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Before I begin:
Book Answer: 4%
My Answer: 4.2%
Here's the Question:
Mike buys a perpetuity immediate with varying annual payments. DUring the first 5 years, the payment is constant and equal to 10.. Beginning in year 6, the payemnts start to increase. For year 6 and all future years, the current year's payment is K% larger than the previous year's payment. At an annual effective interest rate of 9.2%, the perpetuity has a present value of 167.50. Calculate K, given K<9.2
Now it seems simple enough. We start with a 5 year Annuity-immediate and add the perpetuity (starting at year 6 and discounted to time 0 -- So it's like a deferred perpetuity of sorts).
Thus we have
Annuity: a[5] @ i = 9.2% --> = 38.6955
Perpetuity:
Let J = [(1+k)/(1+i)]^n
PV = 10v^5 * (1-J)/(i-k)
Write this on paper if this doesn't seem clear. I didn't bother making these look pretty, but this equation can be proven when showing the payment series.
Well, with a perpetuity we have infinite payments so we conclude that
n = INFINITY
Thus J = 0 (because 1+k < 1+i)
So we get
167.5 = 38.6955 + 10v^5 * 1/(.092-k)
Solve for k...
k = 4.2%
Anyone have any ideas? Anyone out there think they can solve it properly? The book takes a different approach that doesn't make sense to me.
Refer to ASM P. 243 Question #2.
Book Answer: 4%
My Answer: 4.2%
Here's the Question:
Mike buys a perpetuity immediate with varying annual payments. DUring the first 5 years, the payment is constant and equal to 10.. Beginning in year 6, the payemnts start to increase. For year 6 and all future years, the current year's payment is K% larger than the previous year's payment. At an annual effective interest rate of 9.2%, the perpetuity has a present value of 167.50. Calculate K, given K<9.2
Now it seems simple enough. We start with a 5 year Annuity-immediate and add the perpetuity (starting at year 6 and discounted to time 0 -- So it's like a deferred perpetuity of sorts).
Thus we have
Annuity: a[5] @ i = 9.2% --> = 38.6955
Perpetuity:
Let J = [(1+k)/(1+i)]^n
PV = 10v^5 * (1-J)/(i-k)
Write this on paper if this doesn't seem clear. I didn't bother making these look pretty, but this equation can be proven when showing the payment series.
Well, with a perpetuity we have infinite payments so we conclude that
n = INFINITY
Thus J = 0 (because 1+k < 1+i)
So we get
167.5 = 38.6955 + 10v^5 * 1/(.092-k)
Solve for k...
k = 4.2%
Anyone have any ideas? Anyone out there think they can solve it properly? The book takes a different approach that doesn't make sense to me.
Refer to ASM P. 243 Question #2.
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