Updated April 13, 1999

HOME MORTGAGES, ANNUITIES, AND PERPETUITIES

NOTATION

  • i = annual interest rate = internal rate of return
  • i/m = per period rate of interest; if annual interest rate = 12% and m = 12, then i = .01 = monthly interest rate
  • n = number of years of the loan or mortgage
  • m = number of payments per year = number of compoundings of interest per year
  • mn = total number of payments; on a 30-year mortgage with monthly payments, mn = 360
  • (1 + i/m)mn = tupling (also known as geometric tupling in the compound interest context)
  • PMT = payment (same each period); if m = 12, the payment is monthly
  • INTi = interest charged at the time of the ith payment; i = 1, 2, ..., 360 for a 30-year mortgage with monthly payments
  • PRINi = principal payment for the ith payment == PMT - INTi
  • AB = amount borrowed
  • CPPx = cumulative principal paid after the xth payment (x = 0, 1, 2, ..., 360 in the case of a 30-year mortgage with monthly payments)
  • TPB = Total Pay Back = (m)(n)(PMT) = total number of payments times amount of each payment

    BASIC FORMULAS

    (1.0)

    (2.0)

    EXAMPLE #1

    Given: $100,000 30-year mortgage with monthly payments; annual interest rate = 7%

    Find PMT, TPB (Total PayBack), TI (Total Interest), INT1, PRIN1, CPP0, CPP12 , CPP180 , and CPP360 .

    SOLUTION

    100,000 = [PMT/(.07/12)]{[(1 + .07/12)360 - 1]/(1 + .07/12)360}

    PMT = $665.30

    TPB = (360)($665.30) = $239,508

    TI = $239,508 - $100,000 = $139,508

    INT1 = (.07/12)($100.000) = $583.33

    PRIN1 = $665.30 - $583.33 = $81.97

    CPP0 = [$81.97/(.07/12)]{[1 + (.07/12)]0 - 1} = $0

    CPP12 = [$81.97/(.07/12)]{[(1 + (.07/12)]12 - 1} = $1,015.82

    CPP180 = $25,981.40

    CPP360 = $100,001.20 (bank will refund $1.20)

    Comment: If you are the borrower, you get $100,000 now and pay the $665.30 a month and over the life of the mortgage you pay back $239,508. The situation is reversed for the lender who pays $100,000 now and receives $665.30 a month for the 30-years. Although the $139,508 interest seems huge, the ending value of the lender's investment (assuming non-reinvestment of the $665.30 payments) is $239,508 and the annual rate of return is

    $100,000(1 + i/m)30 = $239,508

    (1 + i/m)30 = 2.39508

    1 + i/m = (2.39508)(1/30)

    i = 2.95418%

    PRESENT VALUE OF AN ANNUITY

    Equation (1.0) also gives the cost/value of a pension (annuity):

    (3.0)

    EXAMPLE #2

    Suppose you are age 65 and about to draw your first social security payment of $665.30 a month, i = 7%, and you expect to live to age 95. What is the present value (worth, cost) of the pension?

    SOLUTION

    PPRI = .07/12 = Per Period Rate of Interest
    PV0 = [$665.30/PPRI]{[(1 + PPRI]360 - 1]/(1 + PPRI)360} = $99,999.62

    Comment

    In the case of social security, you don't have to pay in cash the $100,000 cost -- in a sense, you already paid the cost over your earlier working years. If you had no other assets, your balance sheet would show your wealth to be $100,000 when you are age 65. In this context, wealth and present value are synonyms; your wealth is the present value of your future income payments.

    EXAMPLE #3

    When you are age 65, the capitalized value (present value) of your social security pension (from the example above) will be $100,000 assuming i = 7% and m = 12. If you are now age 25, what is the present value of this pension to be received 40 years later? Assume annual compounding.

    SOLUTION

    PV0 = $100,000/(1 + .07)40 = $6,678.04

    Check:

    $6,678.04(1.07)40 = $100,000 ? (yes)

    Implication:

    If the government on your behalf, at the time of your 25th birthday, deposited $6,678.04 in government bonds paying 7% per year, then your social security pension is pre-paid and you will not have to pay social security taxes during your working years

    EXAMPLE #4

    Suppose you are age 35 and have $200,000 in a bank which pays 7% interest and you want to retire now and award yourself a private 30-year pension (til you are age 65 at which time you collect social security). How much will your monthly pension be? How much will you collect altogether and how much will be the total interest?

