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 ten years and $3000 per month for the next 15 years? Assume that the payment represents a rate of return to the person receiving the payments of 7.6% compounded quarterly.

Correct answer:

x =  430245.06 USD

Step-by-step explanation:

$$\begin{gathered} a=2000\cdot 4=8000 \\ b=3000\cdot 4=12000 \\ p=7.6\%=\frac{7.6}{100}=0.076 \\ r=1+p/4=1+0.076/4=1.019 \\ {n}_1=10\cdot 4=40 \\ {n}_2=15\cdot 4=60 \\ s=a\cdot n_1+b\cdot n_2=8000\cdot 40+12000\cdot 60=1040000 \\ {a}_1=a/r^0=8000/1.019^0=8000 \\ {a}_2=a/r^1=8000/1.019^1\doteq 7850.8342 \\ {a}_3=a/r^2=8000/1.019^2\doteq 7704.4496 \\ {a}_4=a/r^3=8000/1.019^3\doteq 7560.7945 \\ {a}_5=a/r^4=8000/1.019^4\doteq 7419.818 \\ {a}_6=a/r^5=8000/1.019^5\doteq 7281.47 \\ {a}_7=a/r^6=8000/1.019^6\doteq 7145.7017 \\ {a}_8=a/r^7=8000/1.019^7\doteq 7012.4649 \\ {a}_9=a/r^8=8000/1.019^8\doteq 6881.7123 \\ {a}_{10}=a/r^9=8000/1.019^9\doteq 6753.3978 \\ {a}_{11}=a/r^{10}=8000/1.019^{10}\doteq 6627.4757 \\ {a}_{12}=a/r^{11}=8000/1.019^{11}\doteq 6503.9016 \\ {a}_{13}=a/r^{12}=8000/1.019^{12}\doteq 6382.6316 \\ {a}_{14}=a/r^{13}=8000/1.019^{13}\doteq 6263.6228 \\ {a}_{15}=a/r^{14}=8000/1.019^{14}\doteq 6146.833 \\ {a}_{16}=a/r^{15}=8000/1.019^{15}\doteq 6032.2208 \\ {a}_{17}=a/r^{16}=8000/1.019^{16}\doteq 5919.7456 \\ {a}_{18}=a/r^{17}=8000/1.019^{17}\doteq 5809.3676 \\ {a}_{19}=a/r^{18}=8000/1.019^{18}\doteq 5701.0477 \\ {a}_{20}=a/r^{19}=8000/1.019^{19}\doteq 5594.7475 \\ {a}_{21}=a/r^{20}=8000/1.019^{20}\doteq 5490.4293 \\ {a}_{22}=a/r^{21}=8000/1.019^{21}\doteq 5388.0563 \\ {a}_{23}=a/r^{22}=8000/1.019^{22}\doteq 5287.592 \\ {a}_{24}=a/r^{23}=8000/1.019^{23}\doteq 5189.001 \\ {a}_{25}=a/r^{24}=8000/1.019^{24}\doteq 5092.2483 \\ {a}_{26}=a/r^{25}=8000/1.019^{25}\doteq 4997.2996 \\ {a}_{27}=a/r^{26}=8000/1.019^{26}\doteq 4904.1213 \\ {a}_{28}=a/r^{27}=8000/1.019^{27}\doteq 4812.6804 \\ {a}_{29}=a/r^{28}=8000/1.019^{28}\doteq 4722.9444 \\ {a}_{30}=a/r^{29}=8000/1.019^{29}\doteq 4634.8817 \\ {a}_{31}=a/r^{30}=8000/1.019^{30}\doteq 4548.4609 \\ {a}_{32}=a/r^{31}=8000/1.019^{31}\doteq 4463.6515 \\ {a}_{33}=a/r^{32}=8000/1.019^{32}\doteq 4380.4235 \\ {a}_{34}=a/r^{33}=8000/1.019^{33}\doteq 4298.7473 \\ {a}_{35}=a/r^{34}=8000/1.019^{34}\doteq 4218.594 \\ {a}_{36}=a/r^{35}=8000/1.019^{35}\doteq 