2.1 The Present Value of an Annuity One example of a fixed annuity is an agreement to pay someone a fixed amount x for N periods (commonly months or years), e.g. a fixed pension It is assumed that the first payment is to be made at time t 0, thereafter payments are made at unit intervals, i.e. the final payment is made at time t 0 + N 1. Note that the unit of time is assumed to be the time between payments. 1 / 30
The Present Value of an Annuity An individual would rather obtain an amount Nx now than obtain such an annuity (e.g. he/she could invest the money, take x from this account each period and have some money left at the end). As in Chapter 1, we will assume that the present value of a nominal value of money x received t time units in the future is given by V (x; t) = xα t, where α is the discount factor (per period). 2 / 30
The Present Value of an Annuity Very often, the value of money to be obtained in the future is discounted according to the expected inflation rate. This gives a measure of the buying power of the annuity When the expected inflation rate is 100i%, the discount rate applied is α = 1 1 + i. When the expected inflation rate is small ( 5%), we may use the approximation α 1 i. Sometimes the interest rate is used to define the appropriate discount rate (in the above equations i is substituted by R). Such discounting measures the investment power of the annuity. 3 / 30
Formula for the Sum of a Geometric Sequence Let a i = cr i. Then the set a 0, a 1, a 2,... form a geometric sequence c, cr, cr 2,.... Let S k be the sum of the first k elements of this sequence, a 0, a 1,... a k 1. Note that k 1 S k = cr i = c(1 + r + r 2 +... + r k 1 ) i=0 rs k = c(r + r 2 + r 3 +... + r k ). Subtracting the second equation from the first, we obtain S k rs k = c(1 r k ) (1 r)s k = c(1 r k ) 4 / 30
Formula for the Sum of a Geometric Sequence It follows that k 1 S k = In particular, when r < 1 i=0 S = cr i = c(1 r k ). 1 r cr i = i=0 c 1 r Note: c is the starting (initial) term of the sequence and r is the ratio between neighbouring elements r = a i+1 a i. 5 / 30
The Present Value of an Annuity It follows that the present value of such an annuity is given by V A = t 0 +N 1 t=t 0 V (x; t)=xα t 0 + xα t 0+1 +... + xα t 0+N 1 =xα t 0 [1 + α +... + α N 1 ] =xα t 0 N 1 i=0 α i = xαt 0 (1 α N ). 1 α 6 / 30
Result: Present Value of a Fixed Annuity The present value of a fixed annuity to be paid at unit intervals is given by V A = xαt 0 (1 α N ), 1 α where x is the payment each period, α is the discount rate per period, t 0 is the time at which the first payment is to be made and N is the number of payments to be made. 7 / 30
Example 2.1 An annuity of $2000 is to be paid monthly for 10 years. Assume that the estimated annual inflation rate is 5%. Using an appropriate approximation for the annual discount rate, calculate i) The monthly discount rate ii) The present value of such an annuity if the first payment is due immediately. iii) The present value of such an annuity if the first payment is due in two years time. 8 / 30
Example 2.1 9 / 30
Example 2.1 10 / 30
Example 2.1 11 / 30
Present Value of a General Future Stream of Income When the payments each period can vary, in order to calculate the present value, it is simplest to sum the individual discounted payoffs. Such streams of payments are often met in cost-benefit analysis, which is considered in the next chapter. 12 / 30
2.2 Repayment of Loans Suppose that a loan of nominal value M is to be paid off at a nominal annual interest rate of 100R% over a period of T years. We will ignore any price for obtaining a loan (it is assumed that this price is incorporated into the value of the loan). Assume that the interest to be paid will be added at the beginning of a year, based on the amount still owed (other cases will be considered in the tutorial class). It is assumed that a fixed monthly payment is used to pay off the loan. 13 / 30
Repayment of Loans For example, suppose someone is paying off a loan of $100,000 at an annual interest rate of 5% using monthly payments of $600. In year 1, the interest to be paid is $5,000 (5% of $100,000). The debtor pays $7,200 (12 monthly payments of $600). It follows that in the first year the nominal value of the debt falls by $2,200 (=7,200-5,000). Hence, the nominal value of the debt after one year is $97,800 (=100,000-2,200). 14 / 30
Repayment of Loans In year 2, the interest to be paid is $4,890 (0.05 97,800). The debtor pays $7,200 (12 monthly payments of $600). It follows that in the second year the net value of the debt falls by $2,310 (=7,200-4,890). Hence, the nominal value of the debt after two years is $95,490 (=97,800-2,310). It should be noted that the fall in the nominal value of the debt becomes more rapid over time. 15 / 30
