Question

A company can borrow $180 000 for 15 years. They can amortize the debt at j1 = 10%, or they can pay interest on the loan at j1 = 9% and set up a sinking fund at j1 = 7% to repay the loan. Which plan is cheaper and by how much per annum?

Answer 1

Alternative 1: Amortizing the loan:

Yearly payment required= $ 23,665.28 calculated as follows:

15 10 B C D 1 Equated Yearly Installments (1) 2 Assuming payments at the end of each period 3 Equal yearly instalment is calculated using the formula 1=[P*r*(1+r)^n]/[(1+r)^n)-1 4 Where P= Principal (loan amount), 5 n= Number of installments. 6 r= Rate of interest per year in decimals 7 All amounts in $ 8 Given that 9 Principal (loan amount) (P)= 180000 No. of years (N)= 10 Rate of interest (% p.a)=R= 11 12 Calculations: 13 Formula Cell reference Value 14 Interest rate per year in decimals (r = R/(100*12) D10/(100*12) 0.1 15 No. of years (n)= N*12 G9*12 16 (1+r)= 1+G14 1.1 17 (1+r)^n= G16469 4.17724817 [(1+r)^n]-1= G17-1 3.17724817 19 P*r*(1+r)^n= D9*G14*G17 75190.467 20 Equated Yearly Insallments (1)= [P*r*(1+r)^n]/[(1+r)^n)-1 G19/G18 23665.2798 21 Rounded to $ 23,665.28 15 18

Alternative 2: Pay interest at 9% and set up sinking fund at 7% interest.

Yearly interest= Principal*Rate

Loan amount (Principal)= $180,000. Interest rate= 9% (Compounded yearly)

Therefore, yearly interest= $180,000*9%= $16,200.

Yearly payment required towards sinking fund= $7,163.03 calculated as follows:

8 9 А. B C D 3 Amount of periodical payments is calculated using the formula PMT= (PV*r)/(1-(1+r)^-n] 4 Where PMT= Periodical payments required, 5 PV= Present value of future sum which is FV/(1+r)^n 6 n= Number of payments. 7 r= Rate of interest per period in decimals All amounts in $ Given that 10 Sinking fund required FV= $180,000 No. of years (N) 11 Rate of interest (% p.a)=R= 7 No. of payments a year (t) 12 13 Calculations: 14 Formula Cell reference Value 15 Interest rate per period in decimals (r )= R/(100*t) D11/(100*G11) 0.07 16 No. of periods (n)= Nät O'G11*G11 17 (1+r)= 1+G15 1.07 18 (1+r)^n= G174616 2.759031541 19 Present Value of future amount (PV)= FV/(1+r)^n= D10/G18 65240.28354 20 (1+r)^-n= G171-G16 0.36244602 21 [1-(1+r)^-n]= 1-G20 0.63755398 22 PV*r= G19*G15 4566.819847 23 Periodical contribution required (PMT)= (PV*r)/(1-(1+r)^-n] G22/G21 7163.032446 24 Rounded to $7,163.03 15

Total yearly payment under alternative 2= Yearly interest at 9% interest + Yearly payment to sinking fund

=$16,200 + $7,163.03 = $23,363.03

This shows that plan 2 (Pay interest at 9% and set up sinking fund at 7% interest) is cheaper by

$23,665.28 - $23,363.03 = $302.25