Question

Albert Einstein reportedly said, “Compound interest is the eighth wonder of the world. He who understands it, earns it. He who doesn’t, pays it.” Regardless of whether Einstein uttered these exact words, the essence of his statement is still immensely powerful and cannot be disputed. For anyone who wants to build lasting wealth, understanding and harnessing the power of compound interest is essential. For the more visual of you, imagine, if you will, building the bottom part of a snowman. It starts with a snowball (or initial investment). You roll it around in the snow and it slowly gets bigger (interest on the investment). A slow and monotonous process until something wonderful becomes apparent – the snowball not only gets bigger and bigger, but at a faster and faster rate (interest on the interest).

Your friend, Mike Szyslak wants be a millionaire, and he found several ways applicable. But he is still hesitating among the various options and comes to you for financial advice. Complete each of the options, below, with your group.

• Option 1: He is considering to buy Mega Millions lottery. If the current tax rate on earnings of lottery is 40%, how much money will he have to win on a lottery to become a millionaire?
• Option 2: His uncle promised to invest his business $100,000 a year over the next 15 years, and the interest rates over next 15 years are expected be at 5% per annual. Can you help him to know whether the present value of such series of investments make him equivalent to be a millionaire today?
• Option 3: He considers to save money and become a millionaire. Starting at age 22, every night Mike takes $5 out of your pocket and put it in a manila envelope. At the end of the year, you place the money from the envelope in a stock fund with an average interest rate of 10%. Will the amount he has in the account ensure him a millionaire when you retire at age 65? What if he starts saving at age 40?
• Option 4: He sets aside $50,000 into a saving account now, and will deposit $50,000 into the account at the beginning of each year for next 10 years. If the market rate is 10%, Will he become a millionaire in 10 years?
• Option 5: Mike considers to buy 1,000 bonds. The bond is semi-annual coupon bond with 10-year maturity, $1,000 par value bond with a 10 percent annual coupon, and 10 percent annual required rate of return? How much does it cost now if he wants to receive all the coupon payments and par values during the 10-year period? What would be the value of the bond if, just after it had been issued, the expected inflation rate rose by 3 percentage points, causing Mike to require a 13 percent return? What would happen to the bonds' value if inflation fell, and required rate of return declined to 7 percent?

Which of the options would you recommend that Mike choose? Why?

Answer 1

Option -1-If he wants to have $ 1 million left after paying tax, then his win from lottery need to be

x*(1-40%)=1000000

so, x=1000000/(1-40%)=

1666667

So, he has to win $ 1666667 , so that the net amount after-tax is $ 1000000 & he becomes a millionaire

Option-2

We need to find the present value of pmt.=annuity of $ 100000 at end of every year

for n= 15 yrs

at an interest rate, r= 5% p.a.

using the formula, for PV of ordinary annuity,

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

& plugging-in the above values,

PV=100000*(1-(1+0.05)^-15)/0.05=

1037966

YES.

The present value of the above series of investments makes him equivalent to be a millionaire today as $ 1037966 > $ 1000000

Option-3

He saves $ 5 every night for 365 days in a year , ie. Annual PMT.=365*5= $ 1825 per year(at end of yr.)

for n= no.of yrs. From start of 23 to end 65 --ie. 43 yrs., so, n=43

at an interest rate, r= 10 % p.a.

so, the future value of this annuity at end yr. 43 will be =

using FV of ordinary annuity formula,

FV=PMT.*((1+r)^n-1)/r

& plugging-in the above values,

ie.FV=1825*(1.10^43-1)/0.10=

1081131

YES.

The future value of the above series of investments will make him a millionaire at end yr. 65 (his age) as $ 1081131 > $ 1000000

If he starts saving at age 40

using the same FV of ordinary annuity formula,

FV=PMT.*((1+r)^n-1)/r

n will become Start yr. 41- end yr. 65 ---25 yrs.

& plugging-in the rest of the above values,

ie.FV=1825*(1.10^25-1)/0.10=

179483

NO.

Starting at age 40 , will not make him a millionaire at age 65 as $ 179483 < $ 1000000

Option-4

using the FV of annuity due (beginning -of-yr.) formula,

FV=PMT.*((1+r)^n-1)/r*(1+r)

where, pmt.= $ 50000

r= the market interest rate 10%

n= 10

plugging-in the the above values in the formula,

FV=(50000*((1+0.10)^10-1)/0.1)*(1+0.10)=

876558

NO.

He will not become a millionaire in 10 yrs. , as $ 876558 < $ 1000000

Option-5

Present value of the bond=PV of coupons+ PV of face value at maturity

ie. PV of bond= (Coupon amt.*(1-(1+r)^n)/r)+(FV/(1+r)^n

we use all the semi-annual metrics

ie. PV of bond=((1000*10%)*(1-1.10^-10)/0.10)+(1000/1.10^10)=

1000

so, $ value of 1000 bonds = 1000 * $1000/ bond= $ 1000000

If the expected inflation rate rise by 3 percentage points,ie, by 3% causing Mike to require a 13 % return

ie. PV of bond=((1000*10%)*(1-1.13^-10)/0.13)+(1000/1.13^10)=

837.21

so, $ value of 1000 bonds = 1000 * $ 837.21/ bond= $ 837210

If the expected inflation rate fell by 3 percentage points,ie, by 3% causing Mike to require a 7 % return

ie. PV of bond=((1000*10%)*(1-1.07^-10)/0.07)+(1000/1.07^10)=

1210.71

so, $ value of 1000 bonds = 1000 * $ 1210.71/ bond= $ 1210710

Option	To choose In the following order
2	15 yrs.' savings
5	10 yrs.	subject to mkt.int rates & inflation
3	43 yrs.	Longest
1	immediate	lottery---so speculative only
4	ruled out	< 1 mln.