Solution: Let i = 10% and d = 5%. By definition, the respective forces of interest on funds A and B are. i 1 + it. S A (t) = d (1 dt) 2 1. = d 1 dt.

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1 Chapter 9 Revew problems 9.1 Interest rate measurement Example 9.1. Fund A accumulates at a smple nterest rate of 10%. Fund B accumulates at a smple dscount rate of 5%. Fnd the pont n tme at whch the forces of nterest on the two funds are equal. Soluton: Let = 10% and d = 5%. By defnton, the respectve forces of nterest on funds A and B are and δ A (t) = S A (t) S A (t) = δ B (t) = S B (t) S B (t) = d (1 dt) dt 1 + t Thus, we need to solve for t such that δ A (t) = δ B (t),.e., whch yelds t = t = t, = d 1 dt. Example 9.2. Consder a stream of payments, whch pays $100 at tme one, $200 at tme two, and $500 at tme three. Ths stream of payments s to be replaced by a sngle payment at tme three. Fnd the sze of ths payment f δ t = 1/( t). 1

2 Soluton: The accumulated value of the three payments at tme three s 100e R 3 1 δtdt + 200e R 3 2 δtdt =100e 10 ln(1+0.1t) 3 t= e 10 ln(1+0.1t) 3 t= =100e ln( ( 1.3 = = ) e ) ln( ) ) ( Valuaton of annutes Example 9.3. Calculate the present value of an annuty payable contnuously for ten years, under whch the rate of payment at tme t s t, and where the force of nterest s δ t = 0.001(15 t). Soluton: By defnton, the present value of a contnuous annuty s P V = n =10 = 0 n h(t)e R t 0 δudu dt = 0 (15 t)e n (15 u) e n = 10e e (15 t)2 n = (150 10t)e R t (15 u)du dt t n u=0dt = 10 (15 t)e [(15 t) ] dt 0 e (15 t)2 d [ (15 t) 2] t=0 Example 9.4. A loan for amount A s to be amortzed by n annual payments of $1 based on an nterest rate of. P s the present value, at nterest rate, of the prncpal portons of the loan payments. Determne an expresson for (Ia) n n terms of A, P, and only. Soluton: A and P are such that A = a n and P = n v k P R k, k=1 2

3 where P R k =OB k 1 OB k = a n k a n k+1 = 1 vn k+1 =v n k+1. 1 vn k = vn k (1 v) Thus, P = n v k v n k+1 = n v n+1 = nv n+1. Fnally, by defnton, k=1 k=1 (Ia) n =än nv n = (1 + )a n nv n = 1 + (A P ). = 1 + ( an nv n+1) Example 9.5. You are gven the followng seres of payments: () 100 at tme t = 1, 3, 5,..., 19 () 200 at tme t = 2, 4, 6,..., 20. An actuary s asked to determne the tme t, such that the present value of the seres of payments s equal to a sngle payment of 3,000 made at tme t. Derve an exact expresson for t n terms of the present value of annutes and the annual dscount factor. Soluton: Let be the annual effectve nterest rate and j = (1+) 2 1 be the two-year effectve rate. Then the present value of the seres of payments may be calculated by 3000 v t =100a a 10 j =100a v10 j 1 v 20 = 100a j (1 + ) 2 1 =100a a 20 = 100 s a s 20 2 s 2 =100 a 2 + v 2 a a 20 = 100 v + v2 + v a 2 a 20

4 =100 v + 2v2 a 2 a 20. Solvng for t we obtan t = [ ] (v + 2v 2 ) a ln 20 30a 2. ln v 9.3 Loan repayment Example 9.6. A $6,500 loan s to be amortzed by eght semannual payments over four years at a nomnal semannual nterest of 13.5%. Splt the ffth and the sxth payments nto prncpal and nterest. Soluton: The semannual effectve nterest rate s = 0.135/2 = , the term of the loan s n = 8, and the payments are of amount K = L = 6500 a 1 v 8 n = [We may use here ether the prospectve or the retrospectve method. For llustraton purposes, I wll use both.] By the retrospectve method, OB 4 = L(1 + ) 4 Ks 4 = = Thus, the nterest pad by the ffth payment s I 5 = OB 4 = and the prncpal repad s P R 5 = k I 5 = By the prospectve method, P R 6 = OB 5 OB 6 = Ka 3 K 2 = K 1 v3 (1 v 2 ) mplyng that I 6 = K P R 6 = = , Example 9.7. On a debt of $1,000,000, nterest s pad quarterly at (4) = 15% and quarterly deposts are made nto a snkng fund to retre the debt at the end of seven years. If the snkng fund earns nterest at j (4) = 10%, what s the quarterly expense on the debt and the book value of the debt at the end of the thrd year? 4

