Present value involving deferred annuities - is this a typo?

Question

Catfish Hunter’s 1974 baseball contract with the Oakland Athletics called for half of his 100,000 salary to be paid to a life insurance company of his choice for the purchase of a deferred annuity. More precisely, there were to be semi-monthly contributions in Hunter’s name to the Jefferson Insurance Company with the first payment on April 16 and the final payment on September 30. We suppose that the first eleven of these were to be for 4,166.67 and the final payment was to be for four cents less. (12 × $4,166.67 = $50, 000.04.)

A) Using an annual effective interest rate of 6% (a rate that figures in a six-year personal loan of $120,000 that Oakland’s owner Charles Finley had made to Hunter in 1969 and then promptly recalled), find the value of the specified payments to the insurance company at the scheduled time of the last payment.

B) Suppose that the contracted payments had been made to the insurance company from April 16, 1974 through September 30, 1974, and that they accumulated at an annual effective interest rate of 6%. Further suppose that Hunter had drawn a level January 1st salary for twenty years, beginning on January 1, 1980, the first January after his retirement. Find the amount of the annual salary payments. (Hunter died on September 9, 1999, so he would not have received a January 1, 2000 annuity payment.)

The work for A makes sense. It involves finding out what $\frac{i^{(24)}}{24}$ is, which = .0024308208, then finding the product of 4166.67 [$\frac{1.0024308208^{20} - 1}{0.0024308208}$] - .04 = 50673.92. This is the answer in the book and I completely understand how we get there.

The work for B involves using A. My thinking was: find the product of 50673.92(1.06)$^{5.5}$ = 68808.22 because the money accumulates for 5 years from 9/30/74 to 9/30/79 plus another 1/4 year. Then, I set that equal to P[$\frac{1 - (1.06)^{-20}}{0.06}$] to find the payment, which equals 5999.01. However, the book answer is 5659.45. I used some "reverse engineering" to basically figure out that they were doing the exact same thing, except for some reason, they did 50673.92(1.06)$^{4.25}$. I cannot understand what wording in this question would indicate that he doesn't earn interest for that last year.

Answer 1

$\require{enclose}$ At the time of the last payment of premium, the equation of value is $$AV = 4166.67 \left((1+j)^{11} + (1+j)^{10} + \cdots + 1\right) - 0.04 = 4166.67 s_{\enclose{actuarial}{12} j} - 0.04,$$ where $j$ is the effective semimonthly rate of interest; i.e., $$(1 + j)^{24} = 1+i = 1.06.$$ Thus $$j = (1.06)^{1/24} - 1 \approx 0.00243082$$ and the accumulated value of payments is $$4166.67 \frac{(1.00243082)^{12} - 1}{0.00243802} - 0.04 \approx 50673.92.$$

For the second part, the annuity continues to accrue interest at effective annual rate $i = 0.06$, so the accumulated value at the time of the first level annual payment is $$AV(1+j)^6 (1+i)^5 \approx 68808.2187690,$$ as the annuity has had six semimonthly periods from 30 September to 31 December 1974 and five years to 31 Dec 1979 to accrue. Then the first disbursement of $K$ occurs immediately, so the equation of value on 01 January 1980 is $$68808.22 = K (1 + v + v^2 + \cdots + v^{19}) = K \ddot a_{\enclose{actuarial}{20} i} = K (1.06)\frac{1 - (1.06)^{-20}}{0.06} \approx 12.1581K,$$ where $v = (1+i)^{-1}$ is the annual present value discount factor, and there are $20$ disbursements in total. Hence $$K \approx 5659.44723,$$ which agrees with the book's answer. Your error is that you did not set the time of the first disbursement to be 01 January 1980: the formula for the present value of an annuity-due is $$\ddot a_{\enclose{actuarial}{n}} = (1+i) a_{\enclose{actuarial}{n}} = (1+i)\frac{1-(1+i)^{-n}}{i}.$$ The easiest way to see that your formula cannot be correct is that had there been a lump sum payment on that date, then $K$ is just the accumulated value $68808.22$. But your formula would have us solve $$68808.22 = K \frac{1 - (1.06)^{-1}}{0.06} = \frac{K}{1.06} \ne K.$$