Determining the fair value of an asset using mathematical equations.

• Author: Frensidy, Budi
• Position: Report

1. INTRODUCTION

   A young investor is contemplating investment in the following financial assets which are offered at the same price i.e. Rp 100 million. With his budget constraint, he can choose one and only one of the alternatives given.

   1. Zero-coupon bond with the nominal value of Rp 250 million, due in 8 years

   2. No-par bond that pays cash Rp 18 million every year for 10 times starting next year

   3. No-par bond that pays cash Rp 16 million every year for 10 times starting right now

   4. No-par bond with the payoff Rp 50 million every year for five times, but starting in five years

   5. Discretionary fund that gives cash Rp 12.5 million every year for the whole life starting next year

   6. Discretionary fund that gives cash Rp 11.5 million every year for the whole life starting next week

   7. Hedge fund with the payoff Rp 15 million every year for the whole life starting in three years

   8. Hedge fund that pays cash Rp 15 million next year, then becomes Rp 15.9 million the following year and steadily rises 6% every year, and the payments are for ten times only

   9. Banking product with the payoff Rp 13 million next week then Rp 14.04 million next year and rises consistently 8% every year, and the payments are for ten times only

   10. Banking product that promises cash Rp 40 million in five years then becomes Rp 43.2 million a year after and steadily rises 8% every year, and the payments are for five times only

   11. Investment trust that promises cash Rp 4 million next year, and then rises 8% every year for the whole life

   12. Investment trust that gives cash Rp 7 million next week, then Rp 7.35 million next year and rises on at 5% every year for the whole life

   13. Private equity with the payoff Rp 12 million starting in three years and rises to Rp 12.36 million the

   following year and at 3% every year for the whole life n. Private equity that distributes cash Rp 12 million next year and rises Rp 60,000 every year for 50 years

   The security, risk, and certainty of the above financial assets are assumed the same. The investor is expected to act rationally and will base his decision only on the mathematical calculation. Using a different formula for each alternative, which asset should be chosen if the relevant discount rate is 12% p.a.?

2. BASIC EQUATION

   To decide which asset should be chosen, the investor must calculate the present value of all the assets. As long as the present value of the cash flows generated from an asset is more than the cost which is Rp

   100 million, the asset is worth buying. The problem is when there is a budget constraint, as we usually encounter, we have to rank all the choices and choose the highest value. This is what is meant by mutually exclusive projects in finance.

   For all the above assets, we can use the basic present value formula from a single amount in the future namely:

   PV = FV / [(1 + i)).sup.n]

   where: PV = present value

   FV = future cash flow

   i = discount rate per period

   n = number of periods

   Using the above basic equation, we can get the fair value of asset a, which is a zero-coupon bond with the nominal value of Rp 250 million, due in 8 years.

   PV = Rp 250,000,000 / [(1 + 0.12).sup.8]

   PV = Rp 100,970,807

   To get the present value of the other 13 assets, alternatives b to n, if we do not use other equations but the basic one, we have to calculate one by one the present value of future cash flow and sum all the results. This is impractical since some assets give cash flows indefinitely. The question now is whether there is a mathematical equation that can be used to get the present value for each asset mentioned above easily.

   As far as I know, there has not been any books in the literature of financial management, financial mathematics, or investment management that discuss all the mathematical equations thoroughly. This is what motivates me to write this paper.

3. ANNUITY AND PERPETUITY

   Annuity is a series of cash flows to pay or to receive, usually in equal amounts, in the constant time interval. The home loan and bond coupon are examples of annuity. Whereas perpetuity is indefinite annuity; that is when...