\%, r_m = 5\%, t=?$

2. Plug:🔌
   

   $6\% = \frac {5\%}{1-t}$

3. Solve: 🚂
   

   $1-t = \frac {5\%}{6\%} = 5\%/6\% = 83.33\%$
   Recopy: $1-t = 83.33\%$
   $1= 83.33\% +t$
   $1-83.33\% = 0.1667 = t$

4. Reflect: 🧠
   

   With a tax rate of 16.67%, you will be indifferent between those two bonds.
   You can check your answser by plugging your answer back in the original equation:
   $6\% = \frac{5\%}{1-16.67\%}$

Suppose your marginal tax rate is about 16.67%. Which bond would you prefer?

If tax sheltering is a higher priority because your marginal tax rate is MTR > 16.67%, then you will prefer the tax-free muni yielding 5% - that’s why people in high tax brackets are likely to buy munis.

If tax sheltering is a lower priority, because your marginal tax rate is MTR < 16.67%, then you will prefer the corporate yielding 6%.

If your MTR = 16.67%, then you will be indifferent between the two bonds.