Mathematics for Finance
2
Risk-Free Assets
Case 2
Consider a do-it-yourself pension fund based on regular savings invested in a bank account attracting interest at 5% per annum. When you retire after 40 years, you want to receive a pension equal to 50% of your final salary and payable for 20 years. Your earnings are assumed to grow at 2% annually, and you want the pension payments to grow at the same rate.
2.1 Time Value of Money
It is a fact of life that $100 to be received after one year is worth less than the same amount today. The main reason is that money due in the future or locked in a fixed-term account cannot be spent right away. One would therefore expect to be compensated for postponed consumption. In addition, prices may rise in the meantime, and the amount will not have the same purchasing power as it would have at present. Moreover, there is always some risk, even if very low, that the money will never be received. Whenever a future payment is uncertain due to the possibility of default, its value today will be reduced to compensate for this. (However, here we shall consider situations free from default or credit risk.) As generic examples of risk-free assets we shall consider a bank deposit or a bond.
M. Capi´ ski, T. Zastawniak, Mathematics for Finance, n Springer Undergraduate Mathematics Series, © Springer-Verlag London Limited 2011
26
Mathematics for Finance
The way in which money changes its value in time is a complex issue of fundamental importance in finance. We shall be concerned mainly with two questions: What is the future value of an amount invested or borrowed today? What is the present value of an amount to be paid or received at a certain time in the future? The answers depend on various factors, which will be discussed in the present chapter. This topic is often referred to as the time value of money.
2.1.1 Simple Interest
Suppose that an amount is paid into a bank account, where it is to earn interest. The future value of this investment
Risk-Free Assets
Case 2
Consider a do-it-yourself pension fund based on regular savings invested in a bank account attracting interest at 5% per annum. When you retire after 40 years, you want to receive a pension equal to 50% of your final salary and payable for 20 years. Your earnings are assumed to grow at 2% annually, and you want the pension payments to grow at the same rate.
2.1 Time Value of Money
It is a fact of life that $100 to be received after one year is worth less than the same amount today. The main reason is that money due in the future or locked in a fixed-term account cannot be spent right away. One would therefore expect to be compensated for postponed consumption. In addition, prices may rise in the meantime, and the amount will not have the same purchasing power as it would have at present. Moreover, there is always some risk, even if very low, that the money will never be received. Whenever a future payment is uncertain due to the possibility of default, its value today will be reduced to compensate for this. (However, here we shall consider situations free from default or credit risk.) As generic examples of risk-free assets we shall consider a bank deposit or a bond.
M. Capi´ ski, T. Zastawniak, Mathematics for Finance, n Springer Undergraduate Mathematics Series, © Springer-Verlag London Limited 2011
26
Mathematics for Finance
The way in which money changes its value in time is a complex issue of fundamental importance in finance. We shall be concerned mainly with two questions: What is the future value of an amount invested or borrowed today? What is the present value of an amount to be paid or received at a certain time in the future? The answers depend on various factors, which will be discussed in the present chapter. This topic is often referred to as the time value of money.
2.1.1 Simple Interest
Suppose that an amount is paid into a bank account, where it is to earn interest. The future value of this investment