Question

Please help by providing explanations/step-by-step processes for solutions. Thank you in advance!

1. You are 20 years old. Starting at age 70 (in 50 years), it is expected that you will begin to receive a $1,500 monthly check from Social Security. You expect to live until you are 86 years old. Assuming a discount rate of 5%, what is the value today of the expected stream of Social Security payments?

2. 5. Suppose an investment promises to pay you $10,000 annually into perpetuity, but the payment stream will not begin for 7 years. Assuming a discount rate of 6%, what is the value today of the expected stream of payments?

3. You are set to buy your first car. The Lexus IS costs $38,000, and you have decided to make a down payment of $3,500 and finance the remaining balance. The bank tells you that your monthly payment will be $540.54 on a 6-year loan. What is the interest rate on the loan?

4. How much would you have to deposit today to have $1 million in 30 years at 6% interest compounded quarterly?

5. A 25-year old individual begins a savings plan for retirement starting at age 65 (in 40 years). The individual wants to save enough such that she can spend $50,000 per year in retirement. She would like to design a plan to withdraw $12,500 every three months during retirement. Assuming an interest rate of 5% and a life expectancy of 90 years, how much would she have to save annually until retirement to meet her goal?

Answer 1

1.The value today of the expected stream of Social Security payments

is the Present Value at time t=0, of the

Present value at end yr. 70/start yr.71 , of the (86-70)=16 yrs.*12= 192 monthly annuities of $ 1500

at an interest rate 5% p.a ,ie.5%/12=0.4167% p.m.

So,first we need to find the PV of the annuity at end Yr. 70--then find the PV at Yr. 0 of this single lumpsum amt, ie. Dividing the PV of 192 months' annuity BY (1+0.004167)^(50*12)

So,proceeding on the above lines & using the formula,

PV of ordinary annuity=Pmt.*(1-(1+r)^-n)/r

PV (at end yr. 70)=1500*(1-1.004167^-192)/0.004167=

197966.9876

NOW,

PV at Yr. 0,ie. (at age 20)=

197967/1.004167^600=

16331.49

(ANSWER)

2.Almost similar to the above in 1.

Value today of the expected stream of payments is the

PV at yr.0 of the

PV of the Year-end perpetuity, at end Yr. 6(pmt. Starts at end yr. 7) from today.

Proceeding on the above lines & using the formula,

PV of perpetuity= Periodic payment/Interest rate per period

So, PV at end yr. 6=10000/6%=

166667

Now, PV at Yr. 0 of the above amt.=

166667/1.06^6=

117494

So, the answer is:

Value today of the expected stream of payments= $ 117494

3.Interest rate on the loan can be found out by

using the PV of ordinary annuity formula,

& filling up all the known values.

The PV of the bank loan= 38000-3500=34500

Monthly pmt.= 540.54

for a period of 6 yrs.*12= 72 months

at r % monthly rate

so, using the formula for PV of annuities,

PV of ordinary annuity=Pmt.*(1-(1+r)^-n)/r

ie. 34500=540.54*(1-(1+r)^-72)/r

Solving for r ,in an online equation solver,

(I use Wolfram Alpha)

we get the monthly rate of interest as 0.34%

Interest rate on the loan is normally expressed as % per annum

so, converting the above monthly rate into an yearly one,

(1+0.34%)^12-1=

4.16%

The answer is :

Interest rate on the loan= 4.16%

4. We need to find the PV of a single sum

using the formula,

PV=FV/(1+r)^n--- where

PV = Present value of the single sum, in future

FV= the future value given as   $ 1 million= $ 1000000

r= rate per period, ie. 6% p.a.= 6%/4 Quarters per year =1.5% or 0.015 per period(quarter)

n= No.of periods, here quarters= 30 yrs.*4= 120 quarters

So, putting values in the formula,

PV =1000000/(1+0.015)^120=

167523.19

So the amt.

to be deposited today = $ 167523

5. She wants to withdraw 12500 ever quarter(4 times in a year) in a year for her age 65-90, ie. 25 years,

ie. For a total of 25*4=100 quarters

at 5% p.a.,ie. 5%/4= 1.25 % or 0.0125 per quarter

So. First we find the PV of these withdrawals at end of age 65.

so, using the formula for PV of ordinary period-end annuities,

PV of ordinary annuity=Pmt.*(1-(1+r)^-n)/r

& putting te above values,

PV =12500*(1-(1+0.0125)^-100)/0.0125

711266.74

Now, this should be the FUTURE VALUE of

her annual savings for the next 40 years

at 5% p.a.

ie. Using the FV of ordinary annuity formula,

FV(OA)=Pmt.((1+r)^n-1)/r

where FV(OA)= 711266.74

Pmt. =annual savings , we need to find ??

r= the given 5% (here, we need not convert , as Pmt. & n are annual)

n= 40 yrs.

So, substituting the values in the formula,

711266.74=Pmt.*((1+0.05)^40-1)/0.05

Solving for pmt., we get the annual amt. to be saved as

5887.98

So, the answer is:

Amount she would have to save annually until retirement to meet her goal= $ 5888