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 twoyear 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 bweekly payments. The bweekly 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 bweekly payments s m, the bweekly 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 bweekly payments n total, whch equals 19 years and 32 weeks. 9.4 Bond valuaton Example 9.9. A $65,000 threeyear 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 prceplusaccrued 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 prceplusaccrued at tme t = (n k 1) + s where s = 146/183. Namely, Prceplusaccrued 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 nyear 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 wrteup 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 wrteup 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 retrementhome 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 cardelaelershp 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 dollarweghted 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 dollarweghted 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 fouryear $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 npayment annutymmedate 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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