Question

You deposit $10,000 annually into a life insurance fund for the next 12 years, after which time you plan to retire.

a. If the deposits are made at the beginning of the year and earn an interest rate of 8 percent, what will be the amount in the retirement fund at the end of year 12? (Do not round intermediate calculations. Round your answer to 2 decimal places. (e.g., 32.16)) Future value $ ?

b. Instead of a lump sum, you wish to receive annuities for the next 24 years (years 13 through 36). What is the constant annual payment you expect to receive at the beginning of each year if you assume an interest rate of 8 percent during the distribution period? (Do not round intermediate calculations. Round your answer to 2 decimal places. (e.g., 32.16)) Annual payment $ ?

c. Repeat parts (a) and (b) above assuming earning rates of 7 percent and 9 percent during the deposit period and earning rates of 7 percent and 9 percent during the distribution period. (Do not round intermediate calculations. Round your answers to 2 decimal places. (e.g., 32.16))

Answer 1

a ) For this question, we need to know the formula for Future Value of Annuity, when the deposits are being made at the beginning of the year

Future Value, FV = P [ {(1+r)ⁿ - 1}/ r ] * (1+r)

P = Principal amount deposited = $10000

r = Rate of interest = 8% = 0.08

n = Tenure of deposits = 12 years

Substituting values in the formula,

FV = 10000 [ { ( 1 + 0.08)¹² - 1) } /0.08 ] * (1 + 0.08)

FV = $204952.97

b) For this part, we need to find the yearly installments that needs to be paid to distribute the amount collected in the firts part i.e. $204952.97

For this, we will use the formula for calclulating the present value of Annuity from equal future payments

i.e. PV_Annuity = F * [ { 1 - ( 1 + r )^-n } / r ] * (1 + r )

F = denotes the yearly amount to be paid in future = To be determined

r = rate of interest = 8% = 0.08

n = Tenure = 24 years

On Rearranging the above equation, we get

F = [ PV_Annuity / ( 1+r) ] * [ r / { 1 - (1+r)^-n } ]

Substituting the values,

F = [ 204952.97 / ( 1+0.08)] * [ 0.08 / { 1 - (1 + 0.08 )^-24 } ]

F = $18024.09

c ) We just have to change the interest rates in the above formulas

Amount Collected in the future after 12 years

For 7 % =>  FV = P [ {(1+r)ⁿ - 1}/ r ] * (1+r)

FV = 10000 [ { ( 1 + 0.07)¹² - 1) } /0.07 ] * (1 + 0.07)

FV = $191406.43

For 9 % =>  FV = P [ {(1+r)ⁿ - 1}/ r ] * (1+r)

FV = 10000 [ { ( 1 + 0.09)¹² - 1) } /0.09 ] * (1 + 0.09)

FV = $219533.85

Amount paid in yearly installments for 24 years

For 7 % => F = [ PV_Annuity / ( 1+r) ] * [ r / { 1 - (1+r)^-n } ]

F = [ 191406.43 / ( 1+0.07)] * [ 0.07 / { 1 - (1 + 0.07 )^-24 } ]

F = $15596.76

For 9 % => F = [ PV_Annuity / ( 1+r) ] * [ r / { 1 - (1+r)^-n } ]

F = [ 219533.85 / ( 1+0.09)] * [ 0.09 / { 1 - (1 + 0.09 )^-24 } ]

F = $20749.48