Question

1. Erika and Kitty, who are twins, just received $30,000 each for their 25th birthday. They both have aspirations to become millionaires. Each plans to make a $5,000 annual contribution to her “early retirement fund” on her birthday, beginning a year from today. Erika opened an account with the Safety First Bond Fund, a mutual fund that invests in high-quality bonds whose investors have earned 6% per year in the past. Kitty invested in the New Issue Bio-Tech Fund, which invests in small, newly issued bio-tech stocks and whose investors have earned an average of 20% per year in the fund’s relatively short history.
   1. If the two women’s funds earn the same returns in the future as in the past, how old will each be when she becomes a millionaire?
   2. How large would Erika’s annual contributions have to be for her to become a millionaire at the same age as Kitty, assuming their expected returns are realized?
   3. Is it rational or irrational for Erika to invest in the bond fund rather than in stocks?

Solve using a financial calculator and explain why FV is negative.

Answer 1

Please note that the sign of the cash flows such PV, FV or PMT is primarily to tell us whether we are parting away with that particular cash low or we are receiving the same. If we part away with our PV and PMT, then we will be entitled to get the FV. In such a case, PV and PMT should have a negative sign as they signify cash outflow for us. FV is what I get in return so it should then have a positive sign because I am receiving the money.

Alternatively you can have the cash outflows inputs such as PV and PMT as positive number (so a positive sign signify cash outlow, hence an inflow will be represented by a negative sign), then in that case the FV will have a negative value. In a nutshell, your inflows and outflows will have different sign.

Part (a)

I will follow the convention that my outflows have negative sign, so my output FV, which is inflow, will have a positive sign.

PV = - 30,000, PMT = -5 ,000, FV = 1,000,000

For Erika, i = 6% = 0.06

Hence, n = number of years = NPER (Rate, Payment, PV, FV) = NPER (6%, -5000, -30000, 1000000) = 38.74218224

Please note that the same output could have been obtained, if I had used = NPER (6%, 5000, 30000,-1000000).

So FV need not be negative. It just needs to have a different sign in comparison to that of PV and PMT.

Thus, for Erika, n = 38.74.

So her age when she becomes millionaire = 25 + 38.74 = 63.74 i.e. 64 years old.

For Kitty, i = 20% = 0.20

Hence, n = number of years = NPER (Rate, Payment, PV, FV) = NPER (20%, -5000, -30000, 1000000) = 16.04371287

So her age when she becomes millionaire = 25 + 16.04 = 41.04 i.e. 41 years old.

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Part (b)

We now need to calculate the annual payment i. e. PMT for Erika such that she ends up becoming a millionaire in the same time frame as Kitty.

Hence, n = 16.04, i = 6% = 0.06

The desired annual payment = PMT(rate, period, PV, FV) = =PMT(0.06,16.04,-30000, 1000000) = - $ 35,825.33

Please note that the negative sign indicates that Erika needs to part away this much money every year.

Hence,  Erika’s required annual contributions to become a millionaire at the same age as Kitty, assuming their expected returns are realized = $ 35,825.33

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Part (c)

There is nothing rational or irrational here. Every individual has a specific risk reward requirement. Securities which give higher returns do have higher risks and vice versa. Erika's risk profile appears to be "Cautious" and hence she chose a debt fund where the reward is low but so is her risk.

Given that she is just 25 years old and still a way to go, the general behavior is to go for equity funds. But if she chose debt fund, there is nothing irrational about it.