Here are the answers to your questions.
The "real value" of this prize is the present value of the described
flow of payments. Following is a link that gives a good description
and formulae for calculating the present value of a stream of
Calculating the Present and Future Value of Annuities
[check under the section "Calculating the Present Value of an Annuity Due"]
Given that the discount rate is 8%, the present value of the lottery
prize would be (in thousands):
PV = 100 + 100/(1.08)^1 + 100/(1.08)^2 + 100/(1.08)^3 + ... + 100/(1.08)^19
Fortunately, the same result can be achieved using a simpler formula,
which is mentioned in the link I gave above, under the section
"Calculating the Present Value of an Annuity Due". Applying this
formula, we get that the present value of the lottery prize is:
PV = 100*[1 - 1.08^(-20)]*1.08/0.08 = 1,060.36
Thefore, the "real value" of the prize is $1,060,360 rather than 2 million.
In this question, we must again make use of the formula for present
value. The yield to maturity (YTM) is the discount rate that is
implied in the price of the bond.
The value of a bond is the present value of its flow of payments. The
bond from this question gives you $65 every year for 15 years, and
then gives you $1,000 (it's par value) at the end of the 15th year.
Therefore, letting "i" be the discount rate, the value of this bond
PV = 65 + 65/(1+i)^1 + 65/(1+i)^2 + ... + 65/(1+i)^14 + 1000/(1+i)^15
Now, we also know that the market price of this bond is $1,240.
Therefore, the present value of the bond is also $1,240. So we get the
1240 = 65 + 65/(1+i)^1 + 65/(1+i)^2 + ... + 65/(1+i)^15 + 1000/(1+i)^15
From this equation, we should be able to find the discount rate "i".
The value we find for "i" will be the yield to maturity. That is, the
YTM is simply the discount rate that makes the market price be what it
is. Since the above equation cannot be solved analytically, we will
use a bond calculator. You can find an online one at the following
Bond Yield Calculator
A small caveat is that this calculator (and all of the online
calculators I could find) assume that the first coupon payment occurs
one year from now instead of today. So, in order to be able to use
them, we just have to rearrange the equation like this:
1240 - 65 = 65/(1+i)^1 + 65/(1+i)^2 + ... + 65/(1+i)^15 + 1000/(1+i)^15
1175 = 65/(1+i)^1 + 65/(1+i)^2 + ... + 65/(1+i)^15 + 1000/(1+i)^15
Notice that since you received $65 the same day you bought it for
$1,240, then this is equivalent as paying $1,175 and receiving the
first coupon payment one year from today.
So now we can use the calculator. Enter the following data into it:
Current price: 1175
Par Value: 1000
Coupon Rate: 6.5 (because $65 -the coupon value- is 6.5% of the par value)
Years to Maturity: 15
Now click on "Calculate". We get from the calcualtor that the YTM is 4.833%
Google search terms:
yield to maturity
bond price calculator
I hope this helps! If you have any questions regarding my
answer,please don't hesitate to request a clarification. Otherwise I
await your rating and final comments.