Question

​(Nonannual compounding using a calculator​) Should we have bet the​ kids' college fund at the dog​ track? Let's look at one specific case of a college professor​ (let's call him Prof.​ ME) with two young children. Three years​ ago, Prof. ME invested ​$170 000 hoping to have ​$440 000 available 14 years later when his first child started college.​ However, the​ account's balance is now only ​$150 000. ​Let's figure out what is needed to get Prof.​ ME's college savings plan back on track. a. What was the original annual rate of return needed to reach Prof.​ ME's goal when he started the fund 3 years​ ago? b. Now with only ​$150 000 in the fund and 11 years remaining until his first child starts​ college, what APR would the fund have to earn to reach Prof.​ ME's ​$440,000 goal if he adds nothing to the​ account? c. Shocked by his experience of the past 3 ​years, Prof. ME feels the college mutual fund has invested too much in stocks. He wants a​ low-risk fund in order to ensure he has the necessary ​$440 000 in 11 ​years, and he is willing to make​ end-of-the-month deposits to the fund as well. He later finds a fund that promises to pay a guaranteed APR of 4.5 percent compounded monthly. Prof. ME decides to transfer the ​$150 000 to this new fund and make the necessary monthly deposits. How large of a monthly deposit must Prof. ME make into this new fund to meet his ​$440 000 ​goal?

d. Now Prof. ME gets sticker shock from the necessary monthly deposit he has to make into the guaranteed fund in the preceding question. He decides to invest the ​$150 000 today and ​$450 at the end of each month for the next 11 years into a fund consisting of 50 percent stock and 50 percent​ bonds, and hope for the best. What APR would the fund have to earn for Prof. ME to reach his ​$440 000 ​goal?

Answer 1

(a)

Future Value (FV)=Present Value (PV) *((1+i)^N)

i=Annual return

N=Number of years to future

In this case,

FV=

$440,000

PV=

$170,000

N=

14

440000=170000*((1+i)^14)

(1+i)^14=440000/170000=

2.5882353

1+i =2.588235^(1/14)=

1.070287

Original annual return needed

0.070287

Original annual return needed

7.029%

(b)

Assume APR needed =R

PV=

$150,000

FV=

$440,000

N=

11

440000=150000*((1+R)^11)

(1+R)^11=440000/150000=

2.9333333

1+R=2.93333^(1/11)=

1.1027762

APR Needed =R=

0.1027762

APR Needed

10.28%

.(c)

Guaranteed APR Compounded Monthly

4.50%

Rate

Monthly return rate =4.5/12=

0.375%

Pv

Initial deposit to the fund

$150,000

Fv

Fututer Value needed

$440,000

Nper

Number of months =11*12=

132

PMT

Amount of monthly deposit required

$1,139.43

(Using PMT function of excel with Rate =0.375%,Nper=132,Pv=150000,Fv=-440000)

(d)

APR Needed to Earn

Pv

Initial deposit to the fund

$150,000

Fv

Fututer Value needed

$440,000

Nper

Number of months =11*12=

132

Pmt

Amount of monthly deposit

$450.00

RATE

Monthly Return Needed from investment

0.6379%

(Using RATE function of excel with Nper=132,Pv=-150000,Pmt=-450,Fv=440000)

APR Needed to Earn=0.6379%*12=

7.66%