5 Soluton: The effectve quarterly nterest rates are = (4) /4 = and j = j (4) /4 = The loan amount s L = 1, 000, 000 and the term s n = 4 7 = 28. The quarterly expense on the debt s L + R = L + L s n j = 37, , = 62, The book value of the debt at the end of the thrd year s L Rs 3 4 j = 653, Example 9.8. Coln buys a house whch costs $240,000. He puts down 25% and mortgages the rest over 25 years at (2) = 4.65%. (a) Calculate Coln s monthly payment. (b) At the tme of the 24th payment Coln makes an addtonal payment of $250. Calculate the amount of prncpal and nterest pad n the next regular payment (.e., n the 25th payment). (c) After the 30th payment Coln swtches from monthly payments to b-weekly payments. The b-weekly payment s one half the sze of the monthly payment. Calculate the remanng amortzaton perod for the mortgage n years and weeks. Soluton: The amount that Coln borrows s L = , 000 = 180, 000. Also, the effectve monthly nterest s = ( ) /6 2 1 = and the term of the mortgage s n = = 300. (a) Coln s monthly payment s P = L a n = 180, v 300 = 1, (b) The outstandng balance after the addtonal payment of $250 s OB 24 = L(1 + ) 24 P s = 171, , whch mples that the nterest pad n the 25th payment s I 25 = OB 24 = and the respectve prncpal repad s P R 25 = P I 25 =

6 (c) We need to amortze the outstandng balance after the 30th payment OB 30 = OB 24 (1 + ) 6 P s 6 = 169, Thus, f the remanng number of b-weekly payments s m, the b-weekly nterest s j = (1 + ) 12 2/52 1 = , and the bweekly payment s R = P/2 = , then m solves the equaton OB 30 = Ra m j,.e., we need to determne m such that 169, = vm j j. Therefore, m = ln ( ) 1 169,574.98j = ln v j There wll be 510 b-weekly payments n total, whch equals 19 years and 32 weeks. 9.4 Bond valuaton Example 9.9. A $65,000 three-year bond wth semannual coupons at 13.5%, redeemable at 110 s sold to yeld 11%. (a) What s the purchase prce for ths bond and s the bond sold at a premum or a dscount? (b) Construct a bond schedule to amortze the premum/dscount. Soluton: The quanttes related to prcng the bond are face value F = 65, 000, redempton amount C = 110 F = 71, 500, semannual coupon rate r = 0.135/2 = , semannual yeld 100 rate j = 0.11/2 = 0.055, and term to maturty n = 3 2 = 6 semannual perods. (a) The prce of the bond s P = F ra n j + Cv n = 73, > C, whch mples that the bond s sold at a premum. (b) The amortzaton schedule of the bond s 6

7 t F r, t = 1,..., n 1 K t = F r + C, t = n I t = job j 1 P R t = K t I t OB t = F [ ] 1 + (r j)a n t j , , , , , , , , , , , , , , , , , , Example A $10,000 bond maturng at 98 on June 1, 2010 and payng 13% coupon s prced to yeld (4) = 12%. Fnd the prce of the bond on October 25, Soluton: We know that the face value s F = 10, 000, the redempton amount s C = F = 9, 800, the semannual coupon rate s r = 0.13/2 = 0.065, and the semannual yeld rate s ] 2 j = [1 + (4) 1 = Also, after October 25, 2008 there are k = 4 more coupons (on December 1, 2008, June 1, 2009, December 1, 2009, and June 1, 2010) untl the maturty of the bond. We frst need to determne the prce-plus-accrued on October 25, To ths end, we calculate that between June 1, 2008 and December 1, 2008 there are = 183 days, and between June 1, 2008 and October 25, 2008 there are = 146 days. We then need to determne the prce-plus-accrued at tme t = (n k 1) + s where s = 146/183. Namely, Prce-plus-accrued t =P t = (1 + j) s P n k 1 = (1 + j) ( s F ra n k 1 j + Cv n k 1) = /183 (650 1 ) v v 4 j =10, Thus, the prce on October 25, 2008 s Prce t =P t sf r = 10,

