Retirement annuity

How much will it cost to purchase a two-level retirement annuity that will pay $2000 at the end of every month for the first 10 years, and $3000 per month for the next 15 years? Assume that the payment represent a rate of return to the person receiving the payments of 7.6% compounded quarterly.

Correct result:

x =  430245.06 USD

Solution:

a=2000 4=8000 b=3000 4=12000  p=7.6%=7.6100=0.076  r=1+p/4=1+0.076/4=1.019  n1=10 4=40 n2=15 4=60  s=a n1+b n2=8000 40+12000 60=1040000    a1=a/r0=8000/1.0190=8000 a2=a/r1=8000/1.01917850.8342 a3=a/r2=8000/1.01927704.4496 a4=a/r3=8000/1.01937560.7945 a5=a/r4=8000/1.01947419.818 a6=a/r5=8000/1.01957281.47 a7=a/r6=8000/1.01967145.7017 a8=a/r7=8000/1.01977012.4649 a9=a/r8=8000/1.01986881.7123 a10=a/r9=8000/1.01996753.3978 a11=a/r10=8000/1.019106627.4757 a12=a/r11=8000/1.019116503.9016 a13=a/r12=8000/1.019126382.6316 a14=a/r13=8000/1.019136263.6228 a15=a/r14=8000/1.019146146.833 a16=a/r15=8000/1.019156032.2208 a17=a/r16=8000/1.019165919.7456 a18=a/r17=8000/1.019175809.3676 a19=a/r18=8000/1.019185701.0477 a20=a/r19=8000/1.019195594.7475 a21=a/r20=8000/1.019205490.4293 a22=a/r21=8000/1.019215388.0563 a23=a/r22=8000/1.019225287.592 a24=a/r23=8000/1.019235189.001 a25=a/r24=8000/1.019245092.2483 a26=a/r25=8000/1.019254997.2996 a27=a/r26=8000/1.019264904.1213 a28=a/r27=8000/1.019274812.6804 a29=a/r28=8000/1.019284722.9444 a30=a/r29=8000/1.019294634.8817 a31=a/r30=8000/1.019304548.4609 a32=a/r31=8000/1.019314463.6515 a33=a/r32=8000/1.019324380.4235 a34=a/r33=8000/1.019334298.7473 a35=a/r34=8000/1.019344218.594 a36=a/r35=8000/1.019354139.9352 a37=a/r36=8000/1.019364062.7431 a38=a/r37=8000/1.019373986.9903 a39=a/r38=8000/1.019383912.6499 a40=a/r39=8000/1.019393839.6957 b1=b/rn1+0=12000/1.01940+05652.1527 b2=b/rn1+1=12000/1.01940+15546.7642 b3=b/rn1+2=12000/1.01940+25443.3407 b4=b/rn1+3=12000/1.01940+35341.8456 b5=b/rn1+4=12000/1.01940+45242.243 b6=b/rn1+5=12000/1.01940+55144.4976 b7=b/rn1+6=12000/1.01940+65048.5746 b8=b/rn1+7=12000/1.01940+74954.4403 b9=b/rn1+8=12000/1.01940+84862.0611 b10=b/rn1+9=12000/1.01940+94771.4044 b11=b/rn1+10=12000/1.01940+104682.4381 b12=b/rn1+11=12000/1.01940+114595.1306 b13=b/rn1+12=12000/1.01940+124509.4511 b14=b/rn1+13=12000/1.01940+134425.369 b15=b/rn1+14=12000/1.01940+144342.8548 b16=b/rn1+15=12000/1.01940+154261.8791 b17=b/rn1+16=12000/1.01940+164182.4132 b18=b/rn1+17=12000/1.01940+174104.4291 b19=b/rn1+18=12000/1.01940+184027.899 b20=b/rn1+19=12000/1.01940+193952.7959 b21=b/rn1+20=12000/1.01940+203879.0931 b22=b/rn1+21=12000/1.01940+213806.7646 b23=b/rn1+22=12000/1.01940+223735.7847 b24=b/rn1+23=12000/1.01940+233666.1283 b25=b/rn1+24=12000/1.01940+243597.7706 b26=b/rn1+25=12000/1.01940+253530.6875 