N-Year Geometric Increasing Annuity

n-year geometric increasing annuity Imagine that payments increase every period by a ratio (1+r). (beware - notation is inconsistent between economics, business, math and actuarial textbooks, especially once we get into inflation and real rates of interest). Also, let’s have the first payment of amount K (not yet increased though it’s a year after the date at which we take the PV0). Use

2 2 3 PV0= K { 1/(1+i) + (1+r)/(1+i) + (1+r) /(1+i)

+…+ (1+r)n-1/(1+i)n}

= [K/(1+i)]

www.utstat.utoronto.ca/sharp Geom increasing annuity in actuarial notation

Being careful about the first term, rewrite as

2 2 PV0 = [K/(1+r)] {(1+r)/(1+i) + (1+r) /(1+i) +

+…(1+r)3/(1+i)3 +…+ (1+r)n/(1+i)n}

= [K/(1+r)] an (at (1+i)/(1+r)-1)

Note that in this formulation we effectively say that payments are based on a notional time 0 amount of K/(1+r) , which to an actuary makes intuitive sense. But finance people tend to think of the first actual payment of K at time 1. Both are correct.

www.utstat.utoronto.ca/sharp Value of a stock: Gordon growth model (finance) PV0= K = K/(i-r) if the dividends go on forever. In reality we sell the stock some time for a sale price which reflects the present value then of the then-future dividends. This ends up with the same answer as assuming we never sell. The value of i might be taken as current market Treasury bond yield rate, perhaps adjusted for risk, and the analyst estimates next year’s dividend K and the increase rate r. In real life this formula gets some use as a rough-and-ready tool, maybe to compare two stocks. Or to explain a big stock market fall as a change in business confidence resulting in a small change in the implicitly assumed r.

www.utstat.utoronto.ca/sharp SECTION 7 - PAYMENTS FOLLOW A GEOMETRIC PROGRESSION SOA Exam FM/CAS Exam 2 Study Guide © S. Broverman 2008 www.sambroverman.com 89 Example 27 (SOA): Common stock X pays a dividend of 50 at the end of the first year, with each subsequent annual dividend being 5% greater than the preceding one. John purchases the stock at a theoretical price to earn an expected annual effective yield of 10%. Immediately after receiving the 10th dividend, John sells the stock for a price of P . His annual effective yield over the 10-year period was 8%. Calculate P.

www.utstat.utoronto.ca/sharp www.utstat.utoronto.ca/sharp PROBLEM SET 7 Exam FM/CAS Exam 2 Study Guide © S. Broverman 2008 12. Today is the first day of the month, and it is Smith's 40th birthday, and he has just started a new job today. He will receive a paycheck at the end of each month (starting with this month). His salary will increase by 3% every year (his monthly paychecks during a year are level), with the first increase occurring just after his 41st birthday. He wishes to take c% of each paycheck and deposit that amount into an account earning interest at an annual effective rate of 5%. Just after the deposit on the day before his 65th birthday, Smith uses the full balance in the account to purchase a 15-year annuity. The annuity will make monthly payments starting at the end of the month of Smith's 65th birthday. The monthly payments will be level during each year, and will increase by 5% every year (with the first increase occurring in the year Smith turns 66). The starting monthly payment when Smith is 65 will be 50% of Smith's final monthly salary payment. Find c

www.utstat.utoronto.ca/sharp www.utstat.utoronto.ca/sharp n-year arithmetic increasing annuity

2 3 4 n Ian = v + 2v + 3v + 4v + ……+ nv

2 3 n-1 (1+i) Ian =1 + 2v + 3v + 4v + ..…+ nv

Subtract

2 3 n-1 n i Ian = 1 + v + v + v + ..…+ v - nv

Ian =

www.utstat.utoronto.ca/sharp This arithmetic increasing annuity is less common in real life than the geometric. But it can be useful if e.g. regular coupon payments are accumulated in a fund which pays interest (on the arithmetically increasing balance) into another fund. Example 33 (SOA): (Broverman FM manual): Coco invests 2000 at the beginning of the year in a fund which credits interest at an annual effective rate of 9%. Coco reinvests each interest payment in a separate fund accumulating at an annual effective rate of 8%. The interest payments from this fund accumulate in a bank account that guarantees an annual effective rate of 7%. Determine the sum of the principal and interest at the end of 10 years.

www.utstat.utoronto.ca/sharp www.utstat.utoronto.ca/sharp n-year arithmetic decreasing annuity

2 3 n Dan = nv + (n-1)v + (n-2)v ……+ v

2 3 n (n+1) an = (n+1)v + (n+1)v +(n+1)v ++ (n+1) v

Subtract

Dan = (n+1) an - Ian

www.utstat.utoronto.ca/sharp