4139.9352 \\ {a}_{37}=a/r^{36}=8000/1.019^{36}\doteq 4062.7431 \\ {a}_{38}=a/r^{37}=8000/1.019^{37}\doteq 3986.9903 \\ {a}_{39}=a/r^{38}=8000/1.019^{38}\doteq 3912.6499 \\ {a}_{40}=a/r^{39}=8000/1.019^{39}\doteq 3839.6957 \\ {b}_1=b/r^{n_1+0}=12000/1.019^{40+0}\doteq 5652.1527 \\ {b}_2=b/r^{n_1+1}=12000/1.019^{40+1}\doteq 5546.7642 \\ {b}_3=b/r^{n_1+2}=12000/1.019^{40+2}\doteq 5443.3407 \\ {b}_4=b/r^{n_1+3}=12000/1.019^{40+3}\doteq 5341.8456 \\ {b}_5=b/r^{n_1+4}=12000/1.019^{40+4}\doteq 5242.243 \\ {b}_6=b/r^{n_1+5}=12000/1.019^{40+5}\doteq 5144.4976 \\ {b}_7=b/r^{n_1+6}=12000/1.019^{40+6}\doteq 5048.5746 \\ {b}_8=b/r^{n_1+7}=12000/1.019^{40+7}\doteq 4954.4403 \\ {b}_9=b/r^{n_1+8}=12000/1.019^{40+8}\doteq 4862.0611 \\ {b}_{10}=b/r^{n_1+9}=12000/1.019^{40+9}\doteq 4771.4044 \\ {b}_{11}=b/r^{n_1+10}=12000/1.019^{40+10}\doteq 4682.4381 \\ {b}_{12}=b/r^{n_1+11}=12000/1.019^{40+11}\doteq 4595.1306 \\ {b}_{13}=b/r^{n_1+12}=12000/1.019^{40+12}\doteq 4509.4511 \\ {b}_{14}=b/r^{n_1+13}=12000/1.019^{40+13}\doteq 4425.369 \\ {b}_{15}=b/r^{n_1+14}=12000/1.019^{40+14}\doteq 4342.8548 \\ {b}_{16}=b/r^{n_1+15}=12000/1.019^{40+15}\doteq 4261.8791 \\ {b}_{17}=b/r^{n_1+16}=12000/1.019^{40+16}\doteq 4182.4132 \\ {b}_{18}=b/r^{n_1+17}=12000/1.019^{40+17}\doteq 4104.4291 \\ {b}_{19}=b/r^{n_1+18}=12000/1.019^{40+18}\doteq 4027.899 \\ {b}_{20}=b/r^{n_1+19}=12000/1.019^{40+19}\doteq 3952.7959 \\ {b}_{21}=b/r^{n_1+20}=12000/1.019^{40+20}\doteq 3879.0931 \\ {b}_{22}=b/r^{n_1+21}=12000/1.019^{40+21}\doteq 3806.7646 \\ {b}_{23}=b/r^{n_1+22}=12000/1.019^{40+22}\doteq 3735.7847 \\ {b}_{24}=b/r^{n_1+23}=12000/1.019^{40+23}\doteq 3666.1283 \\ {b}_{25}=b/r^{n_1+24}=12000/1.019^{40+24}\doteq 3597.7706 \\ {b}_{26}=b/r^{n_1+25}=12000/1.019^{40+25}\doteq 3530.6875 \\ {b}_{27}=b/r^{n_1+26}=12000/1.019^{40+26}\doteq 3464.8553 \\ {b}_{28}=b/r^{n_1+27}=12000/1.019^{40+27}\doteq 3400.2505 \\ {b}_{29}=b/r^{n_1+28}=12000/1.019^{40+28}\doteq 3336.8504 \\ {b}_{30}=b/r^{n_1+29}=12000/1.019^{40+29}\doteq 3274.6324 \\ {b}_{31}=b/r^{n_1+30}=12000/1.019^{40+30}\doteq 3213.5744 \\ {b}_{32}=b/r^{n_1+31}=12000/1.019^{40+31}\doteq 3153.655 \\ {b}_{33}=b/r^{n_1+32}=12000/1.019^{40+32}\doteq 3094.8528 \\ {b}_{34}=b/r^{n_1+33}=12000/1.019^{40+33}\doteq 3037.147 \\ {b}_{35}=b/r^{n_1+34}=12000/1.019^{40+34}\doteq 