    SOLUTION

    PPRI = .07/12 = Per Period Rate of Interest
    $200,000 = (PMT/PPRI)[(1 + PPRI)360 - 1]/(1 + PPRI)360
    PMT = $1,330.60
    TPB = (360)(PMT) = $479,016
    TI = $279,016

    EXAMPLE #5 PRESENT VALUE OF A PERPETUITY

    This example assumes a very long life. Suppose you are going to live forever and have $200,000 in a bank paying 7% per annum interest, compounded annually. How much of an annual "pension forever" can you get?

    Comment

    As the value of N = mn approaches infinity, the value of

    [(1 + i)N - 1]/(1 + i)N

    approaches a value of 1, so the formula will be

    (4.0)

    PV0(perpetuity) = PMT/i

    A perpetuity is a special case of a pension in which the payment is perpetual (lasts forever).

    SOLUTION

    i = .07
    $200,000 = PMT/i
    PMT = $14,000 per year

    INCOME

    If, in equation (4.0) you solve for the payment, you get

    (5.0)

    PMT = (i)(PV0)

    Since present value is the same as present worth or wealth, equation (5.0) can be written as

    (6.0)

    Y = iW

    where Y = income and W = wealth. Equation (6.0) can be read as "Income is the yield on wealth." Its companion equation is

    (7.0)

    W = Y/i

    Equation (7.0) can be read as "Wealth is the capitalized value of income."

    EXAMPLE #6

    You are age 25 and I = 7% compounded annually. You expect to work 40 years with a real income (adjusted for inflation) that averages out at $40,000 a year. What is the capitalized value of your human capital (how much is your wealth?)

    SOLUTION

    PV0 = ($40,000/.07)[(1.07)40 - 1}/(1.07)40 = $533,268.35

    TERMINAL VALUE OF AN ANNUITY

    Since present value times the geometric tupling is equal to terminal value, the terminal value of annuity is

    (8.0)

    EXAMPLE #7

    You are age 25 and want to retire at age 45 with $200,000 in the bank to live on until you qualify for social security at age 65. Assume i = 7% and m = 12. How much should you put monthly into a stock investment plan?

    SOLUTION

    PPRI = .07/12 = Per Period Rate of Interest
    TV(Annuity) = (PMT/PPRI)[(1 + PPRI)240 - 1] = $200,000
    PMT = $383.94

    EXAMPLE #8

    Five-year $40,000 car loan at 8% per annum (per year), monthly payments. How much will the payment be and how much will be the total interest?

    PPRI = .08/12
    $40,000 = (PMT/PPRI)[(1 + PPRI)60 - 1]/(1 + PPRI)60
    PMT = $811.06
    TI = $8,663.60

    EXAMPLE #9

    You can afford $300 a month for a car payment, I = 8%, 5-year financing..How expensive a car can you buy?
    SOLUTION

    PPRI = .08/12 = Per Period Rate of Interest
    PV0 = ($300/PPRI)[(1 + PPRI)60 - 1]/(1 + PPRI)60 = $14,795.53

    EXAMPLE #10

    You are buying a home for $150,000; i = 8% compounded monthly, 30-year financing. How much will your monthly payment be and how much the total interest?

    PPRI = .08/12 = Per Period Rate of Interest
    $150,000 = (PMT/PPRI)[(1 + PPRI)360 - 1]/(1 + PPRI)360
    PMT = $1100.65
    TI = $246,234.00

    EXAMPLE #11

    Suppose the house in Example #10 increases in value at a rate of 7% a year (over and above inflation). Assume monthly compounding. How much will the house be worth in 40 years?

    SOLUTION

    PPRI = .07/12 = Per Period Rate of Interest
    $150,000(1 + PPRI)480 = $2,446,711.72

    EXAMPLE #12

    You want to be a millionaire 40 years hence. How much of a one-time-only investment must you make now if i =- 8%, compounded monthly?

    SOLUTION

    PPRI = .08/12 = Per Period Rate of Interest
    PV0(1 + PPRI)480 = $1,000,000
    PV0 = $41,197.38

    EXAMPLE #13

    You want to be a millionaire 40 years hence. How much of a monthly payment (adjusted for inflation) must you put into a stock investment fund that pay 8%, compounded monthly?

