A perpetuity paying 1 every 6 months has present value of 20. A perpetuity paying X every 2 years has the same present value. Assuming equal effective annual rates, what is X?
I know X is 3.71 from the answer key of my text, but I don't know how to get there.
On
First thing you have to notice is that you have a perpetuity due. So you may want to use either $$ PV = \frac{1 + i}{i}~~~\text{or}~~~ PV = \frac{1}{d} $$ where $i$ and $d$ are the effective annual rates of interest and discount respectively. The above formulas represents the present values of a perpetuity paying $1$ at the beginning of the year.
You are told that the PV of a perpetuity paying $1$ every six months is 20.
Thus $$ 20 = \frac{1}{D} \implies D = 0.05.$$ But this represents a semiannual discount rate. You need to convert this discount rate every two years. You can do this by recalling that $$ 1-d = (1-D)^2$$ and if you let D' represent the discount rate every two years, then you have $$ 1 - D' = (1-D)^4 = 0.95^4$$ and hence $D' = 1 - 0.95^4$, because $ 1 - D' = (1-d)^2.$ Now all that is left to do is to solve the following $$ 20 = \frac{X}{D'}$$ for $X$.
As an exercise, try solving the same problem using effective annual rates instead. Hope this helps.
Let the six-monthly interest rate be $r$ so the discount rate $d=1+r$. You need to sum a couple of geometric progressions. I've also assumed that there is a payment now at time zero.
The value of the future payments of the first stream is $1+\frac 1d+\frac 1{d^2}+\dots=\frac d{d-1}=20$ so that $d=\frac {20}{19}=1.0526$ and $r=0.0526$
Now for the two yearly stream with payments $a$, we have to discount over two year intervals, or four six-month periods giving $a+\frac a{d^4}+\frac a{d^8}+\dots =\frac {ad^4}{d^4-1}=20$
So that $a=\frac{((1.0526)^4-1)\times 20}{(1.0526)^4}=3.71$