To determine which investment is more convenient based on the REA (Rate of Equivalent Annuity) of 8%, we need to compare the present value of the future cash flows for both options using the given rates and durations.
Let's calculate the equivalent annuities for both options and compare them:
Option (a): Investment: 10000 euros Deferred Annual Return: 4000 euros for 4 years REA: 8%
The equivalent annuity formula is: Equivalent Annuity=Annual Return(1−(1+REA)−Number of Years)/REAEquivalent Annuity=(1−(1+REA)−Number of Years)/REAAnnual Return
Plugging in the values: Equivalent Annuity (Option a)=4000(1−(1+0.08)−4)/0.08≈3176.65 eurosEquivalent Annuity (Option a)=(1−(1+0.08)−4)/0.084000≈3176.65 euros
Option (b): Investment: 20000 euros Annual Interest Rate: 10% Duration: 3 years REA: 8%
For Option (b), we need to calculate the annual interest payments and the principal repayment: Interest Payment (Year 1)=20000×0.10=2000 eurosInterest Payment (Year 1)=20000×0.10=2000 euros Interest Payment (Year 2)=20000×0.10=2000 eurosInterest Payment (Year 2)=20000×0.10=2000 euros Interest Payment (Year 3)=20000×0.10=2000 eurosInterest Payment (Year 3)=20000×0.10=2000 euros Principal Repayment (Year 3)=20000 eurosPrincipal Repayment (Year 3)=20000 euros
Now, we can calculate the equivalent annuity using the same formula as above:
[math]Equivalent Annuity (Option b)= (1−(1+REA) −Number of Years )/REA Interest Payment (Year 1)+Interest Payment (Year 2)+Interest Payment (Year 3)+Principal Repayment (Year 3) Plugging in the values: Equivalent Annuity (Option b) = 2000 + 2000 + 2000 + 20000 ( 1 − ( 1 + 0.08 ) − 3 ) / 0.08 ≈ 6875.03 euros Equivalent Annuity (Option b)= (1−(1+0.08) −3 )/0.08 2000+2000+2000+20000 ≈6875.03 euros[/math]
Comparing the two equivalent annuities, we see that Option (a) has a lower equivalent annuity (3176.65 euros) compared to Option (b) (6875.03 euros). Therefore, based on the given REA of 8%, Option (a) is the more convenient choice.