    SOLUTION

    PPRI = .08/12 = Per Period Rate of Interest
    TV(Annuity) = PV0(Annuity)(1 + PPRI)480 = (PMT/PPRI)[(1 + PPRI )480 - 1] = $1,000,000
    PMT = $286.45

    Comment

    In the above example, you paid $137,496 and the interest earned came to $862,504

    CONFIRMATION OF THE FORMULAS

    EXAMPLE

    i = 25% compounded annually. Three-year annuity. Payment is $1000 a year, starting one year hence.

    (a) PV0(Annuity)

    Solution

    (i) PV0(Annuity) = $1000/1.25 + $1000/(1.25)2 + 1000/(1.25)3 = $800 + $640 + $512 = $1952

    (ii) (using the formula)
    PV0(Annuity) = ($1000/.25)[(1.25)3 - 1)]/(1.25)3 = $1952

    (b) TV(Annuity)

    Solution

    (i) TV(Annuity) = $1000 + $1000(1.25) + $1000(1.25)2 = $3812.50

    (ii) (using the formula)
    TV(Annuity) = $1000/.25[(1.25)3 - 1] = $3812.50

    PRACTICE PROBLEMS

    PROB 1

    You want to be a millionaire 40 years hence. How much of a monthly payment (adjusted for inflation) must you put into a stock investment fund that pays 10% per annum, compounded monthly?

    SOLUTION

    $158.13

    PROB 2

    You want to be a millionaire 40 years hence. How much of a one-time-only investment must you make now if i = 10%, compounded monthly?

    SOLUTION

    $18,621.74

    PROB 3

    Suppose that a house purchased for $150,000 increases in value at a rate of 10% a year (over and ablove inflation). Assume monthly compounding. How much will the house be worth 40 years later?

    SOLUTION

    $8,055,099.46

    PROB 4

    You are buying a home for $150,000; i = 10%, compounded monthly, 30-year financing. How much will your monthly payment be and how much the total interest?

    PMT = $1316.36; TI = $323,889.60

    PROB 5

    You can afford $300 a month for a car payment, i = 10%, 5-year financing. How expensive a car can you buy?

    SOLUTION

    $14,119.61

    PROB 6

    Five-year $40,000 car loan at 10% per annum , monthly payments. How much will the payment be and how much will be the total interest?

    PMT = $849.88; TI = $10,992.80

    PROB 7

    You are age 25 and want to retire at age 45 with $200,000 in the bank to live on until you qualify for social security at age 65. Assume i = 10% and m = 12. How much should you put monthly into a stock investment plan?

    SOLUTION

    PMT = $263.38

    PROB 8

    You are age 25 and i = 10% compounded annually. You expect to work 40 years with a real income (adjusted for inflation) that averages out at $40,000 a year. What is the capitalized value of your human capital (how much is your wealth?)

    SOLUTION

    $391,162.03

    PROB 9

    Suppose you are going to live forever and have $200,000 in a bank paying 10% per annum interest. How much of an annual "pension forever" can you get?

    SOLUTION

    $20,000 a year

    PROB 10

    Suppose you are age 35 and have $200,000 in a bank which pays 10% interest and you want to retire now and award yourself a private 30-year pension (til you are age 65 at which time you collect social security). (a) How much will your monthly pension be? (b) How much will you collect altogether and (c) how much will be the total interest?

    SOLUTION

    (a) $1755.14
    (b) TPB = $631,850.40
    (c) TI = $431,850.40

    PROB 11

    When you are age 65, assume that the capitalized value (present value) of your social security pension will be $100,000 and i = 10%, m = 12. If you are now age 25, what is the present value of this pension to be received 40 years later? Assume annual com pounding.

    SOLUTION

    $2,209.49

    PROB 12

    Given: $100,000 30-year mortgage with monthly payments; i = 10%.

    Find PMT, TPB (Total PayBack), TI (Total Interest), INT1, PRIN1, CPP0, CPP12 , CPP180 , and CPP360 .

    SOLUTION

    100,000 = [PMT/(.10/12)]{[(1 + .10/12)360 - 1]/(1 + .10/12)360}

    PMT = $877.57

    TPB = (360)($665.30) = $315,925.20

    TI = $215,925.20

    INT1 = $833.33

    PRIN1 = $44.24

    CPP0 = $0

    CPP12 = $555.90

    CPP180 = $18,336.17

    CPP360 = $100,003.99

    PROB 13

    Suppose you are age 65 and about to draw your first social security payment of $877.57 a month, i = 10%, and you expect to live to age 95. What is the present value (worth, cost) of the pension?

    SOLUTION

    $99,999.82