91Ó°ÊÓ

Bond obligation = 18,000,000

Interest rate at the time of issue = 10%

Interest rate after decline = 8.5%

Time = 20 years

Time remaining of bonds = 10 years

Call premium on old issue =9%

Underwriting cost of new issue =$530,000

Underwriting cost on old issue = $380,000

Discount rate = 8%

Tax rate = 35%

The present value of interest savings is $1,177,619.

$$\begin{gathered} \mathbf{\text{Interest savings}}\mathbf{=}\mathbf{(}\mathbf{\text{Interest on old bonds}}\mathbf{-}\mathbf{\text{Interest on new bonds}}\mathbf{)}\mathbf{(}\mathbf{1}\mathbf{-}\mathbf{\text{Tax rate}}\mathbf{)} \\ \mathbf{=}\mathbf{(}\left(\$18,000,000×10\%\right)\mathbf{-}\left(\$18,000,000×8.5\%\right)\mathbf{)}\mathbf{×}\mathbf{(}\mathbf{1}\mathbf{-}\mathbf{35}\mathbf{\%}\mathbf{)} \\ \mathbf{=}\mathbf{(}\mathbf{\$}\mathbf{1}\mathbf{,}\mathbf{800}\mathbf{,}\mathbf{000}\mathbf{-}\mathbf{\$}\mathbf{1}\mathbf{,}\mathbf{530}\mathbf{,}\mathbf{000}\mathbf{)}\mathbf{×}\mathbf{(}\mathbf{1}\mathbf{-}\mathbf{35}\mathbf{\%}\mathbf{)} \\ \mathbf{=}\mathbf{\$}\mathbf{175}\mathbf{,}\mathbf{500} \end{gathered}$$

$$\begin{gathered} \mathbf{\text{Present value of interest savings}}\mathbf{=}\mathbf{\text{Interest savings}}\mathbf{×}{\mathbf{\text{PVAF}}}_{\left(i=8\%,n=10years\right)} \\ \mathbf{=}\mathbf{\$}\mathbf{175}\mathbf{,}\mathbf{500}\mathbf{×}\mathbf{6}\mathbf{.}\mathbf{71008} \\ \mathbf{=}\mathbf{\$}\mathbf{1}\mathbf{,}\mathbf{177}\mathbf{,}\mathbf{619} \end{gathered}$$

The present value of gain on the underwriting cost is $21,878.

$$\begin{gathered} \mathbf{\text{Unamortized underwriting cost}}\mathbf{=}\mathbf{\text{Original amount}}\mathbf{-}\mathbf{\text{Amount written off}} \\ \mathbf{=}\mathbf{\$}\mathbf{380}\mathbf{,}\mathbf{000}\mathbf{-}\frac{\mathbf{\$}\mathbf{380}\mathbf{,}\mathbf{000}}{\mathbf{20}}\mathbf{×}\mathbf{10} \\ \mathbf{=}\mathbf{\$}\mathbf{190}\mathbf{,}\mathbf{000} \end{gathered}$$

role="math" localid="1649771203961" $$\begin{gathered} \mathbf{\text{PV of deferred future underwriting cost}}\mathbf{=}\frac{\mathbf{\text{Future underwriting cost}}}{\mathbf{\text{Time remaining}}}\mathbf{×}{\mathbf{\text{PVAF}}}_{(i=8\%,n=10years)} \\ \mathbf{=}\frac{\mathbf{\$}\mathbf{190}\mathbf{,}\mathbf{000}}{\mathbf{10}}\mathbf{×}\mathbf{6}\mathbf{.}\mathbf{71008} \\ \mathbf{=}\mathbf{\$}\mathbf{19}\mathbf{,}\mathbf{000}\mathbf{×}\mathbf{6}\mathbf{.}\mathbf{71008} \\ \mathbf{=}\mathbf{\$}\mathbf{127}\mathbf{,}\mathbf{492} \end{gathered}$$

role="math" localid="1649771258403" $$\begin{gathered} \mathbf{\text{Gain in old underwriting cost}}\mathbf{=} \\ \mathbf{\text{Unamortized amount}}\mathbf{-}\mathbf{\text{PV of deferred future underwriting}} \\ \mathbf{=}\mathbf{\$}\mathbf{190}\mathbf{,}\mathbf{000}\mathbf{-}\mathbf{\$}\mathbf{127}\mathbf{,}\mathbf{492} \\ \mathbf{=}\mathbf{\$}\mathbf{62}\mathbf{,}\mathbf{508} \end{gathered}$$

$$\begin{gathered} \mathbf{\text{After tax value of old underwriting cost}}\mathbf{=} \\ \mathbf{\text{Gain in old underwriting cost}}\mathbf{×}\mathbf{\text{Tax rate}} \\ \mathbf{=}\mathbf{\$}\mathbf{62}\mathbf{,}\mathbf{508}\mathbf{×}\mathbf{35}\mathbf{\%} \\ \mathbf{=}\mathbf{\$}\mathbf{21}\mathbf{,}\mathbf{878} \end{gathered}$$

The present value of inflows is $1,199,497.

$$\begin{gathered} \mathbf{\text{Present value of inflows}}\mathbf{=} \\ \mathbf{\text{PV of interest savings}}\mathbf{+}\mathbf{\text{Present value of gain in old underwriting cost}} \\ \mathbf{=}\mathbf{\$}\mathbf{1}\mathbf{,}\mathbf{177}\mathbf{,}\mathbf{619}\mathbf{+}\mathbf{\$}\mathbf{21}\mathbf{,}\mathbf{878} \\ \mathbf{=}\mathbf{\$}\mathbf{1}\mathbf{,}\mathbf{199}\mathbf{,}\mathbf{497} \end{gathered}$$