Problem 1. 

Year	Cashflow (at year end) (in thousands)	Present Value (at 8%)	Value at the end of year 15 (at 8%)
0	250	250	25 x 3.172 = 793.042
1-15	80	80 x 8.559 = 684.758	684.758 x 3.172 = 2172.1682
16-40	60	60 x 10.6748/(1.08)15 = 201.9081	201.9081 x 3.172 = 640.4866

Present values of annuities are computed using the annuity formula (for example in the last two rows of the table; column 3); 3.172 is simply equal to (1.08)¹⁵

a. The answer from the above table is 640.4866.

b. The already available $250 will grow to $793.042; so no more additional saving is necessary.  From part (a), we know that on ly $640.4866 is required at the end of yr. 15 (when he is 70).  He can even dissave.  How much can he dissave (or consume in excess of his salary) during the fifteen years?  Suppose he chooses to dissave $x per year.  Then, we need PV(annuity of x per year for 15 years at 8%) = (793.042-640.4866)/(1.08)¹⁵.  Solving, we find x = $5.6185

c. It's five years from the beginning of the events described.  The $2 that the client saved each year would have amounted to PV(annuity of 2 per year for 5 years at 8%) x (1.08)⁵ = 11.7332 and the initial 250 would have grown to 367.332, for a total of 379.065.  From this point on, the monies will grow at 5%.  So these already accumulated funds will grow to 379.065(1.05)¹⁰ = 617.457 by the time the client retires.  Now, he will need $60 per year for the next 25 years.  It's value at the time of retirement can be computed as PV(annuity of 60 per year for 25 years at 5%).  This works out to 845.63667, which is greater than the 617.457, which the client has accumulated.  Hence, he needs to plan to save $y every year for the next ten years, such that PV($y per year for 10 years at 5%)(1.05)¹⁰ = (845.63667 - 617.457).  Solving, we find 18.141328 a year.

d. If, on the other hand, the client only wants to save $2 per year for the next 10 years, he would have 617.457 + PV(annuity of $2 for 10 years at 5%)(1.05)¹⁰ = 617.457 + 25.15579.  Equating this to the present value of an annuity of $x for 25 years at 5%, we find $x = $45.595 a year.

Problem 2:

Suppose we pick GE.  Go to http://www.zacks.com and pick the first Institutional Broker, who has EPS estimates of $3.70 for 2000.  The estimated five-year growth rate is 15%. 

From http://biz.yahoo.com/p/g/ge.html,  we see that dividends last year were $1.64, with earnings of $3.22 for a payout ratio of  approximately 50%.  (Keep in mind that earnings are year-to-date, while dividends seem to be for the last fiscal year (somewhat unclear from the information provided on the site.)  The stock beta is given as 1.16.  The return on 1 year T-bills (http://www.bloomberg.com/markets/C13.html?sidenav=front) as of Feb. 8, 2000 is 6.19%.

Using this payout ratio, dividends next year would be 3.70/2 = 1.85 next year and is expected to grow at 15% for the next five years, with growth tapering off to 6%, as provided for in the question.

The required rate of return is given by 6.19 + 1.16(5.5) = 12.57%

Discounting the dividends for the next 5 years using the information generated, we have a present value of 8.5796.  Dividends at the end of the sixth year would be 1.85(1.15)⁴(1.06) = 3.4298.  Discounting the value (at the end of year 5) of the dividends from the future that would be expected to grow at 6% p.a., we get 3.4298/(.1257 - 0.06) = 52.204.  Discounting the value to the beginning of year 1, we have 52.204(1.1257)⁵ = 28.8795.  Adding in the 8.5796, we get an estimated stock price of $37.46.  The actual stock price at 11 am on Feb. 8, 2000 was $138!

We assumed that all dividends on a yearly basis, starting one year from now, i.e. as of Feb. 8, 2001.  In practice, GE pays dividends four times a year starting in March.  We would need to adjust our computations for this.  However, the estimated price and the actual price are far apart!  How are we to explain this?