Financial Economerics FinMerics02) Reurns, Yields, Compounding, and Horizon Nelson Mark Universiy of Nore Dame Fall 2017 Augus 30, 2017 1 Conceps o cover Yields o mauriy) Holding period) reurns Compounding Reurn horizons, muliperiod reurns. Annual raes Effecive reurns Porfolio reurns 2 n-period yield o mauriy We re looking a a discoun bond. Time is measured in years. The curren ime is. The bond asse) pays F +n dollars n years from now, a +n. F +n is called he face value. The presen ime is n years away from mauriy. P 1
is he curren price of he bond. r n) is he curren yield o mauriy. F +n P = ) n 1) 1 + r n) ) n F +n = 1 + r n) P 2) P dollars invesed for n years a he ineres rae r n) compounded annually pays off F +n. for n years The yield o mauriy, r n) 0.04. is a raw number. 4% a an annual rae is 3 Gross reurn and rae of reurn Gross reurn = 1 + r Rae of reurn = r The log approximaion: ln1 + r) r for small r Explanaion: Firs-order aylor approximaion of an arbirary, coninuously differeniable funcion abou he poin x 0, f x) = f x 0 ) + f x 0 ) x x 0 ) + R 3) where R is he approximaion error. Le x = 1 + r, and f be he log funcion, and expand abou x 0 = 1 i.e., r 0 = 0), hen ln 1 + r) r for small r. 4 Rule of 70 A rule of humb for quick menal calculaions. Rule is if somehing grows a rae r 100r percen) per year, i akes years for i o double in size. 2 70 100r
e.g., If he marke reurns, on average, 8 percen per year, a 100K invesmen will double in 8.75 years. Explanaion:Sar wih 1. Find he value of n ha makes his rue: Solve for n, }{{} 1 1 + r) n = 2 4) Saring n = ln 2) ln 1 + r) 0.693 r = 69.3 r 100) 70 r 100) 5) r n 0.01 70.00 0.02 35.00 0.03 23.33 0.04 17.50 0.05 14.00 0.06 11.67 0.07 10.00 0.08 8.75 0.09 7.78 0.10 7.00 0.11 6.36 0.12 5.83 5 Yields compounded m imes per year 1. n is years, m is number of imes reurn yield) is compounded. m = 4 for quarerly m = 12 for monhly m = for coninuous compounding P = F +n 1 + r ) mn 6) m 3
2. The coninuous compounding resul is obained by working wih he fuure face) value, and leing m. F +n = P 1 + r ) mn P e rn 7) m This is a resul from calculus, where lim 1 + x ) mn = e xn m m 3. The effecive annual ineres rae is he simple annual rae of reurn. Call i r a. I s he soluion o 1 + r a ) n = 1 + m) r mn 9) r a = 1 + m) r m 1 10) 8) How big a difference is i? Say you borrowed 10K, a r = 0.10. m r a End of year annual 1 0.10000 1100 quarerly 4 0.10381 1103.81 weekly 52 0.10506 1105.06 coninuous infy 0.10517 1105.17 6 Coupon Bond Bond pays C a year for n 1 years, hen reurns face value F +n. C are coupon paymens. Face value also known as par value. If bond sells a par, issuer is rening F, wih paymens of C per year. P = C C + 1 + r 1 + r ) 2 + + C 1 + r ) n 1 + F +n 1 + r ) n 11) Compuing he yield o mauriy r is a lile more involved now. 4
7 The erm srucure of ineres raes The yield curve, he relaion beween ime-o-mauriy and yield-omauriy, summarizes he erm srucure of ineres raes. Level, slope, curvaure. 8 Holding Period Reurns Annual holding periods: One period gross holding period bond reurn. P +1 + C = P +1 C + P P P }{{} coupon rae and subrac 1 o ge he rae of reurn. 12) P +1 + C P 1 13) One period holding period sock reurn, where D is he dividend paid in he inerval o + 1 P e +1 + D P e and subrac 1 o ge he rae of reurn, D = P +1 e + P e P }{{} e dividend yield 14) P+1 e + D 1 15) P e Le r if be he holding period reurn. Wha is he annualized reurn r a 1. Holding period is a quarer? 1 + r a ) = 1 + r ) 4 r a 4r 16) using he log approximaion 5
2. Holding period is 2 years? Muliperiod reurns 1. Two-period reurns: 1 + r a ) 2 = 1 + r r a = 1 + r ) 1/2 1 17) r,+2 = P +2 P 1 = P +2 P +1 }{{} P +1 P }{{} 1+r +2 )1+r +1 ) 1 = 1 + r +1 ) 1 + r +2 ) 1 Now add 1 o boh sides o ge he wo-period gross reurn. 18) 1 + r,+2 ) = 1 + r +1 ) 1 + r +2 ) 19) This is called he geomeric average of one-period reurns. If reurns are small, we can use he log approximaion Obviously, for k period holding periods, r,+2 r +1 + r +2 20) 1 + r,+k ) = k 1 + r +j ) 21) Over long periods of ime, reurns end o be prey large, so you don wan o use he log approximaion. 9 Porfolio reurns Consider a porfolio wih 2 asses. The porfolio reurn is he weighed average of he reurns on he individual asses. The weighs are he invesmen shares in each asse. We have asses 1 and 2, wih raes of reurn r,1 and r,2. You ve invesed w,1 and w,2 dollars in hem. Porfolio wealh is j=1 W = w,1 + w,2. 22) Le r p be he rae of reurn on he porfolio. Nex period, W +1 = W 1 + r p +1) = w,1 1 + r,1 ) + w,2 1 + r,2 ) 23) 6
Divide by W W +1 = ) 1 + r p +1 = ω,1 1 + r,1 ) + 1 ω,1 ) 1 + r,2 ) 24) W ω,1 = w,1 W 25) ω,2 = 1 ω,1 ) = w,2 W 26) The ω s are porfolio weighs or porfolio shares. The shares sum o 1. For a porfolio of n asses, ) n 1 + r p +1 = ω,j 1 + r,j ) 27) 1 = j=1 n ω,j 28) j=1 The reurn on he Sandard and Poors 500 has n = 500, ω,j is he marke value of firm j divided by he oal S&P marke value. 10 Real reurns and inflaion adjusmen Deflae by CPI inflaion. Denoe eh CPI by CP I. P real = P nom CP I r real = P real P 1 real The log approximaion gives 1 = P nom P 1 nom CP I 1 CP I }{{} 1+π ) 1 29) 1 30) = 1 + rnom 1 31) 1 + π r real = r nom π 32) 7