Question

Can you please explain the equation you use, I am having trouble understanding which ones to use and why.

Problem 5: Betty and Bob are interested in borrowing money for 4 years as per compound interest. They have shopped around and have found the following plans: Plan 1: 8% per annum continuously compounded for the first year and 9% per annum continuously compounded for the next 3 years. Plans 2: 10% per annum continuously compounded for the first 18 months and 6% per annum compounded monthly for the remainder of the time. Plan 3: 9% per annum compounded semiannually for the first 9 months and 8% per annum compounded quarterly for the remainder of the time. Plan 4: 8% per compounding period of 8 months. This is to last for the entire 4 years. For each plan algebraically find the effective rate of interest per annum and the equivalent continuous rate of interest per annum. Rank the plans from Best to Worst for Betty and Bob. Your final numerical answers should be correct to 3 places after the decimal point.

Answer 1

The future value of an amount A today at a continuously compounded rate r% after time t = A* e^(rt)

Plan 1

The FV of the amount A after 1 year = A* e^(0.08*1)

After 4 year = A* e^(0.08*1) * e^ (0.09*3) = A*e^ (0.35) = A * 1.419068

So, the effective rate of interest r can be found out using the formula

A* (1+r)⁴ = A * 1.419068

=> 1+r= 1.091442

=> r=9.144%

The equivalent continuous rate of interest can be found using the formula

A* e^(r*4) = A*e^ (0.35)

=> r= 0.35/4 = 0.0875 = 8.750%

Plan 2

The FV of the amount A after 18 months (1.5years) = A* e^(0.1*1.5)

After 4 year =A* e^(0.1*1.5)   * (1+0.06/12)^2.5*12 = A * 1.349354

So, the effective rate of interest r can be found out using the formula

A* (1+r)⁴ = A * 1.349354

=> 1+r= 1.077783

=> r=7.778%

The equivalent continuous rate of interest can be found using the formula

A* e^(r*4) = A*1.349354

=> 4* r= ln(1.349354) = 0.299626

=> r =  0.299626/4

= 0.074907 = 7.490%

Plan 3 (For first 6 months the account will earn compound interest and for next 3 months Simple interest)

The FV of the amount A after 6 months (0.5years) = A* (1+0.09/2)¹  

After 9 months =A* 1.045 * (1+0.09/4) = A * 1.068513

After 4 years (for remaining 13 quarters) = A*1.068513* (1+0.08/4)¹³ =A* 1.382235

So, the effective rate of interest r can be found out using the formula

A* (1+r)⁴ = A * 1.382235

=> 1+r= 1.08429

=> r= 8.429%

The equivalent continuous rate of interest can be found using the formula

A* e^(r*4) = A*1.382235

=> 4* r= ln(1.382235) = 0.323702

=> r =  0.323702/4

= 0.080925 = 8.092%

Plan 4 (There are six periods of 8 months in 4 years)

The FV of the amount A after 48 months (8*6 months= 4years) = A* (1+0.08)⁶  

=A*1.586874

So, the effective rate of interest r can be found out using the formula

A* (1+r)⁴ = A * 1.586874

=> 1+r= 1.122369

=> r = 12.236%

The equivalent continuous rate of interest can be found using the formula

A* e^(r*4) = A*1.586874

=> 4* r= ln(1.586874) = 0.461766

=> r =  0.461766/4

= 0.115442 = 11.544%

So, the best plan (Rank 1) is for Plan 2 because it is the cheapest

Rank 2 goes to plan 3 as the second-best plan

Rank 3 goes to plan 1 as the third-best plan

Rank 4 goes to plan 4 as the costliest plan