Lecture 3: Force of Interest, Real Interest Rate, Annuity


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1 Lecture 3: Force of Interest, Real Interest Rate, Annuty Goals: Study contnuous compoundng and force of nterest Dscuss real nterest rate Learn annutymmedate, and ts present value Study annutydue, and compare wth annutymmedate Learn contnuous annuty and perpetuty. Suggested Textbook Readngs: Chapter 1: ; Chapter 2: Practce Problems: All exercses n and $ 2.1, wthout an astersk, and Secton 2.2: 15, 710, 1217
2 Lecture 3: Force of Interest, Real Interest Rate, Annuty 2 Contnuous Compoundng At nomnal rate (m), the accumulated value of an ntal nvestment of a dollar after a year s ) m a(m) = (1 + (m) m Wth fxed (m), the more often compoundng takng place durng the year, the larger the accumulated value s. Example 1: Suppose that the nterest s compounded contnuously. Fnd the accumulated value a(m) when m. Force of Interest If the nterest s compounded contnuously, the accumulated amount functon A(t) s a contnuous functon of t. The nomnal rate s called the force of nterest and denoted by δ t (sometmes ( ) ). The notaton ( ) makes sense, snce we can thnk of the force of nterest as the lmt as the number of tmes we credt the compound nterest goes to nfnty. that s, δ t = ( ) = lm m (m). Force of Interest For an nvestment that grows accordng to accumulated amount functon A(t), the force of nterest at tme t, s defned to be δ t = A (t) A(t)
3 Lecture 3: Force of Interest, Real Interest Rate, Annuty 3 Example 2: (Example 1.13, page 40) Derve an expresson for δ t f accumulaton s based on 1. smple nterest at annual rate, and 2. compound nterest at annual rate. Force of Interest and Accumulaton functon We can recover the accumulaton functon from the force of nterest δ t = A (t) A(t). We may also calculate the nterest earned from tme t = 0 to tme t = n from the force of nterest δ t. Inflaton, Real Rate of Interest Inflaton s the growth n prces from year to year. It s generally measured va the rate of change n the prce of a basket of goods. The rate of nflaton s denoted by r. Real rate of nterest Wth annual nterest rate and annual nflaton rate r, the real rate f nterest for the year s real = value of amount of real return (yrend dollars) value of nvested amount (yrend dollars) = r 1 + r Note that when r s small, r s a good estmate for the real rate of nterest.
4 Lecture 3: Force of Interest, Real Interest Rate, Annuty 4 Annutes An annuty s a fnte sequence of payments made at fxed perods of tme over a gven nterval. The fxed perods of tme that we consder wll always be of equal length. Example 3: Smth wants to save $100 each month for a vacaton n the summer of At the end of each month he makes deposts nto an account earnng a nomnal rate of (12) = 9% startng from May 31, How much wll Smth have saved after hs last depost on Aprl 30, 2013? Geometrc Sequence AnnutyImmedate 1 + x + x x n 1 = xn 1 x 1 = 1 xn 1 x The seres of payments n the example s referred to as an accumulated annutymmedate. The notaton s n s used to express the accumulated value at the tme of (and ncludng) the fnal payment of a seres of payment of 1 each made at equally spaced ntervals of tme, where the rate of nterest per payment perod s. s n = (1 + ) n 1 + (1 + ) n (1 + ) + 1 = (1 + )n 1 The notaton s n can be used to express the accumulated value of an annuty f the followng condtons are satsfed: 1. the payments are of equal amount; 2. the payments are made at equal ntervals of tme, wth the same frequency as the nterest rate s compounded; 3. the accumulated value s found at the tme of and ncludng the fnal payment.
5 Lecture 3: Force of Interest, Real Interest Rate, Annuty 5 Example 4: (Example 3 contnued) If (12) = 9%, how much wll Smth have to save each month to reach $2000 on Aprl 30, 2013? Present value of an annuty Example 5: Suppose Smth has decded to borrow money to go on vacaton now. If he has to pay back bank $110 each month startng one month from now for 12 months to pay back the loan. How much money dd Smth borrow? Suppose (12) = 9%. Present value of an npayment annutymmedate of 1 per perod The symbol a n s used to denote the present value of a seres of equally spaced payments of amount 1 each when the valuaton pont s one payment perod before the payments begn. a n = v + v v n = 1 vn In order to use the above notaton and formula, the followng condtons have to be satsfed. 1. the n payments are of equal amount; 2. the payments are made at equal ntervals of tme, wth the same frequency as the nterest rate s compounded; 3. the valuaton pont s one perod before the frst payment s made.