2980.5172 \\ {b}_{36}=b/r^{n_1+35}=12000/1.019^{40+35}\doteq 2924.9433 \\ {b}_{37}=b/r^{n_1+36}=12000/1.019^{40+36}\doteq 2870.4056 \\ {b}_{38}=b/r^{n_1+37}=12000/1.019^{40+37}\doteq 2816.8847 \\ {b}_{39}=b/r^{n_1+38}=12000/1.019^{40+38}\doteq 2764.3619 \\ {b}_{40}=b/r^{n_1+39}=12000/1.019^{40+39}\doteq 2712.8183 \\ {b}_{41}=b/r^{n_1+40}=12000/1.019^{40+40}\doteq 2662.2358 \\ {b}_{42}=b/r^{n_1+41}=12000/1.019^{40+41}\doteq 2612.5965 \\ {b}_{43}=b/r^{n_1+42}=12000/1.019^{40+42}\doteq 2563.8827 \\ {b}_{44}=b/r^{n_1+43}=12000/1.019^{40+43}\doteq 2516.0773 \\ {b}_{45}=b/r^{n_1+44}=12000/1.019^{40+44}\doteq 2469.1632 \\ {b}_{46}=b/r^{n_1+45}=12000/1.019^{40+45}\doteq 2423.1238 \\ {b}_{47}=b/r^{n_1+46}=12000/1.019^{40+46}\doteq 2377.9429 \\ {b}_{48}=b/r^{n_1+47}=12000/1.019^{40+47}\doteq 2333.6044 \\ {b}_{49}=b/r^{n_1+48}=12000/1.019^{40+48}\doteq 2290.0927 \\ {b}_{50}=b/r^{n_1+49}=12000/1.019^{40+49}\doteq 2247.3922 \\ {b}_{51}=b/r^{n_1+50}=12000/1.019^{40+50}\doteq 2205.4879 \\ {b}_{52}=b/r^{n_1+51}=12000/1.019^{40+51}\doteq 2164.365 \\ {b}_{53}=b/r^{n_1+52}=12000/1.019^{40+52}\doteq 2124.0088 \\ {b}_{54}=b/r^{n_1+53}=12000/1.019^{40+53}\doteq 2084.4051 \\ {b}_{55}=b/r^{n_1+54}=12000/1.019^{40+54}\doteq 2045.5399 \\ {b}_{56}=b/r^{n_1+55}=12000/1.019^{40+55}\doteq 2007.3993 \\ {b}_{57}=b/r^{n_1+56}=12000/1.019^{40+56}\doteq 1969.9699 \\ {b}_{58}=b/r^{n_1+57}=12000/1.019^{40+57}\doteq 1933.2383 \\ {b}_{59}=b/r^{n_1+58}=12000/1.019^{40+58}\doteq 1897.1917 \\ x=a_1+a_2+a_3+a_4+a_5+a_6+a_7+a_8+a_9+a_{10}+a_{11}+a_{12}+a_{13}+a_{14}+a_{15}+a_{16}+a_{17}+a_{18}+a_{19}+a_{20}+a_{21}+a_{22}+a_{23}+a_{24}+a_{25}+a_{26}+a_{27}+a_{28}+a_{29}+a_{30}+a_{31}+a_{32}+a_{33}+a_{34}+a_{35}+a_{36}+a_{37}+a_{38}+a_{39}+a_{40}+b_1+b_2+b_3+b_4+b_5+b_6+b_7+b_8+b_9+b_{10}+b_{11}+b_{12}+b_{13}+b_{14}+b_{15}+b_{16}+b_{17}+b_{18}+b_{19}+b_{20}+b_{21}+b_{22}+b_{23}+b_{24}+b_{25}+b_{26}+b_{27}+b_{28}+b_{29}+b_{30}+b_{31}+b_{32}+b_{33}+b_{34}+b_{35}+b_{36}+b_{37}+b_{38}+b_{39}+b_{40}+b_{41}+b_{42}+b_{43}+b_{44}+b_{45}+b_{46}+b_{47}+b_{48}+b_{49}+b_{50}+b_{51}+b_{52}+b_{53}+b_{54}+b_{55}+b_{56}+b_{57}+b_{58}+b_{59}=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\text{USD}\doteq 4.3\cdot 10^5\text{USD} \end{gathered}$$

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