Question

You wish to borrow $1,000 for 6 years to start a business. You approach different lenders who have the following offers. Which is the best lender to borrow from? All percentage rates are rounded to 6 decimal places.

Select one:

a. 11.494938% pa APR compounding quarterly where there are exactly 4 quarters per year.

b. 11.660105% pa APR compounding semi-annually where there are exactly 2 semi-annual periods per year.

c. 2.873734% as an effective quarterly rate where there are exactly 4 quarters per year.

d. 0.028096% as an effective daily rate where there are exactly 365 days per year.

e. I'll lend you the $1,000 now if you pay me a single payment of $1,973.82 in 6 years, with no other payments in between.

Answer 1

a. 11.494938% per annum APR compounding quarterly:

Effective Quarterly interest rate =(11.494938/4)%=2.873734%

b.11.660105% pa APR compounding semi-annually

Effective Semi annual rate =(11.660105/2)%=5.830053%

Effective Quarterly Rate=R

(1+R)^2=1+0.05830053=1.05830053

1+R=1.05830053^(1/2)=1.028737345

Effective Quarterly Rate =0.0287373=2.873734%

c. 2.873734% as an effective quarterly rate

d. 0.028096% as an effective daily rate

Effective annual Rate =r

(1+r)=(1+0.00028096)^365=1.107977184

Effective Annual Rate =0.107977184

Effective Quarterly Rate=R

(1+R)^4=1.107977184

1+R=1.107977184^(1/4)=1.025965376

Effective Quarterly Rate=0.025965376=2.596538%

e. Number of Quarters =6*4=24

Effective quarterly Rate =R

1000=1973.82/((1+R)^24)

(1+R)^24=1973.82/1000=1.97382

1+R=1.97382^(1/24)=1.028737286

R=0.028737286=2.873729%

Best lender to borrow from: d. 0.028096% as an effective daily rate

Effective Quarterly Rate=0.025965376=2.596538%

This is the lowest rate