Question

A. Derek plans to retire on his 65th birthday. However, he plans to work part-time until he turns 74.00. During these years of part-time work, he will neither make deposits to nor take withdrawals from his retirement account. Exactly one year after the day he turns 74.0 when he fully retires, he will begin to make annual withdrawals of $138,548.00 from his retirement account until he turns 94.00. He he will make contributions to his retirement account from his 26th birthday to his 65th birthday. To reach his goal, what must the contributions be? Assume a 6.00% interest rate.

B. Derek plans to retire on his 65th birthday. However, he plans to work part-time until he turns 75.00. During these years of part-time work, he will neither make deposits to nor take withdrawals from his retirement account. Exactly one year after the day he turns 75.0 when he fully retires, he will begin to make annual withdrawals of $128,654.00 from his retirement account until he turns 87.00. After this final withdrawal, he wants $1.76 million remaining in his account. He he will make contributions to his retirement account from his 26th birthday to his 65th birthday. To reach his goal, what must the contributions be? Assume a 7.00% interest rate.

C. A bank offers 8.00% on savings accounts. What is the effective annual rate if interest is compounded semi-annually?

Answer 1

Part A

Withdrawals span over 20 years. Amount required on 74th birthday, in order to enable these withdrawals is the present value of annuity as on that date. It is ascertained at $1,589,134.64 using PV function of Excel as follows:

C6 =PV(C1,C2,C3,C4,C5)*-1 6% d B Rate Nper Pmt FV Type PV 20 $ 1 Interest rate 2 Number of withdrawals 3 Amount of withdrawal 4 Future value 5 Timing of withdrawal 6 Present value 138,548 $1,589,134.64 1

Value of this amount as on the 65th birthday is $940,606.35 as follows:

In order to accumulate this amount, contributions to be made annually from 26th birthday to 65th birthday is $6,077.76 as follows:

B C D A 1 Present Value 2 3 Present value of an amount is calculated using the formula V=F/(1+r)^n 4 Where F= Amount after the specified period (Future Value), 5 n= Number of compounbding periods. 6 r=Rate of interest per period in decimals 7 All amounts in $ 8 Given that 9 Future value F= 1589134.64 No. of years (N)= 10 Discount Rate (% p.a)=R= 6 No. of times compounded a year (t) 11 12 Calculations: 13 Formula Cell reference Value 14 Discount rate per period in decimals (r )= R/(100*t) D10/(100*G10) 15 No. of periods (n)= N*t G9*610 16 (1+r)= 1+G14 1.06 17 (1+r)^n= G164G15 1.68947896 18 Present value = F/(1+r)^n D9/617 940606.355 19 Rounded to $940,606.35 0.06 20

C60 6% A 1 Interest rate 2 Deposits 3 Present value 4 Future value 5 Timing of Deposits 6 Amount of yearly deposits =PMT(C1,C2,C3,C4,C5)*-1 B Rate Nper 40 Pv FV $ 940,606.35 Type Pmt $ 6,077.76

Part B:

Withdrawals will span over 12 years (from 76 years to 87 years). Value of these withdrawals, an on 75th birthday is $1,803,319.41 as follows:

C6 =PV(C1,C2,C3,C4,C5)*-1 fo в Rate Nper Pmt А 1 Interest rate 2 Deposits 3 Amount of withdrawal 4 Future value 5 Timing of withdrawal 6 Present value 7 7% 12 128,654 1,760,000 $ $ FV Type Pv $1,803,319.41 |

Value of this amount on 65th birthday is $916,716.15 as follows:

G 7 A B C D E 1 1 Present Value 2 3 Present value of an amount is calculated using the formula V=F/(1+r)^n 4 Where F= Amount after the specified period (Future Value), 5 n= Number of compounbding periods. 6 r= Rate of interest per period in decimals All amounts in $ 8 Given that 9 Future value F= 1803319.41 No. of years (N)= 10 10 Discount Rate (% p.a)= R= 7 No. of times compounded a year (t) 11 12 Calculations: 13 Formula Cell reference Value 14 Discount rate per period in decimals (r )= R/(100*t) D10/(100*G10) 0.07 15 No. of periods (n)= G9*G10 10 16 (1+r)= 1+G14 1.07 17 (1+r)^n= G164615 1.96715136 18 Present value = F/(1+r)^n D9/G17 916716.146 19 Rounded to $916,716.15 N*t

Yearly contributions needed from 26 to 65 years, in order to accumulate this amount is $4,591.96 as follows:

Part C:

Effective annual rate of semi annually compounded rate is

EAR= [(1+r/2)^2]-1 Where r= nominal rate (given as 8%)

Therefore, Effective annual rate= [(1+8%/2)^2]-1 = 91.04^2)-1 = 1.0816-1 = 8.16%