8 =9, Example Becky buys and n-year 1,000 par value bond wth 6.5% annual coupons at a prce of The prce assumes an annual effectve nterest rate of. The total wrte-up n book value of the bond durng the frst two years after purchase s Calculate. Soluton: Recall from Example 4.5 part (c), that when a bond s bought at a dscount, ts amortzaton s called wrtng up and that the prncpal repad before the term of the bond s negatve. Thus, the wrte-up n book value of the bond durng the frst two years after purchase s P R 1 + P R 2 = Also, we know that F = C = 1000, r = 6.5%, and P = Therefore, =P R 1 + P R 2 = F r P + F r [P (F r P )] =2F r (2P F r) 2 P, whch s a quadratc equaton whose solutons are 1,2 = 2P F r ± (2P F r) 2 + 4P (2F r ) 2P = ± ( F r) ( ), whch has a unque postve root = Measurng the rate of return of an nvestment Example An nvestor needs to decde between nvestng n a retrement home and nvestng n a car dealershp. Both nvestments requre a $1,000,000 ntal nvestment; however, the retrement home wll requre addtonal nvestment of $20,000 at the end of the frst year and $30,000 at the end of the second. In the thrd year profts are estmated to start at $70,000 and ncrease by 7% per year forever. The car dealershp wll return $190,000 at the end of the frst year, $170,000 at the end of the second year and $80,000 every year after that. (a) If the nvestor can borrow money at an annual nterest rate of 10%, use the net present value crteron to decde f she should make one, both, or none of these nvestments. (b) Determne the nternal rate of return for each nvestment and compare them. Does ths lead to the same concluson? 8

9 Soluton: (a) The net present value of the retrement-home nvestment s P 1 =70, 000 ( v v v ) 1, 000, , 000v 30, 000v 2 =70, 000v 3 =885, v 1, 000, , 000v 30, 000v2 (9.5.1) The net present value of the dealershp nvestment s P 2 =190, 000v + 170, 000v , 000v 2 1 1, 000, 000 (9.5.2) 0.1 = 25, Therefore, the nvestor should choose only the retrement home snce the car dealershp s not proftable. (b) By equaton (9.5.1), the nternal rate of return 1 on the retrement home s the soluton to 70, 000(1 + 1 ) 3 or equvalently, = 1, 000, , 000(1 + 1 ) , 000(1 + 1 ) 2, (1 + 1 ) = (1 + 1 ). 2 Multplyng the latter equaton by (1 + 1 ) 2 ( ) yelds 7 = 100 ( ) (1 0.07) + 2(1 + 1 )( ) + 3( ). Regroupng the terms leads to the cubc equaton = 0, whose only admssble value for 1 s %. Smlarly, by equaton (9.5.2), the nternal rate of return 2 on the car-delaelershp nvestment s 190, 000(1 + 2 ) , 000(1 + 2 ) , 000(1 + 2 ) = 1, 000, 000, 9

10 whch s equvalent to = 0, yeldng 2 = 9.7%. Snce 1 > 2, the retrement home s a more proftable nvestment than the car dealershp. Example One unt of a Mutual Fund s sellng for $10 on January 1, 2009, $15 on July 1, 2009, and $10 on January 1, On January 1, 2009, you purchase $100,000 of the fund, and on July 1, 2009, you purchase an addtonal $150,000 worth of the fund. Fnd the dollar-weghted effectve annual rate of return on your nvestment for year Soluton: There are 10,000 unts of the fund purchased on January 1, 2009 and another 10,000 unts purchased on July 1, Thus, the fund value on January 1, 2010 s 20, 000 $10 = $200, 000. Consequently, the dollar-weghted rate of return s = 200, 000 (100, , 000) 100, , = = , whch mples a loss of 28.57%. 9.6 The term structure of nterest rates Example The yeld on strps s as follows: Term (n years) Strp nterest rate % % % % % % % % What s the maxmum prce of a four-year $1,000 8% par bond that an nvestor should pay? Soluton: The semannual coupon rate of the bond s r = 4%, the term to maturty s n = 2 4, and the face value s F = C = 1, 000. Snce the bond s an nvestment that s at least as rsky 10

11 as the purchase of the strps, the maxmum prce that should be pad for the bond s P = n F r C ( k=1 1 + s ) [0,k/2] k + ( 1 + s ) [0,n/2] n = Cashflow duraton and mmunzaton Example (Exercse n the textbook) Develop an expresson for the duraton of a level n-payment annuty-mmedate of 1 per perod wth nterest rate per perod and show that t s equal to 1 d n. s n Soluton: By defnton, the duraton s ( ) d d a d 1 v n n d a n = a n = nvn + 1 a n d = 1 d as needed. = nv n+1 (1 v n ) a 2 n 1 + n s n 9.8 Addtonal topcs n fnance and nvestment Example (SOA Course 1, May 2001) A stock pays annual dvdends. The frst dvdend s 8 and each dvdend thereafter s 7% larger than the pror dvdend. Let m be the number of dvdends pad by the stock when the cumulatve amount pad frst exceeds 500. Calculate m. (A) 23 (B) 24 (C) 25 (D) 26 (E) 27 Soluton: We need to fnd the smallest nteger number m that solves ( m 1) = 8s m 7% =8 1.07m Snce equalty s acheved at m = , we choose m = = 25. Answer: C 11

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