b27=b/rn1+26=12000/1.01940+263464.8553 b28=b/rn1+27=12000/1.01940+273400.2505 b29=b/rn1+28=12000/1.01940+283336.8504 b30=b/rn1+29=12000/1.01940+293274.6324 b31=b/rn1+30=12000/1.01940+303213.5744 b32=b/rn1+31=12000/1.01940+313153.655 b33=b/rn1+32=12000/1.01940+323094.8528 b34=b/rn1+33=12000/1.01940+333037.147 b35=b/rn1+34=12000/1.01940+342980.5172 b36=b/rn1+35=12000/1.01940+352924.9433 b37=b/rn1+36=12000/1.01940+362870.4056 b38=b/rn1+37=12000/1.01940+372816.8847 b39=b/rn1+38=12000/1.01940+382764.3619 b40=b/rn1+39=12000/1.01940+392712.8183 b41=b/rn1+40=12000/1.01940+402662.2358 b42=b/rn1+41=12000/1.01940+412612.5965 b43=b/rn1+42=12000/1.01940+422563.8827 b44=b/rn1+43=12000/1.01940+432516.0773 b45=b/rn1+44=12000/1.01940+442469.1632 b46=b/rn1+45=12000/1.01940+452423.1238 b47=b/rn1+46=12000/1.01940+462377.9429 b48=b/rn1+47=12000/1.01940+472333.6044 b49=b/rn1+48=12000/1.01940+482290.0927 b50=b/rn1+49=12000/1.01940+492247.3922 b51=b/rn1+50=12000/1.01940+502205.4879 b52=b/rn1+51=12000/1.01940+512164.365 b53=b/rn1+52=12000/1.01940+522124.0088 b54=b/rn1+53=12000/1.01940+532084.4051 b55=b/rn1+54=12000/1.01940+542045.5399 b56=b/rn1+55=12000/1.01940+552007.3993 b57=b/rn1+56=12000/1.01940+561969.9699 b58=b/rn1+57=12000/1.01940+571933.2383 b59=b/rn1+58=12000/1.01940+581897.1917  x=a1+a2+a3+a4+a5+a6+a7+a8+a9+a10+a11+a12+a13+a14+a15+a16+a17+a18+a19+a20+a21+a22+a23+a24+a25+a26+a27+a28+a29+a30+a31+a32+a33+a34+a35+a36+a37+a38+a39+a40+b1+b2+b3+b4+b5+b6+b7+b8+b9+b10+b11+b12+b13+b14+b15+b16+b17+b18+b19+b20+b21+b22+b23+b24+b25+b26+b27+b28+b29+b30+b31+b32+b33+b34+b35+b36+b37+b38+b39+b40+b41+b42+b43+b44+b45+b46+b47+b48+b49+b50+b51+b52+b53+b54+b55+b56+b57+b58+b59=8000+7850.8342+7704.4496+7560.7945+7419.818+7281.47+7145.7017+7012.4649+6881.7123+6753.3978+6627.4757+6503.9016+6382.6316+6263.6228+6146.833+6032.2208+5919.7456+5809.3676+5701.0477+5594.7475+5490.4293+5388.0563+5287.592+5189.001+5092.2483+4997.2996+4904.1213+4812.6804+4722.9444+4634.8817+4548.4609+4463.6515+4380.4235+4298.7473+4218.594+4139.9352+4062.7431+3986.9903+3912.6499+3839.6957+5652.1527+5546.7642+5443.3407+5341.8456+5242.243+5144.4976+5048.5746+4954.4403+4862.0611+4771.4044+4682.4381+4595.1306+4509.4511+4425.369+4342.8548+4261.8791+4182.4132+4104.4291+4027.899+3952.7959+3879.0931+3806.7646+3735.7847+3666.1283+3597.7706+3530.6875+3464.8553+3400.2505+3336.8504+3274.6324+3213.5744+3153.655+3094.8528+3037.147+2980.5172+2924.9433+2870.4056+2816.8847+2764.3619+2712.8183+2662.2358+2612.5965+2563.8827+2516.0773+2469.1632+2423.1238+2377.9429+2333.6044+2290.0927+2247.3922+2205.4879+2164.365+2124.0088+2084.4051+2045.5399+2007.3993+1969.9699+1933.2383+1897.1917=430245.06 USD



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