6 Lecture 3: Force of Interest, Real Interest Rate, Annuty 6 More Annutes Example 6: Smth wants to take a vacaton n May, Startng from May 31, 2011, he deposts $100 each month from hs monthly paycheck nto an account earnng nomnal rate (12) = 9%. He was lad off on August 31, 2012 when he made hs fnal depost. If the money he has deposted contnues to accumulate untl Aprl 30, 2013, what s the balance n the account at that tme? Example 7: Smth bought a large TV on January 1, He worked out the fnancng so that he does not have to pay for the frst 12 months, and then pays $50 at the end of each month for 36 months (hs frst depost s on January 31, 2013). If (12) = 9%, what s the prce of the TV? Such an annuty s called a deferred annuty. The present value of a kperod deferred, npayments annuty of 1 per perod s v k a n = a n+k a k
7 Lecture 3: Force of Interest, Real Interest Rate, Annuty 7 AnnutyDue Another form of annuty s that of annutydue. Total of n equal payments occur at tme 0, 1, 2,, n 1. In the case of present value, the annutydue refers to the valuaton at the tme of the frst payment, and n the case of future value (accumulated value), the annutydue refers to the valuaton one perod after the last payment. AnnutyDue For npayment annutes wth payment of amount 1 each, the annutydue present value s at the tme of the frst payment, ä n = 1 + v + v v n 1 = 1 vn 1 v = 1 vn d and the accumulated value s one perod after the fnal payment, s n = (1 + ) + (1 + ) (1 + ) n = (1 + )n 1 d where d s the dscount rate correspondng to. AnnutyImmedate v.s. AnnutyDue AnnutyImmedate AnnutyDue Present value a n = 1 vn ä n = 1 vn d Accumulate Value s n = (1 + )n 1 s n = (1 + )n 1 d
8 Lecture 3: Force of Interest, Real Interest Rate, Annuty 8 Dfferng nterest and payment perods It may happen that the quoted nterest rate has a compoundng perod that doesn t concde wth the annuty payment perod. For the purpose of evaluaton we can fnd the nterest rate per payment perod that s equvalent to the quoted nterest rate, or fnd the equvalent payment n each quoted nterest perod. Example 8: (Example 2.12 (a)) On the last day of March, June, September and December, Smth makes a depost of $1000 nto a savng account that earns nomnal rate (12) = 9%. The frst depost s on Mar 31, 1995 and the last s December 31, What s the balance on January 1, 2011? mthly payable annutes Example 9: (Example 2.12 (b)) In the above example, f the nterest rate s quoted at an effectve annual rate of 10%, what s the balance n Smth s account on January 1, 2011? m thly payable annutymmedate If the effectve annual nterest rate s, and m payments of X are made each year, then the accumulated value over n years s Ks (m) n = K (1 + )n 1 (m) = Ks n (m) where K = mx. The present value of ths seres of payments s Ka (m) n = Ka n (m)
9 Lecture 3: Force of Interest, Real Interest Rate, Annuty 9 Perpetutes If an annuty has no end pont, t s called a perpetuty. We cannot fnd the future value of a perpetuty, but we can always calculate the present value. Annutymmedate: a = lm n a n = 1 Smlarly, for annutydue: ä = 1 d and for m thly payable annutymmedate: a (m) = 1 (m) Contnuous Annuty If payments are made more frequently, t s more convenent to approxmate the calculaton by assumng the payments are made contnuously. The accumulated value of the contnuous annuty, pad at 1 per perod for n perods, denoted by s n, s s n = δ s n Smlarly a n = δ a n. If accumulaton s based on force of nterest δ r, then and s n δr = n 0 a n δr = n 0 e R t 0 δr dr dt e R n t δr dr dt. Also s n δr = a n δr e R t 0 δr dr.
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