Repayment of Loans It is relatively easy to find bounds on the payments to be made based on the two following facts: 1. Lower Bound - The monthly payoff must be greater than that required to simply pay the interest. 2. Upper Bound - The monthly payoff must be smaller than that required to ensure that the nominal value of the loan after one period of a T period loan of nominal value M is M(T 1) T. The second bound follows from the fact that the rate at which the nominal value of the debt falls becomes more rapid, i.e. if 1/T of the loan is paid off in the first period, the amount paid off in T periods will be more than the nominal value of the loan. 16 / 30
Bounds on Monthly Payments For example, when $100,000 is borrowed for 20 years at an interest rate of 10% per annum with annual capitalization. In order to pay off the interest ($10,000), the monthly payment must be 10,000 12 = 833.33. To pay off 1/20 of the loan ($5,000) in the first year, a total of 10,000+5,000=15,000 must be paid (interest+reduction). In this case, the monthly payment is 15,000 12 = 1, 250. 17 / 30
Bounds on Monthly Payments It follows that the monthly payment required to pay off a loan of $100,000 in 20 years is between $833.33 and $1,250. These are not particularly tight bounds, but give a good idea of what payment is required. 18 / 30
An Inductive Formula for the Nominal Value of the Debt Let the nominal value of the debt after t periods be x t, t = 0, 1,..., T, where T is the length of the loan and p is the monthly payment, x 0 = M is the initial value of the debt. The unit of time is the time between capitalizations of the debt (here, one year). It follows that the debt after t + 1 periods is given by the difference equation x t+1 = (1 + R p )x t Fp, x 0 = M, where R p is the interest rate per period, F is the number of payments (months) per period. The first term, (1 + R p )x t, is the value of the debt after the interest has been added, the second term is the amount paid in the course of a period. Using this formula, the changes in the debt can be calculated very easily in any spreadsheet. 19 / 30
Calculation of the Monthly Payment The difference equation on the previous slide can be solved to find an explicit formula for the amount owned after t years. Since by definition x T = 0 (the debt is paid off after T periods), we thus obtain an equation for appropriate monthly payment, which gives p = R pm(1 + R p ) T F [(1 + R p ) T 1], where F is the number of payments per period, here 12. 20 / 30
Example 2.2 Suppose $100,000 is borrowed at an interest rate of 10% per annum to be paid over 20 years. i) Calculate the appropriate monthly payment. ii) Based on this monthly payment, calculate the nominal value of the debt after t years, where t = 1, 2, 3. 21 / 30
Example 2.2 22 / 30
Example 2.2 23 / 30
The Effective Cost of Borrowing It should be noted that interest is calculated based on the value of the debt at the beginning of each period, i.e. the nominal interest rate does not take into account the fact that part of the debt is paid off over the course of each period. It follows that the effective rate of interest is greater than the quoted (nominal) rate of interest. 24 / 30
The Effective Cost of Borrowing The real cost of borrowing may be expressed by approximating the effective annual percentage rate (APR). When constant, regular payments are made, a general formula for the approximate APR, R E, is given by R E = 2AF M(N + 1), where A is the excess paid in comparison to the nominal value of the loan, M, F is the frequency of payments per year and N is the total number of payments. 25 / 30
Approximating the APR In the case of the loan repayments considered, F = 12 (12 payments per year), N = 12T (payments for T years) The excess paid is A = 12Tp M (i.e. the total amount paid minus the nominal value of the loan). Hence, R E = 24(12Tp M) M(12T + 1). 26 / 30
The Real Interest Rate The real interest rate takes into account that the real value of money decreases over time due to inflation. Suppose the (nominal or effective) interest rate is 100R% per annum and the inflation rate is 100i% per annum. The real interest rate, r%, satisfies the equation 1 + r = 1 + R 1 + i. When the inflation rate is low, we may use the equation r R i. 27 / 30
Example 2.3 i) Approximate the APR for the loan considered in Example 2.2. ii) Suppose the expected inflation rate is 3%. Based on the APR, estimate the real interest rate using the a) the exact formula for the real interest rate. b) the approximation. 28 / 30
Example 2.3 29 / 30
Example 2.3 30 / 30